Page 1

AES

ANAGNORISIS

Educational Services

HSC / A LEVEL

MATHEMATICS sub (p1,p2)

2009 AES

up to

AES

2013

*book available on Tablet and Mobile

*video solution available for maths papers


Assessment at a glance

Assessment at a glance The 7 units in the scheme cover the following subject areas: •

Pure Mathematics (units P1, P2 and P3);

Mechanics (units M1 and M2);

Probability and Statistics (units S1 and S2).

Centres and candidates may: •

take all four Advanced (A) Level components in the same examination series for the full Cambridge International A Level;

follow a staged assessment route to the Cambridge International A Level by taking two Advanced Subsidiary (AS) papers (P1 & M1 or P1 & S1) in an earlier examination series;

take the Advanced Subsidiary (AS) qualification only.

Cambridge International AS Level candidates take: Paper 1: Pure Mathematics 1 (P1) 1¾ hours About 10 shorter and longer questions 75 marks weighted at 60% of total plus one of the following papers: Paper 2: Pure Mathematics 2 (P2)

Paper 4: Mechanics 1 (M1)

Paper 6: Probability and Statistics 1 (S1)

1¼ hours

1¼ hours

1¼ hours

About 7 shorter and longer questions

About 7 shorter and longer questions

About 7 shorter and longer questions

50 marks weighted at 40% of total

50 marks weighted at 40% of total

50 marks weighted at 40% of total

Cambridge International AS and A Level Mathematics 9709


Syllabus aims and objectives

Syllabus aims and objectives Aims The aims of the syllabus are the same for all students. These are set out below and describe the educational purposes of any course based on the Mathematics units for the Cambridge International AS and A Level examinations. The aims are not listed in order of priority. The aims are to enable candidates to: •

develop their mathematical knowledge and skills in a way which encourages confidence and provides satisfaction and enjoyment;

develop an understanding of mathematical principles and an appreciation of mathematics as a logical and coherent subject;

acquire a range of mathematical skills, particularly those which will enable them to use applications of mathematics in the context of everyday situations and of other subjects they may be studying;

develop the ability to analyse problems logically, recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and, where necessary, select an appropriate mathematical method to solve the problem;

use mathematics as a means of communication with emphasis on the use of clear expression;

acquire the mathematical background necessary for further study in this or related subjects.

2 Assessment objectives The abilities assessed in the examinations cover a single area: technique with application. The examination will test the ability of candidates to: •

understand relevant mathematical concepts, terminology and notation;

recall accurately and use successfully appropriate manipulative techniques;

recognise the appropriate mathematical procedure for a given situation;

apply combinations of mathematical skills and techniques in solving problems;

present mathematical work, and communicate conclusions, in a clear and logical way.

Cambridge International AS and A Level Mathematics 9709


Curriculum content

Curriculum content The mathematical content for each unit in the scheme is detailed below. The order in which topics are listed is not intended to imply anything about the order in which they might be taught. As well as demonstrating skill in the appropriate techniques, candidates will be expected to apply their knowledge in the solution of problems. Individual questions set may involve ideas and methods from more than one section of the relevant content list. For all units, knowledge of the content of Cambridge O Level/Cambridge IGCSE Mathematics is assumed. Candidates will be expected to be familiar with scientific notation for the expression of compound units, e.g. 5 m s –1 for 5 metres per second. Unit P1: Pure Mathematics 1 (Paper 1) Candidates should be able to: 1. Quadratics

2. Functions

carry out the process of completing the square for a quadratic polynomial ax 2 + bx + c, and use this form, e.g. to locate the vertex of the graph of y = ax 2 + bx + c or to sketch the graph;

find the discriminant of a quadratic polynomial ax 2 + bx + c and use the discriminant, e.g. to determine the number of real roots of the equation ax 2 + bx + c = 0;

solve quadratic equations, and linear and quadratic inequalities, in one unknown;

solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic;

recognise and solve equations in x which are quadratic in some function of x, e.g. x 4 – 5x 2 + 4 = 0.

understand the terms function, domain, range, one-one function, inverse function and composition of functions;

identify the range of a given function in simple cases, and find the composition of two given functions;

determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases;

illustrate in graphical terms the relation between a one-one function and its inverse.

Cambridge International AS and A Level Mathematics 9709


Curriculum content

3. Coordinate geometry

4. Circular measure

find the length, gradient and mid-point of a line segment, given the coordinates of the end-points;

find the equation of a straight line given sufficient information (e.g. the coordinates of two points on it, or one point on it and its gradient);

understand and use the relationships between the gradients of parallel and perpendicular lines;

interpret and use linear equations, particularly the forms y = mx + c and y – y1 = m(x – x1);

understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations (including, in simple cases, the correspondence between a line being tangent to a curve and a repeated root of an equation).

understand the definition of a radian, and use the relationship between radians and degrees;

use the formulae s = r θ and A = 1 r 2θ in solving problems concerning 2

the arc length and sector area of a circle. 5. Trigonometry

sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians);

use the exact values of the sine, cosine and tangent of 30°, 45°, 60°,

• • •

6. Vectors •

and related angles, e.g. cos 150° = – 1 −1

−1

−1

2

3;

use the notations sin x, cos x, tan x to denote the principal values of the inverse trigonometric relations; use the identities sin i ≡ tan θ and sin2 θ + cos2 θ ≡ 1; cos i find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included). x use standard notations for vectors, i.e.   , xi + yj, y  AB , a;

x    y  , xi + yj + zk, z   

carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms;

use unit vectors, displacement vectors and position vectors;

calculate the magnitude of a vector and the scalar product of two vectors;

use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors.

Cambridge International AS and A Level Mathematics 9709


Curriculum content

7. Series

8. Differentiation

use the expansion of (a + b)n , where n is a positive integer (knowledge of the greatest term and properties of the coefficients are not n required, but the notations   and n! should be known); r  recognise arithmetic and geometric progressions;

use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions;

use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

understand the idea of the gradient of a curve, and use the notations 2

dy dy and (the technique of differentiation from first 2 dx dx principles is not required);

f’(x), f’’(x),

9. Integration

use the derivative of xn (for any rational n), together with constant multiples, sums, differences of functions, and of composite functions using the chain rule;

apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change);

locate stationary points, and use information about stationary points in sketching graphs (the ability to distinguish between maximum points and minimum points is required, but identification of points of inflexion is not included).

understand integration as the reverse process of differentiation, and integrate (ax + b)n (for any rational n except –1), together with constant multiples, sums and differences;

solve problems involving the evaluation of a constant of integration, e.g. to find the equation of the curve through (1, –2) for which dy = 2x + 1; dx evaluate definite integrals (including simple cases of ‘improper’

integrals, such as •

y

1 0

1

x- 2 dx and

use definite integration to find

y

3 1

2

x- dx );

the area of a region bounded by a curve and lines parallel to the axes, or between two curves, a volume of revolution about one of the axes.

Cambridge International AS and A Level Mathematics 9709


Curriculum content

Unit P2: Pure Mathematics 2 (Paper 2) Knowledge of the content of unit P1 is assumed, and candidates may be required to demonstrate such knowledge in answering questions. Candidates should be able to: 1. Algebra

2. Logarithmic and exponential functions

3. Trigonometry

4. Differentiation

understand the meaning of x, and use relations such as a  = b  ⇔ a 2 = b 2 and x – a  < b ⇔ a – b < x < a + b in the course of solving equations and inequalities;

divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero);

use the factor theorem and the remainder theorem, e.g. to find factors, solve polynomial equations or evaluate unknown coefficients.

understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base);

understand the definition and properties of ex and In x, including their relationship as inverse functions and their graphs;

use logarithms to solve equations of the form ax = b, and similar inequalities;

use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/ or intercept.

understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude;

use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of: •

sec2 θ ≡ 1 + tan2 θ and cosec2 θ ≡ 1 + cot2 θ ,

the expansions of sin(A ± B), cos(A ± B) and tan(A ± B),

the formulae for sin 2A, cos 2A and tan 2A,

the expressions of a sin θ + b cos θ in the forms R sin(θ ± α) and R cos(θ ± α).

use the derivatives of ex, In x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites;

differentiate products and quotients;

find and use the first derivative of a function which is defined parametrically or implicitly.

Cambridge International AS and A Level Mathematics 9709


Curriculum content

5. Integration

6. Numerical solution of equations

extend the idea of ‘reverse differentiation’ to include the integration of 1 , sin(ax + b), cos(ax + b) and sec2 (ax + b) (knowledge of eax+b, ax + b the general method of integration by substitution is not required);

use trigonometrical relationships (such as double-angle formulae) to facilitate the integration of functions such as cos2 x;

use the trapezium rule to estimate the value of a definite integral, and use sketch graphs in simple cases to determine whether the trapezium rule gives an over-estimate or an under-estimate.

locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change;

understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation;

understand how a given simple iterative formula of the form xn + 1 = F(xn ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy (knowledge of the condition for convergence is not included, but candidates should understand that an iteration may fail to converge).

Cambridge International AS and A Level Mathematics 9709


List of formulae and tables of the normal distribution

List of formulae and tables of the normal distribution PURE MATHEMATICS

Algebra For the quadratic equation ax 2 + bx + c = 0 : − b ± √ (b2 − 4ac) x= 2a For an arithmetic series: un = a + (n − 1)d ,

Sn = 12 n(a + l ) = 12 n{2a + (n − 1)d }

For a geometric series:

un = ar n −1 ,

Sn =

a(1 − r n ) (r ≠ 1) , 1− r

S∞ =

a 1− r

( r < 1)

Binomial expansion: n n n (a + b)n = a n +  a n −1b +  a n −2b2 +  a n −3b3 + L + bn , where n is a positive integer 1 2      3 n! n and   =  r  r! (n − r)! n(n − 1) 2 n(n − 1)(n − 2) 3 x + x L , where n is rational and x < 1 (1 + x)n = 1 + nx + 2! 3! Trigonometry Arc length of circle = rθ ( θ in radians) Area of sector of circle = 12 r 2θ ( θ in radians)

sin θ cosθ cos2 θ + sin2 θ ≡ 1 , 1 + tan 2 θ ≡ sec2 θ , cot 2 θ + 1 ≡ cosec2θ sin( A ± B) ≡ sin A cos B ± cos A sin B cos( A ± B) ≡ cos A cos B m sin A sin B tan A ± tan B tan( A ± B) ≡ 1 m tan A tan B sin 2 A ≡ 2 sin A cos A cos 2 A ≡ cos2 A − sin2 A ≡ 2 cos2 A − 1 ≡ 1 − 2 sin2 A 2 tan A tan 2 A = 1 − tan2 A Principal values: − 12 π ≤ sin −1 x ≤ 12 π tan θ ≡

0 ≤ cos−1 x ≤ π − < tan −1 x < 12 π 1π 2

Cambridge International AS and A Level Mathematics 9709


List of formulae and tables of the normal distribution

Differentiation

f(x)

f ′( x)

xn

nx n −1 1 x ex cos x − sin x sec2 x dv du u +v dx dx du dv v −u dx dx v2

ln x ex sin x cos x tan x

uv u v If x = f(t ) and y = g(t ) then

dy dy dx = ÷ dx dt dt

Integration f(x) xn

1 x ex sin x cos x sec2 x

dv du ⌠  u dx = uv − ⌠  v dx ⌡ dx ⌡ dx ⌠ f ′( x) dx = ln f ( x) + c  f( x) ⌡

∫ f( x) dx

x n +1 + c (n ≠ −1) n +1

ln x + c

ex + c − cos x + c sin x + c tan x + c

Vectors If a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then a.b = a1b1 + a2b2 + a3b3 = a b cosθ Numerical integration Trapezium rule:

b

f( x) dx ≈ 12 h{ y0 + 2( y1 + y2 + L + yn−1 ) + yn } , where h =

a

b−a n

Cambridge International AS and A Level Mathematics 9709


Additional information

Grading and reporting Cambridge International A Level results are shown by one of the grades A*, A, B, C, D or E indicating the standard achieved, Grade A* being the highest and Grade E the lowest. ‘Ungraded’ indicates that the candidate has failed to reach the standard required for a pass at either Cambridge International AS Level or A Level. ‘Ungraded’ will be reported on the statement of results but not on the certificate. If a candidate takes a Cambridge International A Level and fails to achieve grade E or higher, a Cambridge International AS Level grade will be awarded if both of the following apply: •

the components taken for the Cambridge International A Level by the candidate in that series included all the components making up a Cambridge International AS Level

the candidate’s performance on these components was sufficient to merit the award of a Cambridge International AS Level grade.

For languages other than English, Cambridge also reports separate speaking endorsement grades (Distinction, Merit and Pass), for candidates who satisfy the conditions stated in the syllabus. Percentage uniform marks are also provided on each candidate’s statement of results to supplement their grade for a syllabus. They are determined in this way: •

A candidate who obtains… … the minimum mark necessary for a Grade A* obtains a percentage uniform mark of 90%. … the minimum mark necessary for a Grade A obtains a percentage uniform mark of 80%. … the minimum mark necessary for a Grade B obtains a percentage uniform mark of 70%. … the minimum mark necessary for a Grade C obtains a percentage uniform mark of 60%. … the minimum mark necessary for a Grade D obtains a percentage uniform mark of 50%. … the minimum mark necessary for a Grade E obtains a percentage uniform mark of 40%. … no marks receives a percentage uniform mark of 0%.

Candidates whose mark is none of the above receive a percentage mark in between those stated according to the position of their mark in relation to the grade ‘thresholds’ (i.e. the minimum mark for obtaining a grade). For example, a candidate whose mark is halfway between the minimum for a Grade C and the minimum for a Grade D (and whose grade is therefore D) receives a percentage uniform mark of 55%. The percentage uniform mark is stated at syllabus level only. It is not the same as the ‘raw’ mark obtained by the candidate, since it depends on the position of the grade thresholds (which may vary from one series to another and from one subject to another) and it has been turned into a percentage. Cambridge International AS Level results are shown by one of the grades a, b, c, d or e indicating the standard achieved, Grade a being the highest and Grade e the lowest. ‘Ungraded’ indicates that the candidate has failed to reach the standard required for a pass at Cambridge International AS Level. ‘Ungraded’ will be reported on the statement of results but not on the certificate. For languages other than English, Cambridge will also report separate speaking endorsement grades (Distinction, Merit and Pass) for candidates who satisfy the conditions stated in the syllabus. The content and difficulty of a Cambridge International AS Level examination is equivalent to the first half of a corresponding Cambridge International A Level.

Cambridge International AS and A Level Mathematics 9709


Additional information

Percentage uniform marks are also provided on each candidate’s statement of results to supplement their grade for a syllabus. They are determined in this way: •

A candidate who obtains… … the minimum mark necessary for a Grade a obtains a percentage uniform mark of 80%. … the minimum mark necessary for a Grade b obtains a percentage uniform mark of 70%. … the minimum mark necessary for a Grade c obtains a percentage uniform mark of 60%. … the minimum mark necessary for a Grade d obtains a percentage uniform mark of 50%. … the minimum mark necessary for a Grade e obtains a percentage uniform mark of 40%. … no marks receives a percentage uniform mark of 0%.

Candidates whose mark is none of the above receive a percentage mark in between those stated according to the position of their mark in relation to the grade ‘thresholds’ (i.e. the minimum mark for obtaining a grade). For example, a candidate whose mark is halfway between the minimum for a Grade c and the minimum for a Grade d (and whose grade is therefore d) receives a percentage uniform mark of 55%. The percentage uniform mark is stated at syllabus level only. It is not the same as the ‘raw’ mark obtained by the candidate, since it depends on the position of the grade thresholds (which may vary from one series to another and from one subject to another) and it has been turned into a percentage.

Access Reasonable adjustments are made for disabled candidates in order to enable them to access the assessments and to demonstrate what they know and what they can do. For this reason, very few candidates will have a complete barrier to the assessment. Information on reasonable adjustments is found in the Cambridge Handbook which can be downloaded from the website www.cie.org.uk Candidates who are unable to access part of the assessment, even after exploring all possibilities through reasonable adjustments, may still be able to receive an award based on the parts of the assessment they have taken.

Resources Copies of syllabuses, the most recent question papers and Principal Examiners’ reports for teachers are on the Syllabus and Support Materials CD-ROM, which we send to all Cambridge International Schools. They are also on our public website – go to www.cie.org.uk/alevel. Click the Subjects tab and choose your subject. For resources, click ‘Resource List’. You can use the ‘Filter by’ list to show all resources or only resources categorised as ‘Endorsed by Cambridge’. Endorsed resources are written to align closely with the syllabus they support. They have been through a detailed quality-assurance process. As new resources are published, we review them against the syllabus and publish their details on the relevant resource list section of the website. Additional syllabus-specific support is available from our secure Teacher Support website http://teachers.cie.org.uk which is available to teachers at registered Cambridge schools. It provides past question papers and examiner reports on previous examinations, as well as any extra resources such as schemes of work or examples of candidate responses. You can also find a range of subject communities on the Teacher Support website, where Cambridge teachers can share their own materials and join discussion groups.

Cambridge International AS and A Level Mathematics 9709


1 1 5 in the expansion of 2x −  . x x

1

Find the value of the coefficient of

2

Find all the values of x in the interval 0◦ ≤ x ≤ 180◦ which satisfy the equation sin 3x + 2 cos 3x = 0. [4]

3

(a) Differentiate 4x + (b) Find  4x +

6 with respect to x. x2

6  dx. x2

[3]

[2]

[3]

4

In an arithmetic progression, the 1st term is −10, the 15th term is 11 and the last term is 41. Find the sum of all the terms in the progression. [5]

5

The function f is defined by f : x → ax + b, for x ∈ , where a and b are constants. It is given that f(2) = 1 and f(5) = 7.

6

(i) Find the values of a and b.

[2]

(ii) Solve the equation ff(x) = 0.

[3]

(i) Sketch the graph of the curve y = 3 sin x, for −π ≤ x ≤ π .

[2]

The straight line y = kx, where k is a constant, passes through the maximum point of this curve for −π ≤ x ≤ π . (ii) Find the value of k in terms of π .

[2]

(iii) State the coordinates of the other point, apart from the origin, where the line and the curve intersect. [1]

7

8

The line L1 has equation 2x + y = 8. The line L2 passes through the point A (7, 4) and is perpendicular to L1 . (i) Find the equation of L2 .

[4]

(ii) Given that the lines L1 and L2 intersect at the point B, find the length of AB.

[4]

The points A, B, C and D have position vectors 3i + 2k, 2i − 2j + 5k, 2j + 7k and −2i + 10j + 7k respectively. (i) Use a scalar product to show that BA and BC are perpendicular.

[4]

(ii) Show that BC and AD are parallel and find the ratio of the length of BC to the length of AD. [4]

9709/01/M/J/03


9

The diagram shows a semicircle ABC with centre O and radius 8 cm. Angle AOB = θ radians. (i) In the case where θ = 1, calculate the area of the sector BOC.

[3]

(ii) Find the value of θ for which the perimeter of sector AOB is one half of the perimeter of [3] sector BOC. (iii) In the case where θ = 13 π , show that the exact length of the perimeter of triangle ABC is √ (24 + 8 3) cm. [3]

10

√ The equation of a curve is y = (5x + 4). (i) Calculate the gradient of the curve at the point where x = 1.

[3]

(ii) A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of x has the constant value 0.03 units per second. Find the rate of increase of y at the instant when x = 1. [2] (iii) Find the area enclosed by the curve, the x-axis, the y-axis and the line x = 1.

11

[5]

The equation of a curve is y = 8x − x2 . (i) Express 8x − x2 in the form a − (x + b)2 , stating the numerical values of a and b.

[3]

(ii) Hence, or otherwise, find the coordinates of the stationary point of the curve.

[2]

(iii) Find the set of values of x for which y ≥ −20.

[3]

The function g is defined by g : x → 8x − x2 , for x ≥ 4. (iv) State the domain and range of g−1 .

[2]

(v) Find an expression, in terms of x, for g−1 (x).

[3]

9709/01/M/J/03


1

Solve the inequality  x − 4  >  x + 1 .

2

The polynomial x4 − 9x2 − 6x − 1 is denoted by f(x).

[4]

(i) Find the value of the constant a for which

f(x) ≡ (x2 + ax + 1)(x2 − ax − 1). (ii) Hence solve the equation f(x) = 0, giving your answers in an exact form.

[3] [3]

3

The diagram shows the curve y = e2x . The shaded region R is bounded by the curve and by the lines x = 0, y = 0 and x = p. (i) Find, in terms of p, the area of R.

[3]

(ii) Hence calculate the value of p for which the area of R is equal to 5. Give your answer correct to 2 significant figures. [3]

4

(i) Show that the equation

tan(45◦ + x) = 4 tan(45◦ − x) can be written in the form 3 tan2 x − 10 tan x + 3 = 0.

[4]

(ii) Hence solve the equation

tan(45◦ + x) = 4 tan(45◦ − x), for 0◦ < x < 90◦ .

[3]

9709/02/M/J/03


5

(i) By sketching a suitable pair of graphs, show that the equation

ln x = 2 − x2 has exactly one root.

[3]

(ii) Verify by calculation that the root lies between 1.0 and 1.4. (iii) Use the iterative formula

√ xn+1 = (2 − ln xn )

to determine the root correct to 2 decimal places, showing the result of each iteration.

6

The equation of a curve is y =

[2]

[3]

1 . 1 + tan x

(i) Show, by differentiation, that the gradient of the curve is always negative.

[4]

(ii) Use the trapezium rule with 2 intervals to estimate the value of 1 π 4

 0

1 dx, 1 + tan x

giving your answer correct to 2 significant figures.

[3]

(iii)

The diagram shows a sketch of the curve for 0 ≤ x ≤ 14 π . State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in [1] part (ii). 7

The parametric equations of a curve are

x = 2θ − sin 2θ , (i) Show that

y = 2 − cos 2θ .

dy = cot θ . dx

[5]

(ii) Find the equation of the tangent to the curve at the point where θ = 14 π .

[3]

(iii) For the part of the curve where 0 < θ < 2π , find the coordinates of the points where the tangent [3] is parallel to the x-axis. 9709/02/M/J/03


1

2

Find the coordinates of the points of intersection of the line y + 2x = 11 and the curve xy = 12.

(i) Show that the equation 4 sin4 θ + 5 = 7 cos2 θ may be written in the form 4x2 + 7x − 2 = 0, where x = sin2 θ . [1] (ii) Hence solve the equation 4 sin4 θ + 5 = 7 cos2 θ , for 0◦ ≤ θ ≤ 360◦ .

3

[4]

[4]

(a) A debt of $3726 is repaid by weekly payments which are in arithmetic progression. The first payment is $60 and the debt is fully repaid after 48 weeks. Find the third payment. [3] (b) Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4. [3]

4

A curve is such that

dy = 3x2 − 4x + 1. The curve passes through the point (1, 5). dx

(i) Find the equation of the curve.

[3]

(ii) Find the set of values of x for which the gradient of the curve is positive.

[3]

5

The diagram shows a trapezium ABCD in which BC is parallel to AD and angle BCD = 90◦ . The coordinates of A, B and D are (2, 0), (4, 6) and (12, 5) respectively. (i) Find the equations of BC and CD.

[5]

(ii) Calculate the coordinates of C.

[2]

9709/01/O/N/03


6

The diagram shows the sector OPQ of a circle with centre O and radius r cm. The angle POQ is θ radians and the perimeter of the sector is 20 cm. (i) Show that θ =

20 − 2. r

[2]

(ii) Hence express the area of the sector in terms of r .

[2]

(iii) In the case where r = 8, find the length of the chord PQ.

[3]

7

The diagram shows a triangular prism with a horizontal rectangular base ADFC, where CF = 12 units and DF = 6 units. The vertical ends ABC and DEF are isosceles triangles with AB = BC = 5 units. The mid-points of BE and DF are M and N respectively. The origin O is at the mid-point of AC. Unit vectors i, j and k are parallel to OC , ON and OB respectively. (i) Find the length of OB.

[1]

−−−→ −−−→ (ii) Express each of the vectors MC and MN in terms of i, j and k.

[3]

−−−→ −−−→ (iii) Evaluate MC . MN and hence find angle CMN , giving your answer correct to the nearest degree. [4]

9709/01/O/N/03


8

A solid rectangular block has a base which measures 2x cm by x cm. The height of the block is y cm and the volume of the block is 72 cm3 . (i) Express y in terms of x and show that the total surface area, A cm2 , of the block is given by

A = 4x2 +

216 . x

[3]

Given that x can vary, (ii) find the value of x for which A has a stationary value,

[3]

(iii) find this stationary value and determine whether it is a maximum or a minimum.

[3]

9

The diagram shows points A (0, 4) and B (2, 1) on the curve y = crosses the x-axis at C. The point D has coordinates (2, 0).

8 . The tangent to the curve at B 3x + 2

(i) Find the equation of the tangent to the curve at B and hence show that the area of triangle BDC is 43 . [6] (ii) Show that the volume of the solid formed when the shaded region ODBA is rotated completely [5] about the x-axis is 8π . 10

Functions f and g are defined by f : x → 2x − 5, g : x →

4 , 2−x

x ∈ , x ∈ , x ≠ 2.

(i) Find the value of x for which fg(x) = 7.

[3]

(ii) Express each of f −1 (x) and g−1 (x) in terms of x.

[3]

(iii) Show that the equation f −1 (x) = g−1 (x) has no real roots.

[3]

(iv) Sketch, on a single diagram, the graphs of y = f(x) and y = f −1 (x), making clear the relationship between these two graphs. [3] 9709/01/O/N/03


1

Find the set of values of x satisfying the inequality  8 − 3x  < 2.

[3]

2

ln y 3

2

1

1

0

1

2

3

4

x

Two variable quantities x and y are related by the equation

y = k(a−x ), where a and k are constants. Four pairs of values of x and y are measured experimentally. The result of plotting ln y against x is shown in the diagram. Use the diagram to estimate the values of a and k. [5]

3

The polynomial x4 − 6x2 + x + a is denoted by f(x). (i) It is given that (x + 1) is a factor of f(x). Find the value of a.

[2]

(ii) When a has this value, verify that (x − 2) is also a factor of f(x) and hence factorise f(x) completely. [4]

4

√ (i) Express cos θ + ( 3) sin θ in the form R cos(θ − α ), where R > 0 and 0 < α < 21 π , giving the exact value of α . [3] (ii) Hence show that one solution of the equation √ √ cos θ + ( 3) sin θ = 2

is θ =

7 π, 12

and find the other solution in the interval 0 < θ < 2π .

9709/02/O/N/03

[4]


5

(i) By sketching a suitable pair of graphs, for x < 0, show that exactly one root of the equation x2 = 2x is negative. [2] (ii) Verify by calculation that this root lies between −1.0 and −0.5. (iii) Use the iterative formula

[2]

 x xn+1 = − 2 n 

to determine this root correct to 2 significant figures, showing the result of each iteration.

[3]

6

y M P

B

p

O

A

x

The diagram shows the curve y = (4 − x)ex and its maximum point M . The curve cuts the x-axis at A and the y-axis at B. (i) Write down the coordinates of A and B.

[2]

(ii) Find the x-coordinate of M .

[4]

(iii) The point P on the curve has x-coordinate p. The tangent to the curve at P passes through the [5] origin O. Calculate the value of p.

7

(i) By differentiating

(ii) Hence show that 

dy cos x , show that if y = cot x then = − cosec2 x. sin x dx 1 π 2 1 π 6

cosec2 x dx =

3.

[3]

[2]

By using appropriate trigonometrical identities, find the exact value of (iii) 

1 π 2 1 π 6

(iv) 

cot2 x dx,

1 π 2

1 π 6

[3]

1 dx. 1 − cos 2x

[3]

9709/02/O/N/03


1

2

A geometric progression has first term 64 and sum to infinity 256. Find (i) the common ratio,

[2]

(ii) the sum of the first ten terms.

[2]

Evaluate 

1

√ (3x + 1) dx.

[4]

0

3

(i) Show that the equation sin2 θ + 3 sin θ cos θ = 4 cos2 θ can be written as a quadratic equation in [2] tan θ . (ii) Hence, or otherwise, solve the equation in part (i) for 0◦ ≤ θ ≤ 180◦ .

4

[3]

Find the coefficient of x3 in the expansion of (i) (1 + 2x)6 ,

[3]

(ii) (1 − 3x)(1 + 2x)6 .

[3]

5

In the diagram, OCD is an isosceles triangle with OC = OD = 10 cm and angle COD = 0.8 radians. The points A and B, on OC and OD respectively, are joined by an arc of a circle with centre O and radius 6 cm. Find (i) the area of the shaded region,

[3]

(ii) the perimeter of the shaded region.

[4]

9709/01/M/J/04


6

The curve y = 9 −

6 and the line y + x = 8 intersect at two points. Find x

(i) the coordinates of the two points,

[4]

(ii) the equation of the perpendicular bisector of the line joining the two points.

[4]

7

18 and the normal to the curve at P (6, 3). This normal x meets the x-axis at R. The point Q on the x-axis and the point S on the curve are such that PQ and SR are parallel to the y-axis. The diagram shows part of the graph of y =

(i) Find the equation of the normal at P and show that R is the point (4 12 , 0).

[5]

(ii) Show that the volume of the solid obtained when the shaded region PQRS is rotated through [4] 360◦ about the x-axis is 18π .

[Questions 8, 9 and 10 are printed overleaf.]

9709/01/M/J/04


8

The diagram shows a glass window consisting of a rectangle of height h m and width 2r m and a semicircle of radius r m. The perimeter of the window is 8 m. (i) Express h in terms of r.

[2]

(ii) Show that the area of the window, A m2 , is given by

A = 8r â&#x2C6;&#x2019; 2r 2 â&#x2C6;&#x2019; 12 Ď&#x20AC; r2 .

[2]

(iii) find the value of r for which A has a stationary value,

[4]

(iv) determine whether this stationary value is a maximum or a minimum.

[2]

Given that r can vary,

9

Relative to an origin O, the position vectors of the points A, B, C and D are given by 1 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OA =  3  , â&#x2C6;&#x2019;1

3 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OB =  â&#x2C6;&#x2019;1  , 3

4 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OC =  2  p

and

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OD =  0  , q

where p and q are constants. Find â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; (i) the unit vector in the direction of AB,

[3]

(ii) the value of p for which angle AOC = 90â&#x2014;Ś , â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; (iii) the values of q for which the length of AD is 7 units. 10

[3] [4]

The functions f and g are defined as follows: f : x â&#x2020;&#x2019; x2 â&#x2C6;&#x2019; 2x, g : x â&#x2020;&#x2019; 2x + 3,

x â&#x2C6;&#x2C6; , x â&#x2C6;&#x2C6; .

(i) Find the set of values of x for which f(x) > 15.

[3]

(ii) Find the range of f and state, with a reason, whether f has an inverse.

[4]

(iii) Show that the equation gf(x) = 0 has no real solutions.

[3]

(iv) Sketch, in a single diagram, the graphs of y = g(x) and y = gâ&#x2C6;&#x2019;1 (x), making clear the relationship between the graphs. [2] 9709/01/M/J/04


1

Given that 2x = 5y , use logarithms to find the value of

2

The sequence of values given by the iterative formula

x correct to 3 significant figures. y

[3]

306 1 xn+1 = 4xn + 4 , 5 xn with initial value x1 = 3, converges to α . (i) Use this iterative formula to find α correct to 3 decimal places, showing the result of each iteration. [3] (ii) State an equation satisfied by α , and hence show that the exact value of α is

3

4

√ 5 306 .

[2]

The cubic polynomial 2x3 + ax2 − 13x − 6 is denoted by f(x). It is given that (x − 3) is a factor of f(x). (i) Find the value of a.

[2]

(ii) When a has this value, solve the equation f(x) = 0.

[4]

(i) Express 3 sin θ + 4 cos θ in the form R sin(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the value [3] of α correct to 2 decimal places. (ii) Hence solve the equation

3 sin θ + 4 cos θ = 4.5, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ , correct to 1 decimal place. (iii) Write down the least value of 3 sin θ + 4 cos θ + 7 as θ varies.

9709/02/M/J/04

[4] [1]


5

The diagram shows the part of the curve y = xe−x for 0 ≤ x ≤ 2, and its maximum point M . (i) Find the x-coordinate of M .

[4]

(ii) Use the trapezium rule with two intervals to estimate the value of 2

xe−x dx, 0

giving your answer correct to 2 decimal places.

[3]

(iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the integral in part (ii).

6

The parametric equations of a curve are

x = 2t + ln t,

4 y=t+ , t

where t takes all positive values. (i) Show that

dy t2 − 4 = . dx t(2t + 1)

[3]

(ii) Find the equation of the tangent to the curve at the point where t = 1.

[3]

(iii) The curve has one stationary point. Find the y-coordinate of this point, and determine whether this point is a maximum or a minimum. [4]

7

(i) By expanding cos(2x + x), show that

cos 3x ≡ 4 cos3 x − 3 cos x.

[5]

(ii) Hence, or otherwise, show that

1 π 2

0

cos3 x dx = 23 .

9709/02/M/J/04

[5]


1

2 5 Find the coefficient of x in the expansion of 3x −  . x

2

Find

[4]

(i) the sum of the first ten terms of the geometric progression 81, 54, 36, . . . ,

[3]

(ii) the sum of all the terms in the arithmetic progression 180, 175, 170, . . . , 25.

[3]

3

In the diagram, AC is an arc of a circle, centre O and radius 6 cm. The line BC is perpendicular to OC and OAB is a straight line. Angle AOC = 13 π radians. Find the area of the shaded region, giving √ [5] your answer in terms of π and 3. 4

(i) Sketch and label, on the same diagram, the graphs of y = 2 sin x and y = cos 2x, for the interval [4] 0 ≤ x ≤ π. (ii) Hence state the number of solutions of the equation 2 sin x = cos 2x in the interval 0 ≤ x ≤ π . [1]

5

The equation of a curve is y = x2 − 4x + 7 and the equation of a line is y + 3x = 9. The curve and the line intersect at the points A and B. (i) The mid-point of AB is M . Show that the coordinates of M are  12 , 7 12 .

[4]

(ii) Find the coordinates of the point Q on the curve at which the tangent is parallel to the line y + 3x = 9. [3] (iii) Find the distance MQ.

6

[1]

The function f : x → 5 sin2 x + 3 cos2 x is defined for the domain 0 ≤ x ≤ π . (i) Express f(x) in the form a + b sin2 x, stating the values of a and b.

[2]

(ii) Hence find the values of x for which f(x) = 7 sin x.

[3]

(iii) State the range of f.

[2] 9709/01/O/N/04


7

A curve is such that

dy 6 =√ and P (3, 3) is a point on the curve. dx (4x − 3)

(i) Find the equation of the normal to the curve at P, giving your answer in the form ax + by = c. [3] (ii) Find the equation of the curve.

8

[4]

The points A and B have position vectors i + 7j + 2k and −5i + 5j + 6k respectively, relative to an origin O. (i) Use a scalar product to calculate angle AOB, giving your answer in radians correct to 3 significant figures. [4]

−−→ −−→ −−→ (ii) The point C is such that AB = 2BC. Find the unit vector in the direction of OC.

9

[4]

The function f : x → 2x − a, where a is a constant, is defined for all real x. (i) In the case where a = 3, solve the equation ff(x) = 11.

[3]

The function g : x → x2 − 6x is defined for all real x. (ii) Find the value of a for which the equation f(x) = g(x) has exactly one real solution.

[3]

The function h : x → x2 − 6x is defined for the domain x ≥ 3.

10

(iii) Express x2 − 6x in the form (x − p)2 − q, where p and q are constants.

[2]

(iv) Find an expression for h−1 (x) and state the domain of h−1 .

[4]

2 A curve has equation y = x2 + . x dy d2 y (i) Write down expressions for and 2 . dx dx

[3]

(ii) Find the coordinates of the stationary point on the curve and determine its nature.

[4]

(iii) Find the volume of the solid formed when the region enclosed by the curve, the x-axis and the [6] lines x = 1 and x = 2 is rotated completely about the x-axis.

9709/01/O/N/04


1

Solve the inequality | x + 1| > | x |.

[3]

2

Solve the equation x3.9 = 11x3.2 , where x ≠ 0.

[3]

3

Find the values of x satisfying the equation 3 sin 2x = cos x, for 0◦ ≤ x ≤ 90◦ .

[4]

4

The cubic polynomial 2x3 − 5x2 + ax + b is denoted by f(x). It is given that (x − 2) is a factor of f(x), [5] and that when f(x) is divided by (x + 1) the remainder is −6. Find the values of a and b.

5

The curve with equation y = x2 ln x, where x > 0, has one stationary point.

6

(i) Find the x-coordinate of this point, giving your answer in terms of e.

[4]

(ii) Determine whether this point is a maximum or a minimum point.

[2]

(i) By sketching a suitable pair of graphs, show that there is only one value of x in the interval 0 < x < 12 π that is a root of the equation

cot x = x.

[2]

(ii) Verify by calculation that this root lies between 0.8 and 0.9 radians.

[2]

(iii) Show that this value of x is also a root of the equation

1 x = tan−1  . x

[1]

(iv) Use the iterative formula

xn+1 = tan−1 

1  xn

to determine this root correct to 2 decimal places, showing the result of each iteration.

9709/02/O/N/04

[3]


7

The diagram shows the curve y = 2ex + 3eâ&#x2C6;&#x2019;2x . The curve cuts the y-axis at A. (i) Write down the coordinates of A.

[1]

(ii) Find the equation of the tangent to the curve at A, and state the coordinates of the point where [6] this tangent meets the x-axis. (iii) Calculate the area of the region bounded by the curve and by the lines x = 0, y = 0 and x = 1, giving your answer correct to 2 significant figures. [4]

8

(i) Express cos θ + sin θ in the form R cos(θ â&#x2C6;&#x2019; Îą ), where R > 0 and 0 < Îą < 12 Ď&#x20AC; , giving the exact values of R and Îą . [3] (ii) Hence show that

1 = 12 sec2 (θ â&#x2C6;&#x2019; 14 Ď&#x20AC; ). 2 (cos θ + sin θ )

(iii) By differentiating

sin x dy , show that if y = tan x then = sec2 x. cos x dx

[1]

[3]

(iv) Using the results of parts (ii) and (iii), show that



1 Ď&#x20AC; 2

0

1 dθ = 1. (cos θ + sin θ )2

9709/02/O/N/04

[3]


1

A curve is such that the curve.

2

3

4

dy = 2x2 − 5. Given that the point (3, 8) lies on the curve, find the equation of dx [4]

Find the gradient of the curve y =

x2

12 at the point where x = 3. − 4x

[4]

(i) Show that the equation sin θ + cos θ = 2(sin θ − cos θ ) can be expressed as tan θ = 3.

[2]

(ii) Hence solve the equation sin θ + cos θ = 2(sin θ − cos θ ), for 0◦ ≤ θ ≤ 360◦ .

[2]

(i) Find the first 3 terms in the expansion of (2 − x)6 in ascending powers of x.

[3]

(ii) Find the value of k for which there is no term in x2 in the expansion of (1 + kx)(2 − x)6 .

[2]

5

The diagram shows a rhombus ABCD. The points B and D have coordinates (2, 10) and (6, 2) respectively, and A lies on the x-axis. The mid-point of BD is M . Find, by calculation, the coordinates [6] of each of M , A and C.

6

A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression. [6]

9709/01/M/J/05


7

A function f is defined by f : x → 3 − 2 sin x, for 0◦ ≤ x ≤ 360◦ . (i) Find the range of f.

[2]

(ii) Sketch the graph of y = f(x).

[2]

A function g is defined by g : x → 3 − 2 sin x, for 0◦ ≤ x ≤ A◦ , where A is a constant. (iii) State the largest value of A for which g has an inverse.

[1]

(iv) When A has this value, obtain an expression, in terms of x, for g−1 (x).

[2]

8

In the diagram, ABC is a semicircle, centre O and radius 9 cm. The line BD is perpendicular to the diameter AC and angle AOB = 2.4 radians.

9

(i) Show that BD = 6.08 cm, correct to 3 significant figures.

[2]

(ii) Find the perimeter of the shaded region.

[3]

(iii) Find the area of the shaded region.

[3]

4 A curve has equation y = √ . x (i) The normal to the curve at the point (4, 2) meets the x-axis at P and the y-axis at Q. Find the [6] length of PQ, correct to 3 significant figures. (ii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 4.

9709/01/M/J/05

[4]


10

The equation of a curve is y = x2 − 3x + 4. (i) Show that the whole of the curve lies above the x-axis.

[3]

(ii) Find the set of values of x for which x2 − 3x + 4 is a decreasing function of x.

[1]

The equation of a line is y + 2x = k, where k is a constant. (iii) In the case where k = 6, find the coordinates of the points of intersection of the line and the curve. [3] (iv) Find the value of k for which the line is a tangent to the curve.

11

[3]

Relative to an origin O, the position vectors of the points A and B are given by −−→ OA = 2i + 3j − k

and

−−→ OB = 4i − 3j + 2k.

(i) Use a scalar product to find angle AOB, correct to the nearest degree.

[4]

−−→ (ii) Find the unit vector in the direction of AB.

[3]

−−→ −−→ (iii) The point C is such that OC = 6j + pk, where p is a constant. Given that the lengths of AB and −−→ AC are equal, find the possible values of p. [4]

9709/01/M/J/05


1

Solve the inequality | x | > |3x − 2|.

2

(a) Use logarithms to solve the equation 3x = 8, giving your answer correct to 2 decimal places. [2]

[4]

(b) It is given that

ln  = ln(y + 2) − 2 ln y, where y > 0. Express  in terms of y in a form not involving logarithms.

3

[3]

The sequence of values given by the iterative formula

xn+1 =

3xn 4

+

2 , x3n

with initial value x1 = 2, converges to α . (i) Use this iteration to calculate α correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3] (ii) State an equation which is satisfied by α and hence find the exact value of α .

4

5

[2]

The polynomial x3 − x2 + ax + b is denoted by p(x). It is given that (x + 1) is a factor of p(x) and that when p(x) is divided by (x − 2) the remainder is 12. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x).

[2]

(i) By differentiating

1 dy , show that if y = sec θ then = sec θ tan θ . cos θ dθ

[3]

(ii) The parametric equations of a curve are

x = 1 + tan θ , for − 12 π < θ < 12 π . Show that

y = sec θ ,

dy = sin θ . dx

(iii) Find the coordinates of the point on the curve at which the gradient of the curve is 12 .

9709/02/M/J/05

[3] [3]


6

ln x for 0 < x â&#x2030;¤ 4. The curve cuts the x-axis at A and its x

The diagram shows the part of the curve y = maximum point is M . (i) Write down the coordinates of A.

[1]

(ii) Show that the x-coordinate of M is e, and write down the y-coordinate of M in terms of e.

[5]

(iii) Use the trapezium rule with three intervals to estimate the value of 4

1

ln x dx, x

correct to 2 decimal places.

[3]

(iv) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the integral in part (iii).

7

(i) By expanding sin(2x + x) and using double-angle formulae, show that

sin 3x = 3 sin x â&#x2C6;&#x2019; 4 sin3 x.

[5]

(ii) Hence show that



1 Ď&#x20AC; 3

sin3 x dx =

0

9709/02/M/J/05

5 . 24

[5]


1

Solve the equation 3 sin2 θ − 2 cos θ − 3 = 0, for 0◦ ≤ θ ≤ 180◦ .

[4]

2

In the diagram, OAB and OCD are radii of a circle, centre O and radius 16 cm. Angle AOC = α radians. AC and BD are arcs of circles, centre O and radii 10 cm and 16 cm respectively. (i) In the case where α = 0.8, find the area of the shaded region.

[2]

(ii) Find the value of α for which the perimeter of the shaded region is 28.9 cm.

[3]

3

In the diagram, ABED is a trapezium√with right angles at E and D, and CED is a straight line. The lengths of AB and BC are 2d and (2 3)d respectively, and angles BAD and CBE are 30◦ and 60◦ respectively. (i) Find the length of CD in terms of d.

[2]

2 (ii) Show that angle CAD = tan−1  √ . 3

[3]

9709/01/O/N/05


4

Relative to an origin O, the position vectors of points P and Q are given by −2 −−→ OP =  3  1

and

2 −−→ OQ =  1  , q

where q is a constant. (i) In the case where q = 3, use a scalar product to show that cos POQ = 17 .

[3]

−−→ (ii) Find the values of q for which the length of PQ is 6 units.

[4]

5

The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm, and the cylinder has radius r cm and height h cm. The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone. (i) Express h in terms of r and hence show that the volume, V cm3 , of the cylinder is given by

V = 12π r2 − 2π r3 . (ii) Given that r varies, find the stationary value of V .

6

[3] [4]

A small trading company made a profit of $250 000 in the year 2000. The company considered two different plans, plan A and plan B, for increasing its profits. Under plan A, the annual profit would increase each year by 5% of its value in the preceding year. Find, for plan A, (i) the profit for the year 2008,

[3]

(ii) the total profit for the 10 years 2000 to 2009 inclusive.

[2]

Under plan B, the annual profit would increase each year by a constant amount $D. (iii) Find the value of D for which the total profit for the 10 years 2000 to 2009 inclusive would be the same for both plans. [3]

9709/01/O/N/05


7

8

9

Three points have coordinates A (2, 6), B (8, 10) and C (6, 0). The perpendicular bisector of AB meets the line BC at D. Find (i) the equation of the perpendicular bisector of AB in the form ax + by = c,

[4]

(ii) the coordinates of D.

[4]

A function f is defined by f : x → (2x − 3)3 − 8, for 2 ≤ x ≤ 4. (i) Find an expression, in terms of x, for f (x) and show that f is an increasing function.

[4]

(ii) Find an expression, in terms of x, for f −1 (x) and find the domain of f −1 .

[4]

The equation of a curve is xy = 12 and the equation of a line l is 2x + y = k, where k is a constant. (i) In the case where k = 11, find the coordinates of the points of intersection of l and the curve. [3] (ii) Find the set of values of k for which l does not intersect the curve.

[4]

(iii) In the case where k = 10, one of the points of intersection is P (2, 6). Find the angle, in degrees [4] correct to 1 decimal place, between l and the tangent to the curve at P.

10

A curve is such that

dy 16 = , and (1, 4) is a point on the curve. dx x3

(i) Find the equation of the curve.

[4]

(ii) A line with gradient − 12 is a normal to the curve. Find the equation of this normal, giving your answer in the form ax + by = c. [4] (iii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2.

9709/01/O/N/05

[4]


1

Solve the inequality (0.8)x < 0.5.

2

The polynomial x3 + 2x2 + 2x + 3 is denoted by p(x).

3

[3]

(i) Find the remainder when p(x) is divided by x − 1.

[2]

(ii) Find the quotient and remainder when p(x) is divided by x2 + x − 1.

[4]

(i) Express 12 cos θ − 5 sin θ in the form R cos(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the [3] exact value of R and the value of α correct to 2 decimal places. (ii) Hence solve the equation

12 cos θ − 5 sin θ = 10, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .

4

[4]

The equation of a curve is x3 + y3 = 9xy. (i) Show that

dy 3 y − x 2 = . dx y2 − 3x

[4]

(ii) Find the equation of the tangent to the curve at the point (2, 4), giving your answer in the form ax + by = c. [3]

5

(i) By sketching a suitable pair of graphs, show that there is only one value of x that is a root of the equation 1 = ln x. [2] x (ii) Verify by calculation that this root lies between 1 and 2.

[2]

(iii) Show that this root also satisfies the equation 1

x = ex .

[1]

(iv) Use the iterative formula 1 xn

xn+1 = e , with initial value x1 = 1.8, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

6

A curve is such that

dy = e2x − 2e−x . The point (0, 1) lies on the curve. dx

(i) Find the equation of the curve.

[4]

(ii) The curve has one stationary point. Find the x-coordinate of this point and determine whether it is a maximum or a minimum point. [5] 9709/02/O/N/05


7

The diagram shows the part of the curve y = sin2 x for 0 ≤ x ≤ π . (i) Show that

dy = sin 2x. dx

[2]

(ii) Hence find the x-coordinates of the points on the curve at which the gradient of the curve is 0.5. [3] (iii) By expressing sin2 x in terms of cos 2x, find the area of the region bounded by the curve and the x-axis between 0 and π . [5]

9709/02/O/N/05


1

A curve has equation y = the constant k.

2

k . Given that the gradient of the curve is −3 when x = 2, find the value of x [3]

Solve the equation sin 2x + 3 cos 2x = 0, for 0◦ ≤ x ≤ 180◦ .

3

[4]

Each year a company gives a grant to a charity. The amount given each year increases by 5% of its value in the preceding year. The grant in 2001 was $5000. Find (i) the grant given in 2011,

[3]

(ii) the total amount of money given to the charity during the years 2001 to 2011 inclusive.

[2]

4

The first three terms in the expansion of (2 + ax)n , in ascending powers of x, are 32 − 40x + bx2 . Find [5] the values of the constants n, a and b.

5

The curve y2 = 12x intersects the line 3y = 4x + 6 at two points. Find the distance between the two points. [6]

6

In the diagram, ABC is a triangle in which AB = 4 cm, BC = 6 cm and angle ABC = 150◦ . The line CX is perpendicular to the line ABX . (i) Find the exact length of BX and show that angle CAB = tan−1  (ii) Show that the exact length of AC is



√ 52 + 24 3 cm.

9709/01/M/J/06

3 √ . 4+3 3

[4] [2]


7

The diagram shows a circle with centre O and radius 8 cm. Points A and B lie on the circle. The tangents at A and B meet at the point T , and AT = BT = 15 cm. (i) Show that angle AOB is 2.16 radians, correct to 3 significant figures.

[3]

(ii) Find the perimeter of the shaded region.

[2]

(iii) Find the area of the shaded region.

[3]

8

The diagram shows the roof of a house. The base of the roof, OABC, is rectangular and horizontal with OA = CB = 14 m and OC = AB = 8 m. The top of the roof DE is 5 m above the base and DE = 6 m. The sloping edges OD, CD, AE and BE are all equal in length. Unit vectors i and j are parallel to OA and OC respectively and the unit vector k is vertically upwards. −−→ (i) Express the vector OD in terms of i, j and k, and find its magnitude.

[4]

(ii) Use a scalar product to find angle DOB.

[4]

9709/01/M/J/06


9

A curve is such that

dy 4 =√ , and P (1, 8) is a point on the curve. dx (6 − 2x)

(i) The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the [5] coordinates of the mid-point of QR. (ii) Find the equation of the curve.

[4]

10

The diagram shows the curve y = x3 − 3x2 − 9x + k, where k is a constant. The curve has a minimum point on the x-axis.

11

(i) Find the value of k.

[4]

(ii) Find the coordinates of the maximum point of the curve.

[1]

(iii) State the set of values of x for which x3 − 3x2 − 9x + k is a decreasing function of x.

[1]

(iv) Find the area of the shaded region.

[4]

Functions f and g are defined by f : x → k − x 9 g : x → x+2

for x ∈ , where k is a constant, for x ∈ , x ≠ −2.

(i) Find the values of k for which the equation f(x) = g(x) has two equal roots and solve the equation [6] f(x) = g(x) in these cases. (ii) Solve the equation fg(x) = 5 when k = 6.

[3]

(iii) Express g−1 (x) in terms of x.

[2]

9709/01/M/J/06


1

2

Solve the inequality |2x − 7| > 3.

(i) Prove the identity

[3]

√ cos(x + 30◦ ) + sin(x + 60◦ ) ≡ ( 3) cos x.

[3]

(ii) Hence solve the equation

cos(x + 30◦ ) + sin(x + 60◦ ) = 1, for 0◦ < x < 90◦ .

[2]

3

The equation of a curve is y = x + 2 cos x. Find the x-coordinates of the stationary points of the curve [7] for 0 ≤ x ≤ 2π , and determine the nature of each of these stationary points.

4

The cubic polynomial ax3 + bx2 − 3x − 2, where a and b are constants, is denoted by p(x). It is given that (x − 1) and (x + 2) are factors of p(x).

5

6

(i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the other linear factor of p(x).

[2]

The equation of a curve is 3x2 + 2xy + y2 = 6. It is given that there are two points on the curve where the tangent is parallel to the x-axis. (i) Show by differentiation that, at these points, y = −3x.

[4]

(ii) Hence find the coordinates of the two points.

[4]

(i) By sketching a suitable pair of graphs, show that there is only one value of x that is a root of the equation x = 9e−2x . [2] (ii) Verify, by calculation, that this root lies between 1 and 2.

[2]

(iii) Show that, if a sequence of values given by the iterative formula

xn+1 = 12 ln 9 − ln xn  converges, then it converges to the root of the equation given in part (i).

[2]

(iv) Use the iterative formula, with x1 = 1, to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

9709/02/M/J/06


7

(i) Differentiate ln(2x + 3).

[2]

(ii) Hence, or otherwise, show that 3

 −1

1 dx = ln 3. 2x + 3

(iii) Find the quotient and remainder when 4x2 + 8x is divided by 2x + 3.

[3]

[3]

(iv) Hence show that 3

 −1

4x2 + 8x dx = 12 − 3 ln 3. 2x + 3

9709/02/M/J/06

[3]


1

2 6 Find the coefficient of x2 in the expansion of x +  . x

2

Given that x = sin−1  25 , find the exact value of

[3]

(i) cos2 x,

[2]

(ii) tan2 x.

[2]

3

In the diagram, AOB is a sector of a circle with centre O and radius 12 cm. The point A lies on the side CD of the rectangle OCDB. Angle AOB = 13 π radians. Express the area of the shaded region in √ [6] the form a( 3) − bπ , stating the values of the integers a and b.

4

−3 −1 6 The position vectors of points A and B are   and  2  respectively, relative to an origin O. 3 4 (i) Calculate angle AOB.

[3]

−−→ −−→ −−→ (ii) The point C is such that AC = 3AB. Find the unit vector in the direction of OC.

[4]

9709/01/O/N/06


5

The three points A (1, 3), B (13, 11) and C (6, 15) are shown in the diagram. The perpendicular from C to AB meets AB at the point D. Find

6

(i) the equation of CD,

[3]

(ii) the coordinates of D.

[4]

(a) Find the sum of all the integers between 100 and 400 that are divisible by 7.

[4]

(b) The first three terms in a geometric progression are 144, x and 64 respectively, where x is positive. Find (i) the value of x, (ii) the sum to infinity of the progression.

[5] 7

The diagram shows the curve y = x(x â&#x2C6;&#x2019; 1)(x â&#x2C6;&#x2019; 2), which crosses the x-axis at the points O (0, 0), A (1, 0) and B (2, 0). (i) The tangents to the curve at the points A and B meet at the point C. Find the x-coordinate of C. [5] (ii) Show by integration that the area of the shaded region R1 is the same as the area of the shaded region R2 . [4] 9709/01/O/N/06


8

The equation of a curve is y =

6 . 5 − 2x

(i) Calculate the gradient of the curve at the point where x = 1.

[3]

(ii) A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of y has a constant value of 0.02 units per second. Find the rate of increase of x when x = 1. [2] (iii) The region between the curve, the x-axis and the lines x = 0 and x = 1 is rotated through 360◦ about the x-axis. Show that the volume obtained is 12 π. [5] 5 9

The diagram shows an open container constructed out of 200 cm2 of cardboard. The two vertical end pieces are isosceles triangles with sides 5x cm, 5x cm and 8x cm, and the two side pieces are rectangles of length y cm and width 5x cm, as shown. The open top is a horizontal rectangle. (i) Show that y =

200 − 24x2 . 10x

[3]

(ii) Show that the volume, V cm3 , of the container is given by V = 240x − 28.8x3 .

[2]

Given that x can vary,

10

(iii) find the value of x for which V has a stationary value,

[3]

(iv) determine whether it is a maximum or a minimum stationary value.

[2]

The function f is defined by f : x → x2 − 3x for x ∈ . (i) Find the set of values of x for which f(x) > 4.

[3]

(ii) Express f(x) in the form (x − a)2 − b, stating the values of a and b.

[2]

(iii) Write down the range of f.

[1]

(iv) State, with a reason, whether f has an inverse.

[1]

√ The function g is defined by g : x → x − 3 x for x ≥ 0. (v) Solve the equation g(x) = 10.

[3]

9709/01/O/N/06


1

2

Solve the inequality |2x − 1| > | x |.

[4]

(i) Express 4x in terms of y, where y = 2x .

[1]

(ii) Hence find the values of x that satisfy the equation

34x  − 102x  + 3 = 0, giving your answers correct to 2 decimal places.

3

4

[5]

The polynomial 4x3 − 7x + a, where a is a constant, is denoted by p(x). It is given that (2x − 3) is a factor of p(x). (i) Show that a = −3.

[2]

(ii) Hence, or otherwise, solve the equation p(x) = 0.

[4]

(i) Prove the identity

tan(x + 45◦ ) − tan(45◦ − x) ≡ 2 tan 2x.

[4]

(ii) Hence solve the equation

tan(x + 45◦ ) − tan(45◦ − x) = 2, for 0◦ ≤ x ≤ 180◦ .

[3]

5

The diagram shows a chord joining two points, A and B, on the circumference of a circle with centre O and radius r. The angle AOB is α radians, where 0 < α < π . The area of the shaded segment is one sixth of the area of the circle. (i) Show that α satisfies the equation

x = 13 π + sin x. (ii) Verify by calculation that α lies between 12 π and 23 π .

[3] [2]

(iii) Use the iterative formula

xn+1 = 13 π + sin xn , with initial value x1 = 2, to determine α correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] 9709/02/O/N/06


6

The diagram shows the part of the curve y =

e2x for x > 0, and its minimum point M . x

(i) Find the coordinates of M .

[5]

(ii) Use the trapezium rule with 2 intervals to estimate the value of 2

 1

e2x dx, x

giving your answer correct to 1 decimal place.

[3]

(iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the integral in part (ii).

7

(i) Given that y = tan 2x, find

dy . dx

[2]

(ii) Hence, or otherwise, show that



1 π 6

sec2 2x dx =

0

1 2

3,

and, by using an appropriate trigonometrical identity, find the exact value of 

1 π 6

tan2 2x dx. [6]

0

(iii) Use the identity cos 4x ≡ 2 cos2 2x − 1 to find the exact value of 1 π 6

 0

1 dx. 1 + cos 4x

9709/02/O/N/06

[2]


1

Find the value of the constant c for which the line y = 2x + c is a tangent to the curve y2 = 4x.

[4]

2

1

The diagram shows the curve y = 3x 4 . The shaded region is bounded by the curve, the x-axis and the lines x = 1 and x = 4. Find the volume of the solid obtained when this shaded region is rotated [4] completely about the x-axis, giving your answer in terms of π . 1 − tan2 x ≡ 1 − 2 sin2 x. 2 1 + tan x

3

Prove the identity

4

Find the real roots of the equation

[4]

18 1 + = 4. x4 x2

[4]

5

In the diagram, OAB is a sector of a circle with centre O and radius 12 cm. The lines AX and BX are tangents to the circle at A and B respectively. Angle AOB = 31 π radians. (i) Find the exact length of AX , giving your answer in terms of

3.

(ii) Find the area of the shaded region, giving your answer in terms of π and

9709/01/M/J/07

[2] √

3.

[3]


6

The diagram shows a rectangle ABCD. The point A is (2, 14), B is (−2, 8) and C lies on the x-axis. Find

7

(i) the equation of BC,

[4]

(ii) the coordinates of C and D.

[3]

The second term of a geometric progression is 3 and the sum to infinity is 12. (i) Find the first term of the progression.

[4]

An arithmetic progression has the same first and second terms as the geometric progression. (ii) Find the sum of the first 20 terms of the arithmetic progression.

8

9

[3]

The function f is defined by f(x) = a + b cos 2x, for 0 ≤ x ≤ π . It is given that f(0) = −1 and f 21 π  = 7. (i) Find the values of a and b.

[3]

(ii) Find the x-coordinates of the points where the curve y = f(x) intersects the x-axis.

[3]

(iii) Sketch the graph of y = f(x).

[2]

Relative to an origin O, the position vectors of the points A and B are given by 4 −−→ OA =  1  −2

and

3 −−→ OB =  2  . −4

−−→ −−→ −−→ (i) Given that C is the point such that AC = 2AB, find the unit vector in the direction of OC.

[4]

1 −−→ The position vector of the point D is given by OD =  4 , where k is a constant, and it is given that k −−→ −−→ −−→ OD = mOA + nOB, where m and n are constants. (ii) Find the values of m, n and k.

[4] 9709/01/M/J/07


10

The equation of a curve is y = 2x +

(i) Obtain expressions for

8 . x2

dy d2 y and 2 . dx dx

[3]

(ii) Find the coordinates of the stationary point on the curve and determine the nature of the stationary point. [3] (iii) Show that the normal to the curve at the point (−2, −2) intersects the x-axis at the point (−10, 0). [3] (iv) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2.

[3]

11

The diagram shows the graph of y = f(x), where f : x →

6 for x ≥ 0. 2x + 3

(i) Find an expression, in terms of x, for f  (x) and explain how your answer shows that f is a decreasing function. [3] (ii) Find an expression, in terms of x, for f −1 (x) and find the domain of f −1 .

[4]

(iii) Copy the diagram and, on your copy, sketch the graph of y = f −1 (x), making clear the relationship between the graphs. [2]

The function g is defined by g : x → 12 x for x ≥ 0. (iv) Solve the equation fg(x) = 32 .

[3]

9709/01/M/J/07


1

Solve the inequality | x − 3| > | x + 2|.

2

The variables x and y satisfy the relation 3y = 4x+2 .

[4]

(i) By taking logarithms, show that the graph of y against x is a straight line. Find the exact value of the gradient of this line. [3] (ii) Calculate the x-coordinate of the point of intersection of this line with the line y = 2x, giving your answer correct to 2 decimal places. [3]

3

The parametric equations of a curve are

x = 3t + ln(t − 1),

(i) Express

y = t2 + 1,

for t > 1.

dy in terms of t. dx

[3]

(ii) Find the coordinates of the only point on the curve at which the gradient of the curve is equal to 1. [4]

4

5

The polynomial 2x3 − 3x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that (x − 2) is a factor of p(x), and that when p(x) is divided by (x + 2) the remainder is −20. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the remainder when p(x) is divided by (x2 − 4).

[3]

(i) By sketching a suitable pair of graphs, show that the equation

sec x = 3 − x, where x is in radians, has only one root in the interval 0 < x < 12 π . (ii) Verify by calculation that this root lies between 1.0 and 1.2.

[2] [2]

(iii) Show that this root also satisfies the equation

x = cos−1 

1 . 3−x

[1]

(iv) Use the iterative formula

xn+1 = cos−1 

1 , 3 − xn

with initial value x1 = 1.1, to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

9709/02/M/J/07


6

(i) Express cos2 x in terms of cos 2x.

[1]

(ii) Hence show that



1 π 3

0

cos2 x dx = 16 π + 18

3.

[4]

(iii) By using an appropriate trigonometrical identity, deduce the exact value of



! π 3

sin2 x dx.

0

[3]

7

The diagram shows the part of the curve y = ex cos x for 0 ≤ x ≤ 12 π . The curve meets the y-axis at the point A. The point M is a maximum point. (i) Write down the coordinates of A.

[1]

(ii) Find the x-coordinate of M .

[4]

(iii) Use the trapezium rule with three intervals to estimate the value of



1 π 2

ex cos x dx,

0

giving your answer correct to 2 decimal places.

[3]

(iv) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the integral in part (iii).

9709/02/M/J/07


1

2

3

Determine the set of values of the constant k for which the line y = 4x + k does not intersect the curve y = x2 . [3] √ Find the area of the region enclosed by the curve y = 2 x, the x-axis and the lines x = 1 and x = 4. [4] (i) Find the first three terms in the expansion of (2 + u)5 in ascending powers of u.

[3]

(ii) Use the substitution u = x + x2 in your answer to part (i) to find the coefficient of x2 in the 5 expansion of 2 + x + x2  . [2]

4

The 1st term of an arithmetic progression is a and the common difference is d, where d ≠ 0. (i) Write down expressions, in terms of a and d, for the 5th term and the 15th term.

[1]

The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.

5

(ii) Show that 3a = 8d.

[3]

(iii) Find the common ratio of the geometric progression.

[2]

(i) Show that the equation 3 sin x tan x = 8 can be written as 3 cos2 x + 8 cos x − 3 = 0.

[3]

(ii) Hence solve the equation 3 sin x tan x = 8 for 0◦ ≤ x ≤ 360◦ .

[3]

6

The three points A (3, 8), B (6, 2) and C (10, 2) are shown in the diagram. The point D is such that [7] the line DA is perpendicular to AB and DC is parallel to AB. Calculate the coordinates of D.

9709/01/O/N/07


7

In the diagram, AB is an arc of a circle, centre O and radius r cm, and angle AOB = θ radians. The point X lies on OB and AX is perpendicular to OB. (i) Show that the area, A cm2 , of the shaded region AXB is given by

A = 12 r 2 (θ − sin θ cos θ ).

[3]

(ii) In the case where r = 12 and θ = 16 π , find the perimeter of the shaded region AXB, leaving your √ [4] answer in terms of 3 and π .

8

The equation of a curve is y = (2x − 3)3 − 6x. (i) Express

dy d2 y and 2 in terms of x. dx dx

[3]

(ii) Find the x-coordinates of the two stationary points and determine the nature of each stationary point. [5]

9

dy = 4 − x and the point P (2, 9) lies on the curve. The normal to the curve at P dx meets the curve again at Q. Find

A curve is such that

(i) the equation of the curve,

[3]

(ii) the equation of the normal to the curve at P,

[3]

(iii) the coordinates of Q.

[3]

9709/01/O/N/07


10

The diagram shows a cube OABCDEFG in which the length of each side is 4 units. The unit vectors −−→ −−→ −−→ i, j and k are parallel to OA, OC and OD respectively. The mid-points of OA and DG are P and Q respectively and R is the centre of the square face ABFE.

11

−−→ −−→ (i) Express each of the vectors PR and PQ in terms of i, j and k.

[3]

(ii) Use a scalar product to find angle QPR.

[4]

(iii) Find the perimeter of triangle PQR, giving your answer correct to 1 decimal place.

[3]

The function f is defined by f : x → 2x2 − 8x + 11 for x ∈ . (i) Express f(x) in the form a(x + b)2 + c, where a, b and c are constants.

[3]

(ii) State the range of f.

[1]

(iii) Explain why f does not have an inverse.

[1]

The function g is defined by g : x → 2x2 − 8x + 11 for x ≤ A, where A is a constant. (iv) State the largest value of A for which g has an inverse.

[1]

(v) When A has this value, obtain an expression, in terms of x, for g−1 (x) and state the range of g−1 . [4]

9709/01/O/N/07


1

Show that 4

 1

2

1 dx = 12 ln 3. 2x + 1

[4]

The sequence of values given by the iterative formula

xn+1 =

2xn 3

+

4 , x2n

with initial value x1 = 2, converges to α . (i) Use this iterative formula to determine α correct to 2 decimal places, giving the result of each iteration to 4 decimal places. [3]

3

(ii) State an equation that is satisfied by α and hence find the exact value of α .

[2]

(i) Solve the inequality | y − 5| < 1.

[2]

(ii) Hence solve the inequality |3x − 5| < 1, giving 3 significant figures in your answer.

[3]

4

The equation of a curve is y = 2x − tan x, where x is in radians. Find the coordinates of the stationary points of the curve for which − 12 π < x < 12 π . [5]

5

The polynomial 3x3 + 8x2 + ax − 2, where a is a constant, is denoted by p(x). It is given that (x + 2) is a factor of p(x).

6

(i) Find the value of a.

[2]

(ii) When a has this value, solve the equation p(x) = 0.

[4]

(i) Express 8 sin θ − 15 cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ , giving the [3] exact value of R and the value of α correct to 2 decimal places. (ii) Hence solve the equation

8 sin θ − 15 cos θ = 14, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .

7

[4]

(i) Prove the identity

(cos x + 3 sin x)2 ≡ 5 − 4 cos 2x + 3 sin 2x.

[4]

(ii) Using the identity, or otherwise, find the exact value of



1 π 4

(cos x + 3 sin x)2 dx.

0

9709/02/O/N/07

[4]


8

The diagram shows the curve y = x2 eâ&#x2C6;&#x2019;x and its maximum point M . (i) Find the x-coordinate of M .

[4]

(ii) Show that the tangent to the curve at the point where x = 1 passes through the origin.

[3]

(iii) Use the trapezium rule, with two intervals, to estimate the value of 3

 x2 eâ&#x2C6;&#x2019;x dx, 1

giving your answer correct to 2 decimal places.

9709/02/O/N/07

[3]


1

2

In the triangle ABC, AB = 12 cm, angle BAC = 60◦ and angle ACB = 45◦ . Find the exact length [3] of BC. (i) Show that the equation 2 tan2 θ cos θ = 3 can be written in the form 2 cos2 θ + 3 cos θ − 2 = 0.

[2]

3

4

(ii) Hence solve the equation 2 tan2 θ cos θ = 3, for 0◦ ≤ θ ≤ 360◦ .

[3]

(i) Find the first 3 terms in the expansion, in ascending powers of x, of (2 + x2 )5 .

[3]

(ii) Hence find the coefficient of x4 in the expansion of (1 + x2 )2 (2 + x2 )5 .

[3]

The equation of a curve C is y = 2x2 − 8x + 9 and the equation of a line L is x + y = 3. (i) Find the x-coordinates of the points of intersection of L and C .

[4]

(ii) Show that one of these points is also the stationary point of C.

[3]

5

O Q

5 cm

P

12 cm

T

The diagram shows a circle with centre O and radius 5 cm. The point P lies on the circle, PT is a tangent to the circle and PT = 12 cm. The line OT cuts the circle at the point Q.

6

(i) Find the perimeter of the shaded region.

[4]

(ii) Find the area of the shaded region.

[3]

The function f is such that f(x) = (3x + 2)3 − 5 for x ≥ 0. (i) Obtain an expression for f  (x) and hence explain why f is an increasing function.

[3]

(ii) Obtain an expression for f −1 (x) and state the domain of f −1 .

[4]

9709/01/M/J/08


7

The first term of a geometric progression is 81 and the fourth term is 24. Find (i) the common ratio of the progression,

[2]

(ii) the sum to infinity of the progression.

[2]

The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression. (iii) Find the sum of the first ten terms of the arithmetic progression.

8

[3]

Functions f and g are defined by f : x → 4x − 2k g : x →

9 2−x

for x ∈ , where k is a constant, for x ∈ , x ≠ 2.

(i) Find the values of k for which the equation fg(x) = x has two equal roots.

[4]

(ii) Determine the roots of the equation fg(x) = x for the values of k found in part (i).

[3]

9

y (1, 18)

P

(4, 3)

O

The diagram shows a curve for which points (1, 18) and (4, 3).

x

1.6

1

k dy = − 3 , where k is a constant. The curve passes through the dx x

(i) Show, by integration, that the equation of the curve is y =

16 + 2. x2

[4]

The point P lies on the curve and has x-coordinate 1.6. (ii) Find the area of the shaded region.

[4]

9709/01/M/J/08


10

Relative to an origin O, the position vectors of points A and B are 2i + j + 2k and 3i − 2j + pk respectively. (i) Find the value of p for which OA and OB are perpendicular.

[2]

(ii) In the case where p = 6, use a scalar product to find angle AOB, correct to the nearest degree. [3]

−−→ (iii) Express the vector AB is terms of p and hence find the values of p for which the length of AB is 3.5 units. [4]

11

y C X

B (2, 2) O

A

x

In the diagram, the points A and C lie on the x- and y-axes respectively and the equation of AC is 2y + x = 16. The point B has coordinates (2, 2). The perpendicular from B to AC meets AC at the point X . (i) Find the coordinates of X .

[4]

The point D is such that the quadrilateral ABCD has AC as a line of symmetry. (ii) Find the coordinates of D.

[2]

(iii) Find, correct to 1 decimal place, the perimeter of ABCD.

[3]

9709/01/M/J/08


1

Solve the inequality |3x − 1| < 2.

2

Use logarithms to solve the equation 4x = 2(3x ), giving your answer correct to 3 significant figures. [4]

3

Find the exact value of 

1 π 6

[3]

(cos 2x + sin x) dx.

[5]

0

4

The polynomial 2x3 + 7x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that (x + 1) is a factor of p(x), and that when p(x) is divided by (x + 2) the remainder is 5. Find the values of a and b. [5]

5

(i) Express 5 cos θ − sin θ in the form R cos(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the exact [3] value of R and the value of α correct to 2 decimal places. (ii) Hence solve the equation

5 cos θ − sin θ = 4, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ . 6

7

[4]

It is given that the curve y = (x − 2)ex has one stationary point. (i) Find the exact coordinates of this point.

[5]

(ii) Determine whether this point is a maximum or a minimum point.

[2]

The equation of a curve is

x2 + y2 − 4xy + 3 = 0. (i) Show that

dy 2y − x = . dx y − 2x

[4]

(ii) Find the coordinates of each of the points on the curve where the tangent is parallel to the x-axis. [5]

9709/02/M/J/08


a

8

1 The constant a, where a > 1, is such that  x +  dx = 6. x 1

(i) Find an equation satisfied by a, and show that it can be written in the form √ a = (13 − 2 ln a).

[5]

√ (ii) Verify, by calculation, that the equation a = (13 − 2 ln a) has a root between 3 and 3.5.

[2]

(iii) Use the iterative formula

√ an+1 = (13 − 2 ln an ),

with a1 = 3.2, to calculate the value of a correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

9709/02/M/J/08


1

x 2 6 Find the value of the coefficient of x2 in the expansion of  +  . 2 x

2

Prove the identity

[3]

1 + sin x cos x 2 + ≡ . cos x 1 + sin x cos x

3

[4]

The first term of an arithmetic progression is 6 and the fifth term is 12. The progression has n terms [4] and the sum of all the terms is 90. Find the value of n.

4

F P

8 cm

E

B

M

C

k

A

N

j

6 cm

O

i

20 cm

D

The diagram shows a semicircular prism with a horizontal rectangular base ABCD. The vertical ends AED and BFC are semicircles of radius 6 cm. The length of the prism is 20 cm. The mid-point of AD is the origin O, the mid-point of BC is M and the mid-point of DC is N . The points E and F are the highest points of the semicircular ends of the prism. The point P lies on EF such that EP = 8 cm. Unit vectors i, j and k are parallel to OD, OM and OE respectively.

5

−−→ −−→ (i) Express each of the vectors PA and PN in terms of i, j and k.

[3]

(ii) Use a scalar product to calculate angle APN .

[4]

The function f is such that f(x) = a − b cos x for 0◦ ≤ x ≤ 360◦ , where a and b are positive constants. The maximum value of f(x) is 10 and the minimum value is −2. (i) Find the values of a and b.

[3]

(ii) Solve the equation f(x) = 0.

[3]

(iii) Sketch the graph of y = f(x).

[2]

9709/01/O/N/08


6

O m 5c

P

Q 9 cm

T In the diagram, the circle has centre O and radius 5 cm. The points P and Q lie on the circle, and the arc length PQ is 9 cm. The tangents to the circle at P and Q meet at the point T . Calculate (i) angle POQ in radians,

[2]

(ii) the length of PT ,

[3]

(iii) the area of the shaded region.

[3]

7

r cm

x cm A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side x cm and the other piece is bent to form a circle of radius r cm (see diagram). The total area of the square and the circle is A cm2 . (i) Show that A =

(π + 4)x2 − 160x + 1600 . π

(ii) Given that x and r can vary, find the value of x for which A has a stationary value.

8

[4] [4]

8 The equation of a curve is y = 5 − . x (i) Show that the equation of the normal to the curve at the point P (2, 1) is 2y + x = 4.

[4]

This normal meets the curve again at the point Q. (ii) Find the coordinates of Q.

[3]

(iii) Find the length of PQ.

[2] 9709/01/O/N/08


9

y Q

2

y = Ö (3x + 1)

1 P

O

1

x

√ The diagram shows the curve y = (3x + 1) and the points P (0, 1) and Q (1, 2) on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 2. (i) Find the area of the shaded region.

[4]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis.

[4]

Tangents are drawn to the curve at the points P and Q. (iii) Find the acute angle, in degrees correct to 1 decimal place, between the two tangents.

10

[4]

The function f is defined by f : x → 3x − 2 for x ∈ . (i) Sketch, in a single diagram, the graphs of y = f(x) and y = f −1 (x), making clear the relationship between the two graphs. [2]

The function g is defined by g : x → 6x − x2 for x ∈ . (ii) Express gf(x) in terms of x, and hence show that the maximum value of gf(x) is 9.

[5]

The function h is defined by h : x → 6x − x2 for x ≥ 3. (iii) Express 6x − x2 in the form a − (x − b)2 , where a and b are positive constants.

[2]

(iv) Express h−1 (x) in terms of x.

[3]

9709/01/O/N/08


1

Solve the inequality | x − 3| > |2x |.

2

The polynomial 2x3 − x2 + ax − 6, where a is a constant, is denoted by p(x). It is given that (x + 2) is a factor of p(x).

[4]

(i) Find the value of a.

[2]

(ii) When a has this value, factorise p(x) completely.

[3]

3

ln y (0, 1.3) (1.6, 0.9)

x

O

The variables x and y satisfy the equation y = A(b−x ), where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 1.3) and (1.6, 0.9), as shown in the diagram. Find the values of A and b, correct to 2 decimal places. [5]

4

(i) Show that the equation

sin(x + 30◦ ) = 2 cos(x + 60◦ ) can be written in the form (3

3) sin x = cos x.

[3]

(ii) Hence solve the equation

sin(x + 30◦ ) = 2 cos(x + 60◦ ), for −180◦ ≤ x ≤ 180◦ .

[3]

2

5

1 4 Show that   −  dx = ln 18 . 25 x 2x + 1

[6]

1

6

− 12 x

Find the exact coordinates of the point on the curve y = x e

9709/02/O/N/08

at which

d2 y = 0. dx2

[7]


7

(i) By sketching a suitable pair of graphs, show that the equation

cos x = 2 − 2x, where x is in radians, has only one root for 0 ≤ x ≤ 12 π .

[2]

(ii) Verify by calculation that this root lies between 0.5 and 1.

[2]

(iii) Show that, if a sequence of values given by the iterative formula

xn+1 = 1 − 12 cos xn converges, then it converges to the root of the equation in part (i).

[1]

(iv) Use this iterative formula, with initial value x1 = 0.6, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

8

(i) (a) Prove the identity

sec2 x + sec x tan x ≡

1 + sin x . cos2 x

sec2 x + sec x tan x ≡

1 . 1 − sin x

(b) Hence prove that

(ii) By differentiating

1 dy , show that if y = sec x then = sec x tan x. cos x dx

[3]

[3]

(iii) Using the results of parts (i) and (ii), find the exact value of



1 π 4

0

1 dx. 1 − sin x

9709/02/O/N/08

[3]


sin x sin x − ≡ 2 tan2 x. 1 − sin x 1 + sin x

[3]

1

Prove the identity

2

Find the set of values of k for which the line y = kx − 4 intersects the curve y = x2 − 2x at two distinct points. [4]

3

(i) Find the first 3 terms in the expansion of (2 + 3x)5 in ascending powers of x.

[3]

(ii) Hence find the value of the constant a for which there is no term in x2 in the expansion of (1 + ax)(2 + 3x)5 . [2]

4

y 9

3

p

O

2p

x

–3

The diagram shows the graph of y = a sin(bx) + c for 0 ≤ x ≤ 2π . (i) Find the values of a, b and c.

[3]

(ii) Find the smallest value of x in the interval 0 ≤ x ≤ 2π for which y = 0.

[3]

5

R1 A

B

q rad O R2

The diagram shows a circle with centre O. The circle is divided into two regions, R1 and R2 , by the radii OA and OB, where angle AOB = θ radians. The perimeter of the region R1 is equal to the length of the major arc AB. (i) Show that θ = π − 1.

[3]

(ii) Given that the area of region R1 is 30 cm2 , find the area of region R2 , correct to 3 significant figures. [4]

9709/01/M/J/09


6

Relative to an origin O, the position vectors of the points A and B are given by −−→ OA = 2i − 8j + 4k

and

−−→ OB = 7i + 2j − k.

−−→ −−→ (i) Find the value of OA . OB and hence state whether angle AOB is acute, obtuse or a right angle. [3] −−→ −−→ (ii) The point X is such that AX = 25 AB. Find the unit vector in the direction of OX .

7

[4]

(a) Find the sum to infinity of the geometric progression with first three terms 0.5, 0.53 and 0.55 .

[3] (b) The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200. Find the sum of all the terms in the progression. [4]

8

y C B

A x

O D (10, –3)

The diagram shows points A, B and C lying on the line 2y = x + 4. The point A lies on the y-axis and AB = BC. The line from D (10, −3) to B is perpendicular to AC. Calculate the coordinates of B and C. [7]

9

y y=

O The diagram shows part of the curve y =

6 3x – 2

1

2

x

6 . 3x − 2

(i) Find the gradient of the curve at the point where x = 2.

[3]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis, giving [5] your answer in terms of π . 9709/01/M/J/09


10

The function f is defined by f : x → 2x2 − 12x + 13 for 0 ≤ x ≤ A, where A is a constant. (i) Express f(x) in the form a(x + b)2 + c, where a, b and c are constants.

[3]

(ii) State the value of A for which the graph of y = f(x) has a line of symmetry.

[1]

(iii) When A has this value, find the range of f.

[2]

The function g is defined by g : x → 2x2 − 12x + 13 for x ≥ 4. (iv) Explain why g has an inverse.

[1]

(v) Obtain an expression, in terms of x, for g−1 (x).

[3]

11

y A

C

D y = x 3 – 6x 2 + 9x

O

B

x

The diagram shows the curve y = x3 − 6x2 + 9x for x ≥ 0. The curve has a maximum point at A and a minimum point on the x-axis at B. The normal to the curve at C (2, 2) meets the normal to the curve at B at the point D. (i) Find the coordinates of A and B.

[3]

(ii) Find the equation of the normal to the curve at C .

[3]

(iii) Find the area of the shaded region.

[5]

9709/01/M/J/09


1

Given that (1.25)x = (2.5)y , use logarithms to find the value of

2

Solve the inequality |3x + 2| < | x |.

x correct to 3 significant figures. [3] y [4]

3

y 1

O

1

The diagram shows the curve y =

2

x

1 √ for values of x from 0 to 2. 1+ x

(i) Use the trapezium rule with two intervals to estimate the value of

ä

2

0

1 √ dx, 1+ x

giving your answer correct to 2 decimal places.

[3]

(ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i). [1]

4

The parametric equations of a curve are

where − 12 π < θ < 12 π . Express 5

6

x = 4 sin θ ,

y = 3 − 2 cos 2θ ,

dy in terms of θ , simplifying your answer as far as possible. dx

Solve the equation sec x = 4 − 2 tan2 x, giving all solutions in the interval 0◦ ≤ x ≤ 180◦ .

[5]

[6]

The polynomial x3 + ax2 + bx + 6, where a and b are constants, is denoted by p(x). It is given that (x − 2) is a factor of p(x), and that when p(x) is divided by (x − 1) the remainder is 4. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the other two linear factors of p(x).

[3]

9709/02/M/J/09


7

y

O

x

M The diagram shows the curve y = xe2x and its minimum point M . (i) Find the exact coordinates of M .

[5]

(ii) Show that the curve intersects the line y = 20 at the point whose x-coordinate is the root of the equation 20 x = 12 ln . [1] x (iii) Use the iterative formula

xn+1 = 12 ln

20 , xn

with initial value x1 = 1.3, to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places. [3]

8

(a) Find the equation of the tangent to the curve y = ln(3x − 2) at the point where x = 1. (b)

[4]

(i) Find the value of the constant A such that

A 6x . ≡2+ 3x − 2 3x − 2 (ii) Hence show that ä

6

2

6x dx = 8 + 83 ln 2. 3x − 2

9709/02/M/J/09

[2] [5]


1

Solve the equation 3 tan(2x + 15◦ ) = 4 for 0◦ ≤ x ≤ 180◦ .

2

The equation of a curve is y = 3 cos 2x. The equation of a line is x + 2y = π . On the same diagram, sketch the curve and the line for 0 ≤ x ≤ π . [4]

3

[4]

(i) Find the first 3 terms in the expansion of (2 − x)6 in ascending powers of x.

[3]

(ii) Given that the coefficient of x2 in the expansion of (1 + 2x + ax2 )(2 − x)6 is 48, find the value of the constant a. [3]

4

The equation of a curve is y = x4 + 4x + 9. (i) Find the coordinates of the stationary point on the curve and determine its nature.

[4]

(ii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 0 and x = 1.

[3]

5

B

A

C 6 cm

O

D

The diagram shows a semicircle ABC with centre O and radius 6 cm. The point B is such that angle BOA is 90◦ and BD is an arc of a circle with centre A. Find

6

(i) the length of the arc BD,

[4]

(ii) the area of the shaded region.

[3]

A curve is such that

dy = k − 2x, where k is a constant. dx

(i) Given that the tangents to the curve at the points where x = 2 and x = 3 are perpendicular, find the value of k. [4] (ii) Given also that the curve passes through the point (4, 9), find the equation of the curve.

9709/11/O/N/09

[3]


7

The equation of a curve is y =

(i) Obtain an expression for

12 . +3

x2

dy . dx

[2]

(ii) Find the equation of the normal to the curve at the point P (1, 3).

[3]

(iii) A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the y-coordinate as the point passes through P. [2]

8

The first term of an arithmetic progression is 8 and the common difference is d, where d ≠ 0. The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is r. (i) Write down two equations connecting d and r . Hence show that r =

9

and find the value of d. [6]

(ii) Find the sum to infinity of the geometric progression.

[2]

(iii) Find the sum of the first 8 terms of the arithmetic progression.

[2]

Relative to an origin O, the position vectors of the points A, B and C are given by −−→ OA =

10

3 4

2 3!, −6

−−→ OB =

0 −6 ! 8

and

−−→ OC =

−2 5!. −2

(i) Find angle AOB.

[4]

−−→ (ii) Find the vector which is in the same direction as AC and has magnitude 30.

[3]

−−→ −−→ −−→ (iii) Find the value of the constant p for which OA + p OB is perpendicular to OC .

[3]

Functions f and g are defined by f : x → 2x + 1, 2x − 1 g : x  → , x+3

x ∈ >, x > 0, x ∈ >, x ≠ −3.

(i) Solve the equation gf(x) = x.

[3]

(ii) Express f −1 (x) and g−1 (x) in terms of x.

[4]

(iii) Show that the equation g−1 (x) = x has no solutions.

[3]

(iv) Sketch in a single diagram the graphs of y = f(x) and y = f −1 (x), making clear the relationship between the graphs. [3]

9709/11/O/N/09


1

The equation of a curve is such that find the equation of the curve.

2

dy 3 = √ − x. Given that the curve passes through the point (4, 6), dx x [4]

(i) Find, in terms of the non-zero constant k, the first 4 terms in the expansion of (k + x)8 in ascending powers of x. [3] (ii) Given that the coefficients of x2 and x3 in this expansion are equal, find the value of k.

3

4

5

[2]

A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases: (i) the progression is arithmetic,

[3]

(ii) the progression is geometric with a positive common ratio.

[3]

The function f is defined by f : x → 5 − 3 sin 2x for 0 ≤ x ≤ π . (i) Find the range of f.

[2]

(ii) Sketch the graph of y = f(x).

[3]

(iii) State, with a reason, whether f has an inverse.

[1]

(i) Prove the identity (sin x + cos x)(1 − sin x cos x) ≡ sin3 x + cos3 x.

[3]

(ii) Solve the equation (sin x + cos x)(1 − sin x cos x) = 9 sin3 x for 0◦ ≤ x ≤ 360◦ .

[3]

6

G

F Q

D

E

C k

B

j P

O

A

i

In the diagram, OABCDEFG is a cube in which each side has length 6. Unit vectors i, j and k are −−→ −−→ −−→ −−→ −−→ parallel to OA, OC and OD respectively. The point P is such that AP = 13 AB and the point Q is the mid-point of DF . −−→ −−→ (i) Express each of the vectors OQ and PQ in terms of i, j and k.

[3]

(ii) Find the angle OQP.

[4] 9709/12/O/N/09


7

P r cm

q rad

O

Q

A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is θ radians (see diagram). (i) Express θ in terms of r and show that the area, A cm2 , of the sector is given by

A = 25r − r2 .

[4]

(ii) Given that r can vary, find the stationary value of A and determine its nature.

8

The function f is such that f(x) =

[4]

3 for x ∈ >, x ≠ −2.5. 2x + 5

(i) Obtain an expression for f ′ (x) and explain why f is a decreasing function.

[3]

(ii) Obtain an expression for f −1 (x).

[2]

(iii) A curve has the equation y = f(x). Find the volume obtained when the region bounded by the [4] curve, the coordinate axes and the line x = 2 is rotated through 360◦ about the x-axis. 9

y C (12, 14)

B

D O

x

A (0, –2) The diagram shows a rectangle ABCD. The point A is (0, −2) and C is (12, 14). The diagonal BD is parallel to the x-axis. (i) Explain why the y-coordinate of D is 6.

[1]

The x-coordinate of D is h. (ii) Express the gradients of AD and CD in terms of h.

[3]

(iii) Calculate the x-coordinates of D and B.

[4]

(iv) Calculate the area of the rectangle ABCD.

[3]

9709/12/O/N/09


10

y y = x 2 â&#x20AC;&#x201C; 4x + 7

2y = x + 5 B A

x

O

(i) The diagram shows the line 2y = x + 5 and the curve y = x2 â&#x2C6;&#x2019; 4x + 7, which intersect at the points A and B. Find (a) the x-coordinates of A and B,

[3]

(b) the equation of the tangent to the curve at B,

[3]

(c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line 2y = x + 5. [3] (ii) Determine the set of values of k for which the line 2y = x + k does not intersect the curve y = x2 â&#x2C6;&#x2019; 4x + 7. [4]

9709/12/O/N/09


1

Solve the inequality |2x + 3| < | x − 3|.

[4]

2

Solve the equation ln(3 − x2 ) = 2 ln x, giving your answer correct to 3 significant figures.

[4]

3

4

5

The polynomial 4x3 − 8x2 + ax − 3, where a is a constant, is denoted by p(x). It is given that (2x + 1) is a factor of p(x). (i) Find the value of a.

[2]

(ii) When a has this value, factorise p(x) completely.

[4]

(i) Show that the equation sin(60◦ − x) = 2 sin x can be written in the form tan x = k, where k is a constant. [4] (ii) Hence solve the equation sin(60◦ − x) = 2 sin x, for 0◦ < x < 360◦ .

[2]

(i) Express cos2 2x in terms of cos 4x.

[2]

(ii) Hence find the exact value of ã

6

1 π 8

cos2 2x dx.

[4]

0

The curve with equation y = x ln x has one stationary point. (i) Find the exact coordinates of this point, giving your answers in terms of e.

[5]

(ii) Determine whether this point is a maximum or a minimum point.

[2]

9709/21/O/N/09


7

y

1

R

p

O

x

The diagram shows the curve y = e−x . The shaded region R is bounded by the curve and the lines y = 1 and x = p, where p is a constant. (i) Find the area of R in terms of p.

[4]

(ii) Show that if the area of R is equal to 1 then

p = 2 − e−p . (iii) Use the iterative formula

[1]

pn+1 = 2 − e−pn ,

with initial value p1 = 2, to calculate the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

8

The equation of a curve is y2 + 2xy − x2 = 2. (i) Find the coordinates of the two points on the curve where x = 1.

[2]

(ii) Show by differentiation that at one of these points the tangent to the curve is parallel to the x-axis. Find the equation of the tangent to the curve at the other point, giving your answer in the form ax + by + c = 0. [7]

9709/21/O/N/09


1

Solve the inequality | x + 3| > |2x |.

2

It is given that ln(y + 5) − ln y = 2 ln x. Express y in terms of x, in a form not involving logarithms. [4]

3

[4]

(i) Use the trapezium rule with two intervals to estimate the value of

ã

1 π 3

sec x dx,

0

giving your answer correct to 2 decimal places.

[3]

(ii) Using a sketch of the graph of y = sec x for 0 ≤ x ≤ 13 π , explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i). [2]

4

The parametric equations of a curve are

x = 1 − e−t , (i) Show that

y = et + e−t .

dy = e2t − 1. dx

[3]

(ii) Hence find the exact value of t at the point on the curve at which the gradient is 2.

5

6

[2]

The polynomial ax3 + bx2 − 5x + 2, where a and b are constants, is denoted by p(x). It is given that (x + 1) and (x − 2) are factors of p(x). (i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the other linear factor of p(x).

[2]

(i) Express 3 cos x + 4 sin x in the form R cos(x − α ), where R > 0 and 0◦ < α < 90◦ , stating the exact [3] value of R and giving the value of α correct to 2 decimal places. (ii) Hence solve the equation

3 cos x + 4 sin x = 4.5,

giving all solutions in the interval 0◦ < x < 360◦ .

9709/22/O/N/09

[4]


7

y

M

1p 2

O

x

The diagram shows the curve y = x2 cos x, for 0 ≤ x ≤ 12 π , and its maximum point M . (i) Show by differentiation that the x-coordinate of M satisfies the equation

tan x =

2 . x

(ii) Verify by calculation that this equation has a root (in radians) between 1 and 1.2.

[4] [2]

(iii) Use the iterative formula xn+1 = tan−1 

2  to determine this root correct to 2 decimal places. xn Give the result of each iteration to 4 decimal places. [3]

8

(a) Find the exact value of ã 4

(b) Show that ä  1

1 π 3

0

(sin 2x + sec2 x) dx.

1 1 +  dx = ln 5. 2x x + 1

[5]

[4]

9709/22/O/N/09


1

The acute angle x radians is such that tan x = k, where k is a positive constant. Express, in terms of k, (i) tan(π − x),

[1]

(ii) tan( 12 π − x),

[1]

(iii) sin x.

[2] 5

2

3 (i) Find the first 3 terms in the expansion of 2x −  in descending powers of x. x (ii) Hence find the coefficient of x in the expansion of 1 +

3

[3]

5

2 3 2x −  . 2 x x

[2]

The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49. (i) Find the first term of the progression and the common difference.

[4]

The nth term of the progression is 46. (ii) Find the value of n.

[2]

4

y y = 6x – x 2

y=5

x

O

The diagram shows the curve y = 6x − x2 and the line y = 5. Find the area of the shaded region.

5

6

[6]

The function f is such that f (x) = 2 sin2 x − 3 cos2 x for 0 ≤ x ≤ π . (i) Express f (x) in the form a + b cos2 x, stating the values of a and b.

[2]

(ii) State the greatest and least values of f (x).

[2]

(iii) Solve the equation f (x) + 1 = 0.

[3]

A curve is such that

1 dy = 3x 2 − 6 and the point (9, 2) lies on the curve. dx

(i) Find the equation of the curve.

[4]

(ii) Find the x-coordinate of the stationary point on the curve and determine the nature of the stationary point. [3]

9709/11/M/J/10


7

y

C

y=2– A O

18 2x + 3 x

B

18 , which crosses the x-axis at A and the y-axis at B. 2x + 3 The normal to the curve at A crosses the y-axis at C .

The diagram shows part of the curve y = 2 −

(i) Show that the equation of the line AC is 9x + 4y = 27.

[6]

(ii) Find the length of BC .

[2]

8

y B (15, 22)

C x

O A (3, –2)

The diagram shows a triangle ABC in which A is (3, −2) and B is (15, 22). The gradients of AB, AC and BC are 2m, −2m and m respectively, where m is a positive constant. (i) Find the gradient of AB and deduce the value of m.

[2]

(ii) Find the coordinates of C.

[4]

The perpendicular bisector of AB meets BC at D. (iii) Find the coordinates of D.

[4]

9709/11/M/J/10


9

The function f is defined by f : x → 2x2 − 12x + 7 for x ∈ >. (i) Express f (x) in the form a(x − b)2 − c.

[3]

(ii) State the range of f.

[1]

(iii) Find the set of values of x for which f (x) < 21.

[3]

The function g is defined by g : x → 2x + k for x ∈ >. (iv) Find the value of the constant k for which the equation gf (x) = 0 has two equal roots.

10

[4]

A B

O C −−→ −−→ The diagram shows the parallelogram OABC . Given that OA = i + 3j + 3k and OC = 3i − j + k, find −−→ [3] (i) the unit vector in the direction of OB,

(ii) the acute angle between the diagonals of the parallelogram,

[5]

(iii) the perimeter of the parallelogram, correct to 1 decimal place.

[3]

9709/11/M/J/10


1

(i) Show that the equation

3(2 sin x − cos x) = 2(sin x − 3 cos x) can be written in the form tan x = − 34 .

[2]

(ii) Solve the equation 3(2 sin x − cos x) = 2(sin x − 3 cos x), for 0◦ ≤ x ≤ 360◦ .

2

[2]

y

a y= x

O

3

1

x

a , where a is a positive constant. Given that the volume x obtained when the shaded region is rotated through 360◦ about the x-axis is 24π , find the value of a. [4]

The diagram shows part of the curve y =

3

The functions f and g are defined for x ∈ > by f : x → 4x − 2x2 , g : x  → 5x + 3. (i) Find the range of f.

[2]

(ii) Find the value of the constant k for which the equation gf (x) = k has equal roots.

[3]

4

y

L1

C

(–1, 3) A

L2 B (3, 1) x

O

In the diagram, A is the point (−1, 3) and B is the point (3, 1). The line L1 passes through A and is parallel to OB. The line L2 passes through B and is perpendicular to AB. The lines L1 and L2 meet at C . Find the coordinates of C . [6] 9709/12/M/J/10


5

Relative to an origin O, the position vectors of the points A and B are given by −−→ OA =

−2 3! 1

and

−−→ OB =

4 1!. p

−−→ −−→ (i) Find the value of p for which OA is perpendicular to OB. −−→ (ii) Find the values of p for which the magnitude of AB is 7.

6

(i) Find the first 3 terms in the expansion of (1 + ax)5 in ascending powers of x.

[2] [4]

[2]

(ii) Given that there is no term in x in the expansion of (1 − 2x)(1 + ax)5 , find the value of the [2] constant a.

7

(iii) For this value of a, find the coefficient of x2 in the expansion of (1 − 2x)(1 + ax)5 .

[3]

(a) Find the sum of all the multiples of 5 between 100 and 300 inclusive.

[3]

(b) A geometric progression has a common ratio of − 23 and the sum of the first 3 terms is 35. Find

8

(i) the first term of the progression,

[3]

(ii) the sum to infinity.

[2]

A solid rectangular block has a square base of side x cm. The height of the block is h cm and the total surface area of the block is 96 cm2 . (i) Express h in terms of x and show that the volume, V cm3 , of the block is given by

V = 24x − 12 x3 .

[3]

Given that x can vary, (ii) find the stationary value of V ,

[3]

(iii) determine whether this stationary value is a maximum or a minimum.

[2]

[Questions 9, 10 and 11 are printed on the next page.]

9709/12/M/J/10


9

y y = (x – 2)

2

A

y + 2x = 7

B x

O

The diagram shows the curve y = (x − 2)2 and the line y + 2x = 7, which intersect at points A and B. Find the area of the shaded region. [8]

10

The equation of a curve is y = 16 (2x − 3)3 − 4x. (i) Find

dy . dx

[3]

(ii) Find the equation of the tangent to the curve at the point where the curve intersects the y-axis. [3] (iii) Find the set of values of x for which 61 (2x − 3)3 − 4x is an increasing function of x. 11

[3]

The function f : x → 4 − 3 sin x is defined for the domain 0 ≤ x ≤ 2π . (i) Solve the equation f (x) = 2.

[3]

(ii) Sketch the graph of y = f (x).

[2]

(iii) Find the set of values of k for which the equation f (x) = k has no solution.

[2]

The function g : x → 4 − 3 sin x is defined for the domain 12 π ≤ x ≤ A. (iv) State the largest value of A for which g has an inverse.

[1]

(v) For this value of A, find the value of g−1 (3).

[2]

9709/12/M/J/10


1

2

3

4

5

The first term of a geometric progression is 12 and the second term is −6. Find (i) the tenth term of the progression,

[3]

(ii) the sum to infinity.

[2]

2 6 (i) Find the first three terms, in descending powers of x, in the expansion of x −  . x

[3]

2 6 (ii) Find the coefficient of x4 in the expansion of (1 + x2 )x −  . x

[2]

The function f : x → a + b cos x is defined for 0 ≤ x ≤ 2π . Given that f (0) = 10 and that f 23 π  = 1, find (i) the values of a and b,

[2]

(ii) the range of f,

[1]

(iii) the exact value of f 56 π .

[2]

(i) Show that the equation 2 sin x tan x + 3 = 0 can be expressed as 2 cos2 x − 3 cos x − 2 = 0.

[2]

(ii) Solve the equation 2 sin x tan x + 3 = 0 for 0◦ ≤ x ≤ 360◦ .

[3]

The equation of a curve is such that

P (2, 11), find

6

6 dy = √ . Given that the curve passes through the point dx (3x − 2)

(i) the equation of the normal to the curve at P,

[3]

(ii) the equation of the curve.

[4]

Relative to an origin O, the position vectors of the points A, B and C are given by −−→ OA = i − 2j + 4k,

−−→ OB = 3i + 2j + 8k,

−−→ OC = −i − 2j + 10k.

(i) Use a scalar product to find angle ABC .

[6]

(ii) Find the perimeter of triangle ABC , giving your answer correct to 2 decimal places.

[2]

9709/13/M/J/10


7

C B

D 10 cm 12 cm

O

A

E

F The diagram shows a metal plate ABCDEF which has been made by removing the two shaded regions from a circle of radius 10 cm and centre O. The parallel edges AB and ED are both of length 12 cm. (i) Show that angle DOE is 1.287 radians, correct to 4 significant figures.

[2]

(ii) Find the perimeter of the metal plate.

[3]

(iii) Find the area of the metal plate.

[3]

8

y

B

C (5, 4) A (â&#x20AC;&#x201C;1, 2) x

O D

The diagram shows a rhombus ABCD in which the point A is (â&#x2C6;&#x2019;1, 2), the point C is (5, 4) and the point B lies on the y-axis. Find (i) the equation of the perpendicular bisector of AC,

[3]

(ii) the coordinates of B and D,

[3]

(iii) the area of the rhombus.

[3] 9709/13/M/J/10


9

y y = x + 4x

A

B

y=5

M

x

O

4 The diagram shows part of the curve y = x + which has a minimum point at M . The line y = 5 x intersects the curve at the points A and B. (i) Find the coordinates of A, B and M .

[5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [6]

10

The function f : x → 2x2 − 8x + 14 is defined for x ∈ >. (i) Find the values of the constant k for which the line y + kx = 12 is a tangent to the curve y = f (x). [4] (ii) Express f (x) in the form a(x + b)2 + c, where a, b and c are constants.

[3]

(iii) Find the range of f.

[1]

The function g : x → 2x2 − 8x + 14 is defined for x ≥ A. (iv) Find the smallest value of A for which g has an inverse.

[1]

(v) For this value of A, find an expression for g−1 (x) in terms of x.

[3]

9709/13/M/J/10


1

2

3

Solve the inequality | 2x − 3 | > 5. Show that ä

6

0

[3]

1 dx = 2 ln 2. x+2

[4]

(i) Show that the equation tan(x + 45◦ ) = 6 tan x can be written in the form

6 tan2 x − 5 tan x + 1 = 0.

(ii) Hence solve the equation tan(x + 45◦ ) = 6 tan x, for 0◦ < x < 180◦ . 4

[3]

The polynomial x3 + 3x2 + 4x + 2 is denoted by f (x).

(i) Find the quotient and remainder when f (x) is divided by x2 + x − 1.

(ii) Use the factor theorem to show that (x + 1) is a factor of f (x). 5

[3]

[4] [2]

(i) Given that y = 2x , show that the equation

2x + 3(2−x ) = 4

can be written in the form

(ii) Hence solve the equation

y2 − 4y + 3 = 0. 2x + 3(2−x ) = 4,

giving the values of x correct to 3 significant figures where appropriate.

6

The equation of a curve is

(i) Show that

dy 6 − 2xy = . dx x2 + 2y

[3]

[3]

x2 y + y2 = 6x. [4]

(ii) Find the equation of the tangent to the curve at the point with coordinates (1, 2), giving your [3] answer in the form ax + by + c = 0.

9709/21/M/J/10


7

(i) By sketching a suitable pair of graphs, show that the equation

e2x = 2 − x has only one root.

[2]

(ii) Verify by calculation that this root lies between x = 0 and x = 0.5.

[2]

(iii) Show that, if a sequence of values given by the iterative formula

xn+1 = 12 ln(2 − xn ) converges, then it converges to the root of the equation in part (i).

[1]

(iv) Use this iterative formula, with initial value x1 = 0.25, to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

8

(i) By differentiating

cos x dy , show that if y = cot x then = − cosec2 x. sin x dx

[3]

(ii) By expressing cot2 x in terms of cosec2 x and using the result of part (i), show that ã

1π 2

cot2 x dx = 1 − 14 π .

[4]

(iii) Express cos 2x in terms of sin2 x and hence show that

1 can be expressed as 12 cosec2 x. 1 − cos 2x

1 π 4

Hence, using the result of part (i), find ä

1 dx. 1 − cos 2x

9709/21/M/J/10

[3]


1

Given that 13x = (2.8)y , use logarithms to show that y = kx and find the value of k correct to 3 significant figures. [3]

2

y

R O

1

2

3

x

The diagram shows part of the curve y = xe−x . The shaded region R is bounded by the curve and by the lines x = 2, x = 3 and y = 0. (i) Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct to 2 decimal places. [3] (ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the area of R.

3

4

Solve the inequality | 2x − 1| < | x + 4 |. (a) Show that ã

1 π 4

0

[4]

cos 2x dx = 12 .

[2]

(b) By using an appropriate trigonometrical identity, find the exact value of ã

5

1 π 3

3 tan2 x dx.

1 π 6

[4]

The equation of a curve is y = x3 e−x . (i) Show that the curve has a stationary point where x = 3.

[3]

(ii) Find the equation of the tangent to the curve at the point where x = 1.

[4]

9709/22/M/J/10


6

(i) By sketching a suitable pair of graphs, show that the equation

ln x = 2 − x2 has only one root.

[2]

(ii) Verify by calculation that this root lies between x = 1.3 and x = 1.4.

[2]

(iii) Show that, if a sequence of values given by the iterative formula √ xn+1 = (2 − ln xn )

converges, then it converges to the root of the equation in part (i).

[1]

√ (iv) Use the iterative formula xn+1 = (2 − ln xn ) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

7

The polynomial 2x3 + ax2 + bx + 6, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x − 3) the remainder is 30, and that when p(x) is divided by (x + 1) the remainder is 18. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, verify that (x − 2) is a factor of p(x) and hence factorise p(x) completely. [4]

8

(i) Prove the identity

√ sin(x − 30◦ ) + cos(x − 60◦ ) ≡ ( 3) sin x.

[3]

(ii) Hence solve the equation

for 0◦ < x < 360◦ .

sin(x − 30◦ ) + cos(x − 60◦ ) = 12 sec x, [6]

9709/22/M/J/10


1

Given that 13x = (2.8)y , use logarithms to show that y = kx and find the value of k correct to 3 significant figures. [3]

2

y

R O

1

2

3

x

The diagram shows part of the curve y = xe−x . The shaded region R is bounded by the curve and by the lines x = 2, x = 3 and y = 0. (i) Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct to 2 decimal places. [3] (ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of [1] the true value of the area of R.

3

4

Solve the inequality | 2x − 1| < | x + 4 |. (a) Show that ã

1 π 4

0

[4]

cos 2x dx = 12 .

[2]

(b) By using an appropriate trigonometrical identity, find the exact value of ã

5

1 π 3

3 tan2 x dx.

1 π 6

[4]

The equation of a curve is y = x3 e−x . (i) Show that the curve has a stationary point where x = 3.

[3]

(ii) Find the equation of the tangent to the curve at the point where x = 1.

[4]

9709/23/M/J/10


6

(i) By sketching a suitable pair of graphs, show that the equation

ln x = 2 − x2 has only one root.

[2]

(ii) Verify by calculation that this root lies between x = 1.3 and x = 1.4.

[2]

(iii) Show that, if a sequence of values given by the iterative formula √ xn+1 = (2 − ln xn )

converges, then it converges to the root of the equation in part (i).

[1]

√ (iv) Use the iterative formula xn+1 = (2 − ln xn ) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

7

The polynomial 2x3 + ax2 + bx + 6, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x − 3) the remainder is 30, and that when p(x) is divided by (x + 1) the remainder is 18. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, verify that (x − 2) is a factor of p(x) and hence factorise p(x) completely. [4]

8

(i) Prove the identity

√ sin(x − 30◦ ) + cos(x − 60◦ ) ≡ ( 3) sin x.

[3]

(ii) Hence solve the equation

for 0◦ < x < 360◦ .

sin(x − 30◦ ) + cos(x − 60◦ ) = 12 sec x, [6]

9709/23/M/J/10


1

1 2 Find ä x +  dx. x

2

In the expansion of (1 + ax)6 , where a is a constant, the coefficient of x is −30. Find the coefficient [4] of x3 .

3

Functions f and g are defined for x ∈ > by

[3]

f : x → 2x + 3,

g : x → x2 − 2x.

Express gf (x) in the form a(x + b)2 + c, where a, b and c are constants. 4

(i) Prove the identity

1 sin x tan x ≡1+ . 1 − cos x cos x

(ii) Hence solve the equation

5

[5]

[3]

sin x tan x + 2 = 0, for 0◦ ≤ x ≤ 360◦ . 1 − cos x

[3]

C

10 cm

B m

8c

k

j

O i

6c

m

A The diagram shows a pyramid OABC with a horizontal base OAB where OA = 6 cm, OB = 8 cm and angle AOB = 90◦ . The point C is vertically above O and OC = 10 cm. Unit vectors i, j and k are parallel to OA, OB and OC as shown. Use a scalar product to find angle ACB.

6

[6]

(a) The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75. Find the first term and the common difference. [4] (b) The first term of a geometric progression is 16 and the fourth term is of the progression.

9709/11/O/N/10

27 4.

Find the sum to infinity [3]


7

A function f is defined by f : x → 3 − 2 tan 12 x for 0 ≤ x < π . (i) State the range of f.

[1]

(ii) State the exact value of f 23 π .

[1]

(iii) Sketch the graph of y = f (x).

[2]

(iv) Obtain an expression, in terms of x, for f −1 (x).

[3]

8

x cm

y cm

x cm

The diagram shows a metal plate consisting of a rectangle with sides x cm and y cm and a quarter-circle of radius x cm. The perimeter of the plate is 60 cm. (i) Express y in terms of x.

[2]

(ii) Show that the area of the plate, A cm2 , is given by A = 30x − x2 .

[2]

Given that x can vary, (iii) find the value of x at which A is stationary,

[2]

(iv) find this stationary value of A, and determine whether it is a maximum or a minimum value. [2]

[Questions 9, 10 and 11 are printed on the next page.]

9709/11/O/N/10


9

C1

P

8 cm

T

C2 Q 2 cm

R

S

The diagram shows two circles, C1 and C2 , touching at the point T . Circle C1 has centre P and radius 8 cm; circle C2 has centre Q and radius 2 cm. Points R and S lie on C1 and C2 respectively, and RS is a tangent to both circles.

10

(i) Show that RS = 8 cm.

[2]

(ii) Find angle RPQ in radians correct to 4 significant figures.

[2]

(iii) Find the area of the shaded region.

[4]

The equation of a curve is y = 3 + 4x − x2 .

(i) Show that the equation of the normal to the curve at the point (3, 6) is 2y = x + 9.

[4]

(ii) Given that the normal meets the coordinate axes at points A and B, find the coordinates of the [2] mid-point of AB. (iii) Find the coordinates of the point at which the normal meets the curve again.

11

The equation of a curve is y =

[4]

9 . 2−x

dy and determine, with a reason, whether the curve has any stationary (i) Find an expression for dx points. [3] (ii) Find the volume obtained when the region bounded by the curve, the coordinate axes and the [4] line x = 1 is rotated through 360◦ about the x-axis.

(iii) Find the set of values of k for which the line y = x + k intersects the curve at two distinct points. [4]

9709/11/O/N/10


1

2

(i) Find the first 3 terms in the expansion, in ascending powers of x, of (1 − 2x2 )8 .

[2]

(ii) Find the coefficient of x4 in the expansion of (2 − x2 )(1 − 2x2 )8 .

[2]

Prove the identity tan2 x − sin2 x ≡ tan2 x sin2 x.

3

[4]

The length, x metres, of a Green Anaconda snake which is t years old is given approximately by the formula √ x = 0.7 (2t − 1), where 1 ≤ t ≤ 10. Using this formula, find (i)

dx , dt

[2]

(ii) the rate of growth of a Green Anaconda snake which is 5 years old.

4

[2]

C

3 cm 2.3 rad

2.3 rad

O 3 cm

3 cm

A

B P

The diagram shows points A, C, B, P on the circumference of a circle with centre O and radius 3 cm. Angle AOC = angle BOC = 2.3 radians.

5

(i) Find angle AOB in radians, correct to 4 significant figures.

[1]

(ii) Find the area of the shaded region ACBP, correct to 3 significant figures.

[4]

(a) The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum [3] of the first m terms is zero. Find the value of m. (b) A geometric progression, in which all the terms are positive, has common ratio r. The sum of [3] the first n terms is less than 90% of the sum to infinity. Show that rn > 0.1.

9709/12/O/N/10


6

A curve has equation y = kx2 + 1 and a line has equation y = kx, where k is a non-zero constant. (i) Find the set of values of k for which the curve and the line have no common points.

[3]

(ii) State the value of k for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve. [4]

7

The function f is defined by f (x) = x2 − 4x + 7 for x > 2. (i) Express f (x) in the form (x − a)2 + b and hence state the range of f.

[3]

(ii) Obtain an expression for f −1 (x) and state the domain of f −1 .

[3]

The function g is defined by g(x) = x − 2 for x > 2. The function h is such that f = hg and the domain of h is x > 0. (iii) Obtain an expression for h(x). 8

[1]

y y=

2 1–x

y = 3x + 4 B

A O

x

2 The diagram shows part of the curve y = and the line y = 3x + 4. The curve and the line meet at 1−x points A and B. (i) Find the coordinates of A and B.

[4]

(ii) Find the length of the line AB and the coordinates of the mid-point of AB.

[3]

9709/12/O/N/10


9

P

G

F

10 cm

a cm

6 cm

6 cm

D

E

k

j

O

i

C

B 10 cm

A

10 cm

The diagram shows a pyramid OABCP in which the horizontal base OABC is a square of side 10 cm and the vertex P is 10 cm vertically above O. The points D, E, F , G lie on OP, AP, BP, CP respectively and DEFG is a horizontal square of side 6 cm. The height of DEFG above the base is a cm. Unit vectors i, j and k are parallel to OA, OC and OD respectively. (i) Show that a = 4.

[2]

−−→ (ii) Express the vector BG in terms of i, j and k.

[2]

(iii) Use a scalar product to find angle GBA.

[4]

10 5 x 4

4 x 5

h

1 x 2

x

The diagram shows an open rectangular tank of height h metres covered with a lid. The base of the tank has sides of length x metres and 21 x metres and the lid is a rectangle with sides of length 45 x metres and 54 x metres. When full the tank holds 4 m3 of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is A m2 . (i) Express h in terms of x and hence show that A = 32 x2 +

24 . x

[5]

(ii) Given that x can vary, find the value of x for which A is a minimum, showing clearly that A is a minimum and not a maximum. [5]

9709/12/O/N/10


11

y

x=5

A

B

at B.

1 1

(3x + 1)4

x

O The diagram shows part of the curve y =

y=

1 1

(3x + 1) 4

. The curve cuts the y-axis at A and the line x = 5

1 (i) Show that the equation of the line AB is y = â&#x2C6;&#x2019; 10 x + 1.

[4]

(ii) Find the volume obtained when the shaded region is rotated through 360â&#x2014;Ś about the x-axis. [9]

9709/12/O/N/10


1 9  . x2

1

Find the term independent of x in the expansion of x −

2

Points A, B and C have coordinates (2, 5), (5, −1) and (8, 6) respectively. (i) Find the coordinates of the mid-point of AB.

[3]

[1]

(ii) Find the equation of the line through C perpendicular to AB. Give your answer in the form ax + by + c = 0. [3]

3

4

Solve the equation 15 sin2 x = 13 + cos x for 0◦ ≤ x ≤ 180◦ .

(i) Sketch the curve y = 2 sin x for 0 ≤ x ≤ 2π .

[4]

[1]

(ii) By adding a suitable straight line to your sketch, determine the number of real roots of the equation

2π sin x = π − x. State the equation of the straight line.

5

A curve has equation y =

(i) Find

[3]

1 + x. x−3

dy d2 y and 2 . dx dx

[2]

(ii) Find the coordinates of the maximum point A and the minimum point B on the curve.

6

[5]

A curve has equation y = f (x). It is given that f ′ (x) = 3x2 + 2x − 5. (i) Find the set of values of x for which f is an increasing function.

[3]

(ii) Given that the curve passes through (1, 3), find f (x).

[4]

9709/13/O/N/10


7

y

x

O The diagram shows the function f defined for 0 ≤ x ≤ 6 by

x → 12 x2

for 0 ≤ x ≤ 2,

x → 12 x + 1 for 2 < x ≤ 6. (i) State the range of f.

[1]

(ii) Copy the diagram and on your copy sketch the graph of y = f −1 (x).

[2]

(iii) Obtain expressions to define f −1 (x), giving the set of values of x for which each expression is valid. [4]

8

A

B

P

Q

C

D

The diagram shows a rhombus ABCD. Points P and Q lie on the diagonal AC such that BPD is an arc of a circle with centre C and BQD is an arc of a circle with centre A. Each side of the rhombus has length 5 cm and angle BAD = 1.2 radians. (i) Find the area of the shaded region BPDQ.

[4]

(ii) Find the length of PQ.

[4] 9709/13/O/N/10


9

(a) A geometric progression has first term 100 and sum to infinity 2000. Find the second term. [3] (b) An arithmetic progression has third term 90 and fifth term 80. (i) Find the first term and the common difference.

[2]

(ii) Find the value of m given that the sum of the first m terms is equal to the sum of the first (m + 1) terms. [2] (iii) Find the value of n given that the sum of the first n terms is zero.

10

[2]

B

A C

O The diagram shows triangle OAB, in which the position vectors of A and B with respect to O are given by −−→ −−→ OA = 2i + j − 3k and OB = −3i + 2j − 4k. −−→ −−→ C is a point on OA such that OC = p OA, where p is a constant. (i) Find angle AOB.

[4]

−−→ (ii) Find BC in terms of p and vectors i, j and k.

(iii) Find the value of p given that BC is perpendicular to OA.

9709/13/O/N/10

[1] [4]


11

y

P y = 9 – x3

Q O

a

b

The diagram shows parts of the curves y = 9 − x3 and y = The x-coordinates of P and Q are a and b respectively.

y=

8 x3

x

8 and their points of intersection P and Q. x3

(i) Show that x = a and x = b are roots of the equation x6 − 9x3 + 8 = 0. Solve this equation and [4] hence state the value of a and the value of b. (ii) Find the area of the shaded region between the two curves.

[5]

(iii) The tangents to the two curves at x = c (where a < c < b) are parallel to each other. Find the value of c. [4]

9709/13/O/N/10


1

Solve the inequality | x + 1| > | x − 4 |.

2

Use logarithms to solve the equation 5x = 22x+1 , giving your answer correct to 3 significant figures. [4]

3

Show that ã (ex + 1)2 dx = 12 e2 + 2e − 32 .

[3]

1

[5]

0

4

The parametric equations of a curve are

x = 1 + ln(t − 2), (i) Show that

9 y=t+ , t

for t > 2.

dy (t2 − 9)(t − 2) = . dx t2

[3]

(ii) Find the coordinates of the only point on the curve at which the gradient is equal to 0.

[3]

5

Solve the equation 8 + cot θ = 2 cosec2 θ , giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .

[6]

6

The curve with equation y =

6 intersects the line y = x + 1 at the point P. x2

(i) Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6. (ii) Show that the x-coordinate of P satisfies the equation q 6 x=  . x+1 (iii) Use the iterative formula

xn+1

r = 

[2]

[2]

6 , xn + 1

with initial value x1 = 1.5, to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

7

The polynomial 3x3 + 2x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that (x − 1) is a factor of p(x), and that when p(x) is divided by (x − 2) the remainder is 10. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, solve the equation p(x) = 0.

[4]

9709/21/O/N/10


8

y

Q

p

O

The diagram shows the curve y = x sin x, for 0 ≤ x ≤ π . The point Q

x 1 1 2π, 2π

(i) Show that the normal to the curve at Q passes through the point (π , 0). (ii) Find

d (sin x − x cos x). dx

(iii) Hence evaluate ã

lies on the curve. [5] [2]

1π 2

x sin x dx.

[3]

0

9709/21/O/N/10


1

Solve the inequality | x + 1| > | x − 4 |.

2

Use logarithms to solve the equation 5x = 22x+1 , giving your answer correct to 3 significant figures. [4]

3

Show that ã (ex + 1)2 dx = 12 e2 + 2e − 32 .

[3]

1

[5]

0

4

The parametric equations of a curve are

x = 1 + ln(t − 2), (i) Show that

9 y=t+ , t

for t > 2.

dy (t2 − 9)(t − 2) = . dx t2

[3]

(ii) Find the coordinates of the only point on the curve at which the gradient is equal to 0.

[3]

5

Solve the equation 8 + cot θ = 2 cosec2 θ , giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .

[6]

6

The curve with equation y =

6 intersects the line y = x + 1 at the point P. x2

(i) Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6. (ii) Show that the x-coordinate of P satisfies the equation q 6 x=  . x+1 (iii) Use the iterative formula

xn+1

r = 

[2]

[2]

6 , xn + 1

with initial value x1 = 1.5, to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

7

The polynomial 3x3 + 2x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that (x − 1) is a factor of p(x), and that when p(x) is divided by (x − 2) the remainder is 10. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, solve the equation p(x) = 0.

[4]

9709/22/O/N/10


8

y

Q

p

O

The diagram shows the curve y = x sin x, for 0 ≤ x ≤ π . The point Q

x 1 1 2π, 2π

(i) Show that the normal to the curve at Q passes through the point (π , 0). (ii) Find

d (sin x − x cos x). dx

(iii) Hence evaluate ã

lies on the curve. [5] [2]

1π 2

x sin x dx.

[3]

0

9709/22/O/N/10


1

Solve the inequality | 3x + 1| > 8.

2

The sequence of values given by the iterative formula

[3]

xn+1 =

with initial value x1 = 1.7, converges to α .

7xn 5 + 4, 8 2xn

(i) Use this iterative formula to determine α correct to 2 decimal places, giving the result of each iteration to 4 decimal places. [3]

√ (ii) State an equation that is satisfied by α and hence show that α = 5 20 .

3

[2]

The polynomial x3 + 4x2 + ax + 2, where a is a constant, is denoted by p(x). It is given that the remainder when p(x) is divided by (x + 1) is equal to the remainder when p(x) is divided by (x − 2). (i) Find the value of a.

[3]

(ii) When a has this value, show that (x − 1) is a factor of p(x) and find the quotient when p(x) is divided by (x − 1). [3] 4

(a) Find ã e1−2x dx.

[2]

(b) Express sin2 3x in terms of cos 6x and hence find ã sin2 3x dx.

[4]

5

ln y

(2.2, 1.2) (1.4, 0.8)

x

O

The variables x and y satisfy the equation y = A(bx ), where A and b are constants. The graph of ln y against x is a straight line passing through the points (1.4, 0.8) and (2.2, 1.2), as shown in the diagram. Find the values of A and b, correct to 2 decimal places. [6]

9709/23/O/N/10


6

(i) Express 2 sin θ − cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ , giving the exact [3] value of R and the value of α correct to 2 decimal places. (ii) Hence solve the equation

2 sin θ − cos θ = −0.4,

giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ . 7

[4]

y M

O

x

1

The diagram shows the curve y =

ln x and its maximum point M . x2

(i) Find the exact coordinates of M .

[5]

(ii) Use the trapezium rule with three intervals to estimate the value of ä

4

1

ln x dx, x2

giving your answer correct to 2 decimal places.

8

The equation of a curve is

[3]

x2 + 2xy − y2 + 8 = 0.

(i) Show that the tangent to the curve at the point (−2, 2) is parallel to the x-axis.

[4]

(ii) Find the equation of the tangent to the curve at the other point on the curve for which x = −2, [5] giving your answer in the form y = mx + c.

9709/23/O/N/10


1

2

3

Find the coefficient of x in the expansion of x +

7

2 . x2

[3]

The volume of a spherical balloon is increasing at a constant rate of 50 cm3 per second. Find the rate [4] of increase of the radius when the radius is 10 cm. [Volume of a sphere = 43 π r3 .] (i) Sketch the curve y = (x − 2)2 .

[1]

(ii) The region enclosed by the curve, the x-axis and the y-axis is rotated through 360◦ about the x-axis. Find the volume obtained, giving your answer in terms of π . [4]

4

Q

B

R

S

P

5 cm

C 6 cm

k

2 cm

j A

i

6 cm

D

The diagram shows a prism ABCDPQRS with a horizontal square base APSD with sides of length 6 cm. The cross-section ABCD is a trapezium and is such that the vertical edges AB and DC are of lengths 5 cm and 2 cm respectively. Unit vectors i, j and k are parallel to AD, AP and AB respectively. −−→ −−→ (i) Express each of the vectors CP and CQ in terms of i, j and k.

(ii) Use a scalar product to calculate angle PCQ.

5

[2] [4]

(i) Show that the equation 2 tan2 θ sin2 θ = 1 can be written in the form

2 sin4 θ + sin2 θ − 1 = 0. (ii) Hence solve the equation 2 tan2 θ sin2 θ = 1 for 0◦ ≤ θ ≤ 360◦ .

9709/11/M/J/11

[2]

[4]


6

The variables x, y and ß can take only positive values and are such that ß = 3x + 2y

(i) Show that ß = 3x +

and

xy = 600.

1200 . x

[1]

(ii) Find the stationary value of ß and determine its nature.

7

8

A curve is such that

[6]

3 dy = and the point (1, 21 ) lies on the curve. dx (1 + 2x)2

(i) Find the equation of the curve.

[4]

(ii) Find the set of values of x for which the gradient of the curve is less than 31 .

[3]

A television quiz show takes place every day. On day 1 the prize money is $1000. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1:

Increase the prize money by $1000 each day.

Model 2:

Increase the prize money by 10% each day.

On each day that the prize money is not won the television company makes a donation to charity. The amount donated is 5% of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity (i) if Model 1 is used,

[4]

(ii) if Model 2 is used.

[3]

9

S

A

B r

P

T 2q

O In the diagram, OAB is an isosceles triangle with OA = OB and angle AOB = 2θ radians. Arc PST has centre O and radius r, and the line ASB is a tangent to the arc PST at S. (i) Find the total area of the shaded regions in terms of r and θ .

[4]

(ii) In the case where θ = 13 π and r = 6, find the total perimeter of the shaded regions, leaving your √ [5] answer in terms of 3 and π . [Questions 10 and 11 are printed on the next page.]

9709/11/M/J/11


10

(i) Express 2x2 − 4x + 1 in the form a(x + b)2 + c and hence state the coordinates of the minimum [4] point, A, on the curve y = 2x2 − 4x + 1.

The line x − y + 4 = 0 intersects the curve y = 2x2 − 4x + 1 at points P and Q. It is given that the coordinates of P are (3, 7).

11

(ii) Find the coordinates of Q.

[3]

(iii) Find the equation of the line joining Q to the mid-point of AP.

[3]

Functions f and g are defined for x ∈ > by f : x → 2x + 1,

g : x → x2 − 2.

(i) Find and simplify expressions for fg(x) and gf (x).

[2]

(ii) Hence find the value of a for which fg(a) = gf (a).

[3]

(iii) Find the value of b (b ≠ a) for which g(b) = b.

[2]

(iv) Find and simplify an expression for f −1 g(x).

[2]

The function h is defined by h : x → x2 − 2,

for x ≤ 0.

(v) Find an expression for h−1 (x).

[2]

9709/11/M/J/11


1

2

Find ä x3 +

1  dx. x3

[3]

(i) Find the terms in x2 and x3 in the expansion of 1 − 32 x . 6

[3]

(ii) Given that there is no term in x3 in the expansion of (k + 2x) 1 − 32 x , find the value of the [2] constant k. 6

3

The equation x2 + px + q = 0, where p and q are constants, has roots −3 and 5. (i) Find the values of p and q.

[2]

(ii) Using these values of p and q, find the value of the constant r for which the equation x2 + px + q + r = 0 has equal roots. [3] 4

5

A curve has equation y =

4 and P (2, 2) is a point on the curve. 3x − 4

(i) Find the equation of the tangent to the curve at P.

[4]

(ii) Find the angle that this tangent makes with the x-axis.

[2]

(i) Prove the identity

cos θ 1 . ≡1+ tan θ (1 − sin θ ) sin θ

(ii) Hence solve the equation

6

cos θ = 4, for 0◦ ≤ θ ≤ 360◦ . tan θ (1 − sin θ )

The function f is defined by f : x → (i) Show that ff (x) = x.

[3]

x+3 , x ∈ >, x ≠ 12 . 2x − 1 [3]

(ii) Hence, or otherwise, obtain an expression for f −1 (x). 7

[3]

[2]

The line L1 passes through the points A (2, 5) and B (10, 9). The line L2 is parallel to L1 and passes through the origin. The point C lies on L2 such that AC is perpendicular to L2 . Find (i) the coordinates of C,

[5]

(ii) the distance AC.

[2]

9709/12/M/J/11


8

Relative to the origin O, the position vectors of the points A, B and C are given by −−→ OA =

2 3!, 5

−−→ OB =

4 2! 3

and

−−→ OC =

10 0!. 6

(i) Find angle ABC .

[6]

The point D is such that ABCD is a parallelogram. (ii) Find the position vector of D. 9

[2]

The function f is such that f (x) = 3 − 4 cosk x, for 0 ≤ x ≤ π , where k is a constant. (i) In the case where k = 2, (a) find the range of f,

[2]

(b) find the exact solutions of the equation f (x) = 1.

[3]

(a) sketch the graph of y = f (x),

[2]

(ii) In the case where k = 1,

(b) state, with a reason, whether f has an inverse. 10

[1]

(a) A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm, find the perimeter of the smallest sector. [6] (b) The first, second and third terms of a geometric progression are 2k + 3, k + 6 and k, respectively. Given that all the terms of the geometric progression are positive, calculate

11

(i) the value of the constant k,

[3]

(ii) the sum to infinity of the progression.

[2]

y

M y = 4 Öx – x

O

A

x

√ The diagram shows part of the curve y = 4 x − x. The curve has a maximum point at M and meets the x-axis at O and A.

(i) Find the coordinates of A and M .

[5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis, giving [6] your answer in terms of π . 9709/12/M/J/11


1

The coefficient of x3 in the expansion of (a + x)5 + (1 − 2x)6 , where a is positive, is 90. Find the value [5] of a.

2

Find the set of values of m for which the line y = mx + 4 intersects the curve y = 3x2 − 4x + 7 at two distinct points. [5]

3

4

x y + = 1, where a and b are positive constants, meets the x-axis at P and the y-axis at Q. a b √ [5] Given that PQ = (45) and that the gradient of the line PQ is − 12 , find the values of a and b. The line

(a) Differentiate

2x3 + 5 with respect to x. x

[3] 1

(b) Find ã (3x − 2)5 dx and hence find the value of ã (3x − 2)5 dx.

[4]

0

5

G

F

P Q D

E C

k O

j

B

R i

A

In the diagram, OABCDEFG is a rectangular block in which OA = OD = 6 cm and AB = 12 cm. The −−→ −−→ − −− → unit vectors i, j and k are parallel to OA, OC and OD respectively. The point P is the mid-point of DG, Q is the centre of the square face CBFG and R lies on AB such that AR = 4 cm. −−→ −−→ (i) Express each of the vectors PQ and RQ in terms of i, j and k.

(ii) Use a scalar product to find angle RQP.

6

[3] [4]

(a) A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term. [4] (b) An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms. [4]

9709/13/M/J/11


7

A

6 cm

1p 3

O

X

B

In the diagram, AB is an arc of a circle, centre O and radius 6 cm, and angle AOB = 13 π radians. The line AX is a tangent to the circle at A, and OBX is a straight line. √ (i) Show that the exact length of AX is 6 3 cm.

Find, in terms of π and

8

3,

(ii) the area of the shaded region,

[3]

(iii) the perimeter of the shaded region.

[4]

(i) Prove the identity 

1 1 2 1 − cos θ −  ≡ . sin θ tan θ 1 + cos θ

(ii) Hence solve the equation 

9

[1]

A curve is such that

[3]

1 1 2 2 −  = 5 , for 0◦ ≤ θ ≤ 360◦ . sin θ tan θ

[4]

2 dy = √ − 1 and P (9, 5) is a point on the curve. dx x

(i) Find the equation of the curve.

[4]

(ii) Find the coordinates of the stationary point on the curve.

[3]

(iii) Find an expression for

d2 y and determine the nature of the stationary point. dx2

[2]

(iv) The normal to the curve at P makes an angle of tan−1 k with the positive x-axis. Find the value [2] of k. 10

Functions f and g are defined by f : x → 3x − 4,

x ∈ >,

g : x → 2(x − 1) + 8, 3

x > 1.

(i) Evaluate fg(2).

[2]

(ii) Sketch in a single diagram the graphs of y = f (x) and y = f −1 (x), making clear the relationship between the graphs. [3] (iii) Obtain an expression for g ′(x) and use your answer to explain why g has an inverse.

[3]

(iv) Express each of f −1 (x) and g−1 (x) in terms of x.

[4]

9709/13/M/J/11


1

Solve the equation | 3x + 4 | = | 2x + 5 |.

2

A curve has parametric equations

[3]

x = 3t + sin 2t,

y = 4 + 2 cos 2t.

Find the exact gradient of the curve at the point for which t = 16 π .

[4]

3

ln y

(6, 10.2)

(0, 2.0)

ln x

O

The variables x and y satisfy the equation y = Kxm , where K and m are constants. The graph of ln y against ln x is a straight line passing through the points (0, 2.0) and (6, 10.2), as shown in the [5] diagram. Find the values of K and m, correct to 2 decimal places.

4

The polynomial f (x) is defined by

f (x) = 3x3 + ax2 + ax + a,

where a is a constant.

5

6

(i) Given that (x + 2) is a factor of f (x), find the value of a.

[2]

(ii) When a has the value found in part (i), find the quotient when f (x) is divided by (x + 2).

[3]

Find the value of

dy when x = 4 in each of the following cases: dx

(i) y = x ln(x − 3),

[4]

(ii) y =

[3]

x−1 . x+1

(a) Find ã 4ex (3 + e2x ) dx. (b) Show that ã

1π 4

− 14 π

[4]

(3 + 2 tan2 θ ) dθ = 12 (8 + π ).

9709/21/M/J/11

[4]


7

(i) By sketching a suitable pair of graphs, show that the equation

e2x = 14 − x2 has exactly two real roots.

[3]

(ii) Show by calculation that the positive root lies between 1.2 and 1.3.

[2]

(iii) Show that this root also satisfies the equation

x = 12 ln(14 − x2 ).

[1]

(iv) Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places. [3]

8

(i) Express 4 sin θ − 6 cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ . Give the exact [3] value of R and the value of α correct to 2 decimal places. (ii) Solve the equation 4 sin θ − 6 cos θ = 3 for 0◦ ≤ θ ≤ 360◦ .

[4]

(iii) Find the greatest and least possible values of (4 sin θ − 6 cos θ )2 + 8 as θ varies.

[2]

9709/21/M/J/11


1

Use logarithms to solve the equation 3x = 2x+2 , giving your answer correct to 3 significant figures. [4]

2

y 3

B

A

2

O

x

√ The diagram shows the curve y = (1 + x3 ). Region A is bounded by the curve and the lines x = 0, x = 2 and y = 0. Region B is bounded by the curve and the lines x = 0 and y = 3.

(i) Use the trapezium rule with two intervals to find an approximation to the area of region A. Give your answer correct to 2 decimal places. [3] (ii) Deduce an approximation to the area of region B and explain why this approximation under[2] estimates the true area of region B.

3

The sequence x1 , x2 , x3 , . . . defined by

x1 = 1,

xn+1 =

1 2

p 3

xn2 + 6

converges to the value α . (i) Find the value of α correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places. [3] (ii) Find, in the form ax3 + bx2 + c = 0, an equation of which α is a root.

4

(a) Find the value of ã

2π 3

0

sin 12 x dx.

[3]

(b) Find ã e−x (1 + ex ) dx.

5

[2]

[3]

A curve has equation x2 + 2y2 + 5x + 6y = 10. Find the equation of the tangent to the curve at the [6] point (2, −1). Give your answer in the form ax + by + c = 0, where a, b and c are integers.

9709/22/M/J/11


6

The curve y = 4x2 ln x has one stationary point. (i) Find the coordinates of this stationary point, giving your answers correct to 3 decimal places. [5] (ii) Determine whether this point is a maximum or a minimum point.

7

[2]

The cubic polynomial p(x) is defined by

p(x) = 6x3 + ax2 + bx + 10,

where a and b are constants. It is given that (x + 2) is a factor of p(x) and that, when p(x) is divided by (x + 1), the remainder is 24.

8

(i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x) completely.

[3]

(i) Prove that sin2 2θ (cosec2 θ − sec2 θ ) ≡ 4 cos 2θ .

[3]

(ii) Hence (a) solve for 0◦ ≤ θ ≤ 180◦ the equation sin2 2θ (cosec2 θ − sec2 θ ) = 3, (b) find the exact value of cosec2 15◦ − sec2 15◦ .

9709/22/M/J/11

[4] [2]


1

Use logarithms to solve the equation 3x = 2x+2 , giving your answer correct to 3 significant figures. [4]

2

y 3

B

A

2

O

x

√ The diagram shows the curve y = (1 + x3 ). Region A is bounded by the curve and the lines x = 0, x = 2 and y = 0. Region B is bounded by the curve and the lines x = 0 and y = 3.

(i) Use the trapezium rule with two intervals to find an approximation to the area of region A. Give your answer correct to 2 decimal places. [3] (ii) Deduce an approximation to the area of region B and explain why this approximation under[2] estimates the true area of region B.

3

The sequence x1 , x2 , x3 , . . . defined by

x1 = 1,

xn+1 =

1 2

p 3

xn2 + 6

converges to the value α . (i) Find the value of α correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places. [3] (ii) Find, in the form ax3 + bx2 + c = 0, an equation of which α is a root.

4

(a) Find the value of ã

2π 3

0

sin 12 x dx.

[3]

(b) Find ã e−x (1 + ex ) dx.

5

[2]

[3]

A curve has equation x2 + 2y2 + 5x + 6y = 10. Find the equation of the tangent to the curve at the [6] point (2, −1). Give your answer in the form ax + by + c = 0, where a, b and c are integers.

9709/23/M/J/11


6

The curve y = 4x2 ln x has one stationary point. (i) Find the coordinates of this stationary point, giving your answers correct to 3 decimal places. [5] (ii) Determine whether this point is a maximum or a minimum point.

7

[2]

The cubic polynomial p(x) is defined by

p(x) = 6x3 + ax2 + bx + 10,

where a and b are constants. It is given that (x + 2) is a factor of p(x) and that, when p(x) is divided by (x + 1), the remainder is 24.

8

(i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x) completely.

[3]

(i) Prove that sin2 2θ (cosec2 θ − sec2 θ ) ≡ 4 cos 2θ .

[3]

(ii) Hence (a) solve for 0◦ ≤ θ ≤ 180◦ the equation sin2 2θ (cosec2 θ − sec2 θ ) = 3, (b) find the exact value of cosec2 15◦ − sec2 15◦ .

9709/23/M/J/11

[4] [2]


1

2

3

Find the term independent of x in the expansion of 2x +

1 6  . x2

[3]

A curve has equation y = 3x3 − 6x2 + 4x + 2. Show that the gradient of the curve is never negative. [3]

(i) Sketch, on a single diagram, the graphs of y = cos 2θ and y =

1 2

for 0 ≤ θ ≤ 2π .

[3]

(ii) Write down the number of roots of the equation 2 cos 2θ − 1 = 0 in the interval 0 ≤ θ ≤ 2π . [1] (iii) Deduce the number of roots of the equation 2 cos 2θ − 1 = 0 in the interval 10π ≤ θ ≤ 20π . 4

5

[1]

A function f is defined for x ∈ > and is such that f ′ (x) = 2x − 6. The range of the function is given by f (x) ≥ −4. (i) State the value of x for which f (x) has a stationary value.

[1]

(ii) Find an expression for f (x) in terms of x.

[4]

C

B q rad

r

O

q rad

r

A

The diagram represents a metal plate OABC, consisting of a sector OAB of a circle with centre O and radius r, together with a triangle OCB which is right-angled at C . Angle AOB = θ radians and OC is perpendicular to OA.

6

(i) Find an expression in terms of r and θ for the perimeter of the plate.

[3]

(ii) For the case where r = 10 and θ = 15 π , find the area of the plate.

[3]

(a) The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term. [4] (b) A geometric progression has first term 1 and common ratio r. A second geometric progression has first term 4 and common ratio 41 r. The two progressions have the same sum to infinity, S. Find the values of r and S. [3]

9709/11/O/N/11


7

x

2y 3y 3x y 4x The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m. (i) Find an expression for y in terms of x.

[1]

(ii) Given that the area of the garden is A m2 , show that A = 48x − 8x2 .

[2]

(iii) Given that x can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value. [4]

8

Relative to an origin O, the point A has position vector 4i + 7j − pk and the point B has position vector 8i − j − pk, where p is a constant. −−→ −−→ (i) Find OA . OB.

[2]

(ii) Hence show that there are no real values of p for which OA and OB are perpendicular to each other. [1] (iii) Find the values of p for which angle AOB = 60◦ . 9

[4]

A line has equation y = kx + 6 and a curve has equation y = x2 + 3x + 2k, where k is a constant. (i) For the case where k = 2, the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB. [5] (ii) Find the two values of k for which the line is a tangent to the curve.

[Questions 10 and 11 are printed on the next page.]

9709/11/O/N/11

[4]


10

y C

y = Ö(1 + 2x)

B

A

x

O

√ The diagram shows the curve y = (1 + 2x) meeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.

(i) Find the coordinates of B and C.

[2]

(ii) Find the equation of the normal to the curve at C.

[4]

(iii) Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis. [5]

11

Functions f and g are defined by f : x → 2x2 − 8x + 10

for 0 ≤ x ≤ 2,

g : x → x

for 0 ≤ x ≤ 10.

(i) Express f (x) in the form a(x + b)2 + c, where a, b and c are constants.

[3]

(ii) State the range of f.

[1]

(iii) State the domain of f −1 .

[1]

(iv) Sketch on the same diagram the graphs of y = f (x), y = g(x) and y = f −1 (x), making clear the relationship between the graphs. [4] (v) Find an expression for f −1 (x).

[3]

9709/11/O/N/11


1

(i) Find the first 3 terms in the expansion of (2 − y)5 in ascending powers of y.

[2] 5

(ii) Use the result in part (i) to find the coefficient of x2 in the expansion of 2 − (2x − x2 ) . 2

[3]

The functions f and g are defined for x ∈ > by f : x → 3x + a, g : x  → b − 2x, where a and b are constants. Given that ff (2) = 10 and g−1 (2) = 3, find

3

(i) the values of a and b,

[4]

(ii) an expression for fg(x).

[2]

Relative to an origin O, the position vectors of points A and B are given by −−→ OA = 5i + j + 2k

and

−−→ OB = 2i + 7j + pk,

where p is a constant. (i) Find the value of p for which angle AOB is 90◦ .

[3]

(ii) In the case where p = 4, find the vector which has magnitude 28 and is in the same direction as −−→ AB. [4]

4

The equation of a curve is y2 + 2x = 13 and the equation of a line is 2y + x = k, where k is a constant. (i) In the case where k = 8, find the coordinates of the points of intersection of the line and the curve. [4]

5

(ii) Find the value of k for which the line is a tangent to the curve.

[3]

(i) Sketch, on the same diagram, the graphs of y = sin x and y = cos 2x for 0◦ ≤ x ≤ 180◦ .

[3]

(ii) Verify that x = 30◦ is a root of the equation sin x = cos 2x, and state the other root of this equation for which 0◦ ≤ x ≤ 180◦ . [2] (iii) Hence state the set of values of x, for 0◦ ≤ x ≤ 180◦ , for which sin x < cos 2x.

9709/12/O/N/11

[2]


6

C2 D

E

C1 6 cm

A

10 cm

1p 3

q

X

B

The diagram shows a circle C1 touching a circle C2 at a point X . Circle C1 has centre A and radius 6 cm, and circle C2 has centre B and radius 10 cm. Points D and E lie on C1 and C2 respectively and DE is parallel to AB. Angle DAX = 13 π radians and angle EBX = θ radians. (i) By considering the perpendicular distances of D and E from AB, show that the exact value of θ √ 3 3 is sin−1  . [3] 10 (ii) Find the perimeter of the shaded region, correct to 4 significant figures.

7

8

[5]

8 dy = 5 − 2 . The line 3y + x = 17 is the normal to the curve at the point P on the dx x curve. Given that the x-coordinate of P is positive, find A curve is such that

(i) the coordinates of P,

[4]

(ii) the equation of the curve.

[4]

√ The equation of a curve is y = (8x − x2 ). Find

(i) an expression for

dy , and the coordinates of the stationary point on the curve, dx

[4]

(ii) the volume obtained when the region bounded by the curve and the x-axis is rotated through [4] 360◦ about the x-axis.

[Questions 9 and 10 are printed on the next page.]

9709/12/O/N/11


9

y

B (3, 6)

C (9, 4) M

O

x

A (–1, –1) D

The diagram shows a quadrilateral ABCD in which the point A is (−1, −1), the point B is (3, 6) and the point C is (9, 4). The diagonals AC and BD intersect at M . Angle BMA = 90◦ and BM = MD. Calculate

10

(i) the coordinates of M and D,

[7]

(ii) the ratio AM : MC .

[2]

(a) An arithmetic progression contains 25 terms and the first term is −15. The sum of all the terms in the progression is 525. Calculate (i) the common difference of the progression,

[2]

(ii) the last term in the progression,

[2]

(iii) the sum of all the positive terms in the progression.

[2]

(b) A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be $4000 in 2012 and will increase by 5% each year. Calculate (i) the value of the grant in 2022,

[2]

(ii) the total amount the college will receive in the years 2012 to 2022 inclusive.

[2]

9709/12/O/N/11


5

1

The coefficient of x2 in the expansion of k + 13 x is 30. Find the value of the constant k.

2

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

[3]

(i) an arithmetic progression,

[2]

(ii) a geometric progression.

[2]

3

y y = 2x B

O

x

A y = 2x 5 + 3x 3 The diagram shows the curve y = 2x5 + 3x3 and the line y = 2x intersecting at points A, O and B. (i) Show that the x-coordinates of A and B satisfy the equation 2x4 + 3x2 â&#x2C6;&#x2019; 2 = 0.

[2]

(ii) Solve the equation 2x4 + 3x2 â&#x2C6;&#x2019; 2 = 0 and hence find the coordinates of A and B, giving your answers in an exact form. [3] 4

D

C

10 cm

P

10 cm

Q

0.8 rad

A

10 cm

B

In the diagram, ABCD is a parallelogram with AB = BD = DC = 10 cm and angle ABD = 0.8 radians. APD and BQC are arcs of circles with centres B and D respectively. (i) Find the area of the parallelogram ABCD.

[2]

(ii) Find the area of the complete figure ABQCDP.

[2]

(iii) Find the perimeter of the complete figure ABQCDP.

[2]

9709/13/O/N/11


5

(i) Given that

3 sin2 x − 8 cos x − 7 = 0, show that, for real values of x, cos x = − 23 .

[3]

(ii) Hence solve the equation

3 sin2 (θ + 70◦ ) − 8 cos(θ + 70◦ ) − 7 = 0 for 0◦ ≤ θ ≤ 180◦ . 6

[4]

Relative to an origin O, the position vectors of points A and B are 3i + 4j − k and 5i − 2j − 3k respectively. (i) Use a scalar product to find angle BOA.

[4]

−−− → −−→ The point C is the mid-point of AB. The point D is such that OD = 2 OB. −−→ (ii) Find DC.

7

[4]

(i) A straight line passes through the point (2, 0) and has gradient m. Write down the equation of the line. [1] (ii) Find the two values of m for which the line is a tangent to the curve y = x2 − 4x + 5. For each [6] value of m, find the coordinates of the point where the line touches the curve. (iii) Express x2 − 4x + 5 in the form (x + a)2 + b and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]

8

9

A curve y = f (x) has a stationary point at P (3, −10). It is given that f ′ (x) = 2x2 + kx − 12, where k is a constant. (i) Show that k = −2 and hence find the x-coordinate of the other stationary point, Q.

[4]

(ii) Find f ′′ (x) and determine the nature of each of the stationary points P and Q.

[2]

(iii) Find f (x).

[4]

Functions f and g are defined by f : x → 2x + 3

g : x → x2 − 6x

for x ≤ 0, for x ≤ 3.

(i) Express f −1 (x) in terms of x and solve the equation f (x) = f −1 (x).

[3]

(ii) On the same diagram sketch the graphs of y = f (x) and y = f −1 (x), showing the coordinates of their point of intersection and the relationship between the graphs. [3] (iii) Find the set of values of x which satisfy gf (x) ≤ 16. 9709/13/O/N/11

[5]


10

y

1

y = Ö(x + 1) y=x+1

–1

O

x

√ The diagram shows the line y = x + 1 and the curve y = (x + 1), meeting at (−1, 0) and (0, 1).

(i) Find the area of the shaded region.

[5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis. [7]

9709/13/O/N/11


1

2

3

Solve the inequality | 4 − 5x | < 3. Show that ä

6

2

[3]

2 dx = ln 35 . 4x + 1

[5]

y

O

1 p 4

1 p 2

x

The diagram shows the part of the curve y = 12 tan 2x for 0 ≤ x ≤ 12 π . Find the x-coordinates of the points on this part of the curve at which the gradient is 4. [5]

4

5

Solve the equation 32x − 7(3x ) + 10 = 0, giving your answers correct to 3 significant figures.

[5]

The polynomial 4x3 + ax2 + 9x + 9, where a is a constant, is denoted by p(x). It is given that when p(x) is divided by (2x − 1) the remainder is 10. (i) Find the value of a and hence verify that (x − 3) is a factor of p(x).

(ii) When a has this value, solve the equation p(x) = 0.

9709/21/O/N/11

[3] [4]


6

(i) Verify by calculation that the cubic equation

x3 − 2x2 + 5x − 3 = 0

has a root that lies between x = 0.7 and x = 0.8.

[2]

(ii) Show that this root also satisfies an equation of the form

x=

ax2 + 3 , x2 + b

where the values of a and b are to be found.

[2]

(iii) With these values of a and b, use the iterative formula

xn+1

ax2n + 3 = 2 xn + b

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

7

The parametric equations of a curve are

(i) Show that

x = e3t ,

dy t(t + 2) = . dx 3e2t

y = t2 et + 3. [4]

(ii) Show that the tangent to the curve at the point (1, 3) is parallel to the x-axis.

[2]

(iii) Find the exact coordinates of the other point on the curve at which the tangent is parallel to the x-axis. [2]

8

(i) By first expanding cos(2x + x), show that

cos 3x ≡ 4 cos3 x − 3 cos x.

[5]

(ii) Hence show that

ã

1π 6

0

(2 cos3 x − cos x) dx =

9709/21/O/N/11

5. 12

[5]


1

Solve the inequality | x + 2 | >

12 x − 2

.

2

Use logarithms to solve the equation 4x+1 = 52x−3 , giving your answer correct to 3 significant figures. [4]

3

[4]

y

M x

O The diagram shows the curve y = x − 2 ln x and its minimum point M . (i) Find the x-coordinate of M .

[2]

(ii) Use the trapezium rule with three intervals to estimate the value of 5

ã (x − 2 ln x) dx, 2

giving your answer correct to 2 decimal places.

[3]

(iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii). [1]

4

Find the exact value of the positive constant k for which k

ã e4x dx = ã 0

5

2k

ex dx. 0

[6]

(i) By sketching a suitable pair of graphs, show that the equation

1 = sin x, x where x is in radians, has only one root for 0 < x ≤ 12 π . (ii) Verify by calculation that this root lies between x = 1.1 and x = 1.2.

[2] [2]

1 to determine this root correct to 2 decimal places. Give sin xn the result of each iteration to 4 decimal places. [3]

(iii) Use the iterative formula xn+1 =

9709/22/O/N/11


6

The parametric equations of a curve are

x = 1 + 2 sin2 θ , (i) Show that

y = 4 tan θ .

1 dy = . dx sin θ cos3 θ

[3]

(ii) Find the equation of the tangent to the curve at the point where θ = 14 π , giving your answer in [4] the form y = mx + c. 7

8

The polynomial ax3 − 3x2 − 11x + b, where a and b are constants, is denoted by p(x). It is given that (x + 2) is a factor of p(x), and that when p(x) is divided by (x + 1) the remainder is 12. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x) completely.

[3]

(i) Express 5 cos θ − 3 sin θ in the form R cos(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the exact value of R and the value of α correct to 2 decimal places. [3] (ii) Hence solve the equation

5 cos θ − 3 sin θ = 4,

giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .

(iii) Write down the least value of 15 cos θ − 9 sin θ as θ varies.

9709/22/O/N/11

[4] [1]


1

Find the gradient of the curve y = ln(5x + 1) at the point where x = 4.

[3]

2

Solve the inequality | 2x − 3 | ≤ | 3x |.

[4]

3

Solve the equation 2 ln(x + 3) − ln x = ln(2x − 2).

[5]

4

(i) Express cos2 x in terms of cos 2x.

[1]

(ii) Hence show that ã

5

6

1π 6

0

(cos2 x + sin 2x) dx =

1 √3 + 1 π 8 12

+ 14 .

[5]

Solve the equation 5 sec2 2θ = tan 2θ + 9, giving all solutions in the interval 0◦ ≤ θ ≤ 180◦ .

[6]

(i) The polynomial x4 + ax3 − x2 + bx + 2, where a and b are constants, is denoted by p(x). It is [5] given that (x − 1) and (x + 2) are factors of p(x). Find the values of a and b. (ii) When a and b have these values, find the quotient when p(x) is divided by x2 + x − 2.

7

[3]

y

x

O P

The diagram shows the curve y = (x − 4)e 2 . The curve has a gradient of 3 at the point P. 1x

(i) Show that the x-coordinate of P satisfies the equation

x = 2 + 6e

− 12 x

.

(ii) Verify that the equation in part (i) has a root between x = 3.1 and x = 3.3. −1x

[4]

[2]

(iii) Use the iterative formula xn+1 = 2 + 6e 2 n to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

9709/23/O/N/11


8

The equation of a curve is 2x2 − 3x − 3y + y2 = 6. (i) Show that

dy 4x − 3 = . dx 3 − 2y

[3]

(ii) Find the coordinates of the two points on the curve at which the gradient is −1.

9709/23/O/N/11

[6]


1

Solve the equation sin 2x = 2 cos 2x, for 0◦ ≤ x ≤ 180◦ .

2

Find the coefficient of x6 in the expansion of 2x3 −

3

[4]

1 . x2 7

[4]

A

2 cm

2 cm

P

R

B

Q

C

2 cm

In the diagram, ABC is an equilateral triangle of side 2 cm. The mid-point of BC is Q. An arc of a circle with centre A touches BC at Q, and meets AB √ at P and AC at R. Find the total area of the shaded regions, giving your answer in terms of π and 3. [5] 4

5

A watermelon is assumed to be spherical in shape while it is growing. Its mass, M kg, and radius, r cm, are related by the formula M = kr3 , where k is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and [5] the mass is 3.2 kg. Find the value of k and the rate at which the mass is increasing on this day.

y y = 6x + k y = 7 Öx B

A

x

O

√ The diagram shows the curve y = 7 x and the line y = 6x + k, where k is a constant. The curve and the line intersect at the points A and B.

(i) For the case where k = 2, find the x-coordinates of A and B.

√ (ii) Find the value of k for which y = 6x + k is a tangent to the curve y = 7 x. 9709/11/M/J/12

[4] [2]


6

7

Two vectors u and v are such that u =

p2 −2 ! and v = 6

2 p − 1 !, where p is a constant. 2p + 1

(i) Find the values of p for which u is perpendicular to v.

[3]

(ii) For the case where p = 1, find the angle between the directions of u and v.

[4]

(a) The first two terms of an arithmetic progression are 1 and cos2 x respectively. Show that the sum of the first ten terms can be expressed in the form a − b sin2 x, where a and b are constants to be found. [3] (b) The first two terms of a geometric progression are 1 and 31 tan2 θ respectively, where 0 < θ < 12 π .

8

(i) Find the set of values of θ for which the progression is convergent.

[2]

(ii) Find the exact value of the sum to infinity when θ = 16 π .

[2]

The function f : x → x2 − 4x + k is defined for the domain x ≥ p, where k and p are constants. (i) Express f (x) in the form (x + a)2 + b + k, where a and b are constants.

[2]

(ii) State the range of f in terms of k.

[1]

(iii) State the smallest value of p for which f is one-one.

[1]

(iv) For the value of p found in part (iii), find an expression for f −1 (x) and state the domain of f −1 , [4] giving your answers in terms of k.

9

10

The coordinates of A are (−3, 2) and the coordinates of C are (5, 6). The mid-point of AC is M and the perpendicular bisector of AC cuts the x-axis at B. (i) Find the equation of MB and the coordinates of B.

[5]

(ii) Show that AB is perpendicular to BC .

[2]

(iii) Given that ABCD is a square, find the coordinates of D and the length of AD.

[2]

It is given that a curve has equation y = f (x), where f (x) = x3 − 2x2 + x. (i) Find the set of values of x for which the gradient of the curve is less than 5.

[4]

(ii) Find the values of f (x) at the two stationary points on the curve and determine the nature of each stationary point. [5]

[Question 11 is printed on the next page.]

9709/11/M/J/12


11

y

y=

2 Ö(x + 1)

y=1

x

O The diagram shows the line y = 1 and part of the curve y = √ (i) Show that the equation y = √ (ii) Find ä 

2 . (x + 1)

2 4 can be written in the form x = 2 − 1. (x + 1) y

4 − 1 dy. Hence find the area of the shaded region. y2

[1]

[5]

(iii) The shaded region is rotated through 360◦ about the y-axis. Find the exact value of the volume of revolution obtained. [5]

9709/11/M/J/12


1

y

y=

O

2

6 2x – 3

3

x

6 , the x-axis and the lines x = 2 and 2x − 3 x = 3. Find, in terms of π , the volume obtained when this region is rotated through 360◦ about the x-axis. [4] The diagram shows the region enclosed by the curve y =

2

√ 2 The equation of a curve is y = 4 x + √ . x

(i) Obtain an expression for

dy . dx

[3]

(ii) A point is moving along the curve in such a way that the x-coordinate is increasing at a constant [2] rate of 0.12 units per second. Find the rate of change of the y-coordinate when x = 4. 3

The coefficient of x3 in the expansion of (a + x)5 + (2 − x)6 is 90. Find the value of the positive constant a. [5]

4

The point A has coordinates (−1, −5) and the point B has coordinates (7, 1). The perpendicular bisector of AB meets the x-axis at C and the y-axis at D. Calculate the length of CD. [6]

5

(i) Prove the identity tan x + (ii) Solve the equation

1 1 ≡ . tan x sin x cos x

2 = 1 + 3 tan x, for 0◦ ≤ x ≤ 180◦ . sin x cos x

9709/12/M/J/12

[2] [4]


6

B

A 2.4 rad

8 cm

O

The diagram shows a metal plate made by removing a segment from a circle with centre O and radius 8 cm. The line AB is a chord of the circle and angle AOB = 2.4 radians. Find

7

(i) the length of AB,

[2]

(ii) the perimeter of the plate,

[3]

(iii) the area of the plate.

[3]

(a) In an arithmetic progression, the sum of the first n terms, denoted by Sn , is given by

Sn = n2 + 8n. Find the first term and the common difference.

[3]

(b) In a geometric progression, the second term is 9 less than the first term. The sum of the second and third terms is 30. Given that all the terms of the progression are positive, find the first term. [5]

8

(i) Find the angle between the vectors 3i − 4k and 2i + 3j − 6k.

[4]

−−→ The vector OA has a magnitude of 15 units and is in the same direction as the vector 3i − 4k. The −−→ vector OB has a magnitude of 14 units and is in the same direction as the vector 2i + 3j − 6k. −−→ −−→ (ii) Express OA and OB in terms of i, j and k.

[3]

−−→ (iii) Find the unit vector in the direction of AB.

[3]

[Questions 9 and 10 are printed on the next page.]

9709/12/M/J/12


9

y A y = –x2 + 8x –10

B

x

O

The diagram shows part of the curve y = −x2 + 8x − 10 which passes through the points A and B. The curve has a maximum point at A and the gradient of the line BA is 2.

10

(i) Find the coordinates of A and B.

[7]

(ii) Find ã y dx and hence evaluate the area of the shaded region.

[4]

Functions f and g are defined by f : x → 2x + 5 for x ∈ >, 8 g : x  → for x ∈ >, x ≠ 3. x−3 (i) Obtain expressions, in terms of x, for f −1 (x) and g−1 (x), stating the value of x for which g−1 (x) is not defined. [4] (ii) Sketch the graphs of y = f (x) and y = f −1 (x) on the same diagram, making clear the relationship between the two graphs. [3] (iii) Given that the equation fg(x) = 5 − kx, where k is a constant, has no solutions, find the set of possible values of k. [5]

9709/12/M/J/12


1

2

(i) Prove the identity tan2 θ − sin2 θ ≡ tan2 θ sin2 θ .

[3]

(ii) Use this result to explain why tan θ > sin θ for 0◦ < θ < 90◦ .

[1]

Relative to an origin O, the position vectors of the points A, B and C are given by −−→ OA =

2 −1 !, 4

−−→ OB =

4 2! −2

and

−−→ OC =

1 3 !. p

Find −−→ (i) the unit vector in the direction of AB,

[3]

(ii) the value of the constant p for which angle BOC = 90◦ . 3

4

5

[2]

The first three terms in the expansion of (1 − 2x)2 (1 + ax)6 , in ascending powers of x, are 1 − x + bx2 . [6] Find the values of the constants a and b. (i) Solve the equation sin 2x + 3 cos 2x = 0 for 0◦ ≤ x ≤ 360◦ .

[5]

(ii) How many solutions has the equation sin 2x + 3 cos 2x = 0 for 0◦ ≤ x ≤ 1080◦ ?

[1]

y A

B (6, 1)

1

x=

8 –2 y2

x

O

8 − 2, crossing the y-axis at the point A. The point B (6, 1) y2 lies on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 1. Find the [6] exact volume obtained when this shaded region is rotated through 360◦ about the y-axis. The diagram shows part of the curve x =

6

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135. (i) Find the common difference of the progression.

[2]

The first term, the ninth term and the nth term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression. (ii) Find the common ratio of the geometric progression and the value of n.

9709/13/M/J/12

[5]


7

10 The curve y = − 2 intersects the x-axis at A. The tangent to the curve at A intersects the y-axis 2x + 1 at C. (i) Show that the equation of AC is 5y + 4x = 8.

[5]

(ii) Find the distance AC .

[2]

8

X

A

O

r

B

In the diagram, AB is an arc of a circle with centre O and radius r. The line XB is a tangent to the circle at B and A is the mid-point of OX . (i) Show that angle AOB = 13 π radians.

[2]

√ Express each of the following in terms of r, π and 3:

9

(ii) the perimeter of the shaded region,

[3]

(iii) the area of the shaded region.

[2]

A curve is such that

d2 y = −4x. The curve has a maximum point at (2, 12). dx2

(i) Find the equation of the curve.

[6]

A point P moves along the curve in such a way that the x-coordinate is increasing at 0.05 units per second. (ii) Find the rate at which the y-coordinate is changing when x = 3, stating whether the y-coordinate is increasing or decreasing. [2]

10

The equation of a line is 2y + x = k, where k is a constant, and the equation of a curve is xy = 6. (i) In the case where k = 8, the line intersects the curve at the points A and B. Find the equation of the perpendicular bisector of the line AB. [6] (ii) Find the set of values of k for which the line 2y + x = k intersects the curve xy = 6 at two distinct points. [3]

9709/13/M/J/12


11

The function f is such that f (x) = 8 − (x − 2)2 , for x ∈ >. (i) Find the coordinates and the nature of the stationary point on the curve y = f (x).

[3]

The function g is such that g(x) = 8 − (x − 2)2 , for k ≤ x ≤ 4, where k is a constant. (ii) State the smallest value of k for which g has an inverse.

[1]

For this value of k, (iii) find an expression for g−1 (x),

[3]

(iv) sketch, on the same diagram, the graphs of y = g(x) and y = g−1 (x).

[3]

9709/13/M/J/12


1

Solve the equation | x3 − 14 | = 13, showing all your working.

2

[4]

ln y

(5, 4.49) (0, 2.14)

x

O

The variables x and y satisfy the equation y = A(bx ), where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 2.14) and (5, 4.49), as shown in the diagram. [5] Find the values of A and b, correct to 1 decimal place.

3

The polynomial p(x) is defined by

p(x) = ax3 − 3x2 − 5x + a + 4,

where a is a constant. (i) Given that (x − 2) is a factor of p(x), find the value of a.

[2]

(ii) When a has this value, (a) factorise p(x) completely,

4

[3]

(b) find the remainder when p(x) is divided by (x + 1).

[2]

(i) Given that 35 + sec2 θ = 12 tan θ , find the value of tan θ .

[3]

(ii) Hence, showing the use of an appropriate formula in each case, find the exact value of (a) tan(θ − 45◦ ),

[2]

(b) tan 2θ .

[2]

9709/21/M/J/12


5

y

M

x

O

The diagram shows the curve y = 4e 2 − 6x + 3 and its minimum point M . 1x

(i) Show that the x-coordinate of M can be written in the form ln a, where the value of a is to be stated. [5] (ii) Find the exact value of the area of the region enclosed by the curve and the lines x = 0, x = 2 and y = 0. [4] 6

A curve has parametric equations 1 , (2t + 1)2

x=

√ y = (t + 2).

The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1. (i) Show that p = (p + 2) 6 − 12 . 1

[6]

(ii) Use an iterative process based on the equation in part (i) to find the value of p correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places. [3]

7

(i) Show that (2 sin x + cos x)2 can be written in the form (ii) Hence find the exact value of ã

1π 4

0

(2 sin x + cos x)2 dx.

9709/21/M/J/12

5 2

+ 2 sin 2x − 32 cos 2x.

[5] [4]


1

2

Solve the inequality | x + 3 | < | 2x + 1|.

[4]

(i) Given that 52x + 5x = 12, find the value of 5x .

[3]

(ii) Hence, using logarithms, solve the equation 52x + 5x = 12, giving the value of x correct to 3 significant figures. [2]

3

(i) Find the quotient when the polynomial

8x3 − 4x2 − 18x + 13

is divided by 4x2 + 4x − 3, and show that the remainder is 4.

[3]

(ii) Hence, or otherwise, factorise the polynomial

8x3 − 4x2 − 18x + 9.

4

[2]

(i) Express 9 sin θ − 12 cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ . Give the value of α correct to 2 decimal places. [3]

Hence (ii) solve the equation 9 sin θ − 12 cos θ = 4 for 0◦ ≤ θ ≤ 360◦ ,

(iii) state the largest value of k for which the equation 9 sin θ − 12 cos θ = k has any solutions. 5

[4] [1]

The parametric equations of a curve are

x = ln(t + 1), (i) Find an expression for

y = e2t + 2t.

dy in terms of t. dx

[4]

(ii) Find the equation of the normal to the curve at the point for which t = 0. Give your answer in [4] the form ax + by + c = 0, where a, b and c are integers.

9709/22/M/J/12


6

y M

O

1 2

a

x

p

sin 2x The diagram shows the curve y = for 0 ≤ x ≤ 12 π . The x-coordinate of the maximum point M x+2 is denoted by α . (i) Find

dy and show that α satisfies the equation tan 2x = 2x + 4. dx

[4]

(ii) Show by calculation that α lies between 0.6 and 0.7.

[2]

(iii) Use the iterative formula xn+1 = 12 tan−1 (2xn + 4) to find the value of α correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]

7

(i) Show that tan2 x + cos2 x ≡ sec2 x + 12 cos 2x − 12 and hence find the exact value of ã

(ii)

1π 4

0

(tan2 x + cos2 x) dx.

[7]

y

1 4

O

x

p

The region enclosed by the curve y = tan x + cos x and the lines x = 0, x = 14 π and y = 0 is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely [4] about the x-axis.

9709/22/M/J/12


1

Solve the equation | x3 − 14 | = 13, showing all your working.

2

[4]

ln y

(5, 4.49) (0, 2.14)

x

O

The variables x and y satisfy the equation y = A(bx ), where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 2.14) and (5, 4.49), as shown in the diagram. [5] Find the values of A and b, correct to 1 decimal place.

3

The polynomial p(x) is defined by

p(x) = ax3 − 3x2 − 5x + a + 4,

where a is a constant. (i) Given that (x − 2) is a factor of p(x), find the value of a.

[2]

(ii) When a has this value, (a) factorise p(x) completely,

4

[3]

(b) find the remainder when p(x) is divided by (x + 1).

[2]

(i) Given that 35 + sec2 θ = 12 tan θ , find the value of tan θ .

[3]

(ii) Hence, showing the use of an appropriate formula in each case, find the exact value of (a) tan(θ − 45◦ ),

[2]

(b) tan 2θ .

[2]

9709/23/M/J/12


5

y

M

x

O

The diagram shows the curve y = 4e 2 − 6x + 3 and its minimum point M . 1x

(i) Show that the x-coordinate of M can be written in the form ln a, where the value of a is to be stated. [5] (ii) Find the exact value of the area of the region enclosed by the curve and the lines x = 0, x = 2 and y = 0. [4] 6

A curve has parametric equations 1 , (2t + 1)2

x=

√ y = (t + 2).

The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1. (i) Show that p = (p + 2) 6 − 12 . 1

[6]

(ii) Use an iterative process based on the equation in part (i) to find the value of p correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places. [3]

7

(i) Show that (2 sin x + cos x)2 can be written in the form (ii) Hence find the exact value of ã

1π 4

0

(2 sin x + cos x)2 dx.

9709/23/M/J/12

5 2

+ 2 sin 2x − 32 cos 2x.

[5] [4]


1

The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first n terms is n. Find the value of the positive integer n. [4]

2

A curve is such that curve.

3

4

8 dy = − 3 − 1 and the point (2, 4) lies on the curve. Find the equation of the dx x [4]

An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday. [4] 6

(i) Find the first 3 terms in the expansion of (2x − x2 ) in ascending powers of x. 6

(ii) Hence find the coefficient of x8 in the expansion of (2 + x)(2x − x2 ) .

5

A curve has equation y = 2x + determine its nature.

[3] [2]

1 . Verify that the curve has a stationary point at x = 2 and (x − 1)2 [5]

6

B

r

O

D

q rad

C

A

The diagram shows a sector OAB of a circle with centre O and radius r. Angle AOB is θ radians. The point C on OA is such that BC is perpendicular to OA. The point D is on BC and the circular arc AD has centre C . (i) Find AC in terms of r and θ .

[1]

(ii) Find the perimeter of the shaded region ABD when θ = 13 π and r = 4, giving your answer as an exact value. [6]

7

(i) Solve the equation 2 cos2 θ = 3 sin θ , for 0◦ ≤ θ ≤ 360◦ .

[4]

(ii) The smallest positive solution of the equation 2 cos2 (nθ ) = 3 sin(nθ ), where n is a positive integer, is 10◦ . State the value of n and hence find the largest solution of this equation in the [3] interval 0◦ ≤ θ ≤ 360◦ .

9709/11/O/N/12


8

y 3y = 2x – 1 y 2 = 2x – 1

O a

1 2

x

The diagram shows the curve y2 = 2x − 1 and the straight line 3y = 2x − 1. The curve and straight line intersect at x = 12 and x = a, where a is a constant.

9

(i) Show that a = 5.

[2]

(ii) Find, showing all necessary working, the area of the shaded region.

[6]

The position vectors of points A and B relative to an origin O are a and b respectively. The position vectors of points C and D relative to O are 3a and 2b respectively. It is given that a=

10

2 1! 2

and

b=

4 0!. 6

−−→ (i) Find the unit vector in the direction of CD.

[3]

(ii) The point E is the mid-point of CD. Find angle EOD.

[6]

The function f is defined by f (x) = 4x2 − 24x + 11, for x ∈ >. (i) Express f (x) in the form a(x − b)2 + c and hence state the coordinates of the vertex of the graph [4] of y = f (x).

The function g is defined by g(x) = 4x2 − 24x + 11, for x ≤ 1. (ii) State the range of g.

[2]

(iii) Find an expression for g−1 (x) and state the domain of g−1 .

[4]

[Question 11 is printed on the next page.]

9709/11/O/N/12


11

y 1

y = (6x + 2)3

A (1, 2) B E

C O

x

1

The diagram shows the curve y = (6x + 2) 3 and the point A (1, 2) which lies on the curve. The tangent to the curve at A cuts the y-axis at B and the normal to the curve at A cuts the x-axis at C. (i) Find the equation of the tangent AB and the equation of the normal AC .

[5]

(ii) Find the distance BC .

[3]

(iii) Find the coordinates of the point of intersection, E, of OA and BC , and determine whether E is [4] the mid-point of OA.

9709/11/O/N/12


1

2

In the expansion of x2 −

a 7  , the coefficient of x5 is −280. Find the value of the constant a. x

A function f is such that f (x) =

r



x+3  + 1, for x ≥ −3. Find 2

[3]

(i) f −1 (x) in the form ax2 + bx + c, where a, b and c are constants,

[3]

(ii) the domain of f −1 .

[1]

3

X

B

40 m

2x m

C

Playground Y xm

A

60 m

D

The diagram shows a plan for a rectangular park ABCD, in which AB = 40 m and AD = 60 m. Points X and Y lie on BC and CD respectively and AX , XY and YA are paths that surround a triangular playground. The length of DY is x m and the length of XC is 2x m. (i) Show that the area, A m2 , of the playground is given by

A = x2 − 30x + 1200. (ii) Given that x can vary, find the minimum area of the playground.

4

The line y =

[2]

[3]

x + k, where k is a constant, is a tangent to the curve 4y = x2 at the point P. Find k

(i) the value of k,

[3]

(ii) the coordinates of P.

[3]

9709/12/O/N/12


5

y B (5, 11)

C

X (4, 4) A (1, 3) x

O

The diagram shows a triangle ABC in which A has coordinates (1, 3), B has coordinates (5, 11) and angle ABC is 90◦ . The point X (4, 4) lies on AC . Find

6

7

(i) the equation of BC,

[3]

(ii) the coordinates of C.

[3]

(i) Show that the equation 2 cos x = 3 tan x can be written as a quadratic equation in sin x.

[3]

(ii) Solve the equation 2 cos 2y = 3 tan 2y, for 0◦ ≤ y ≤ 180◦ .

[4]

The position vectors of the points A and B, relative to an origin O, are given by −−→ OA =

1 0! 2

and

−−→ OB =

k −k !, 2k

where k is a constant.

8

(i) In the case where k = 2, calculate angle AOB.

[4]

−−→ (ii) Find the values of k for which AB is a unit vector.

[4]

(a) In a geometric progression, all the terms are positive, the second term is 24 and the fourth term is 13 12 . Find (i) the first term,

[3]

(ii) the sum to infinity of the progression.

[2]

(b) A circle is divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are 3◦ and 5◦ . Find the value of n. [4]

[Questions 9, 10 and 11 are printed on the next page.]

9709/12/O/N/12


9

y

B (0, 3) y= C

9 2x + 3 A (3, 1) x

O

9 , crossing the y-axis at the point B (0, 3). The point 2x + 3 A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C. The diagram shows part of the curve y =

(i) Find the equation of the tangent to the curve at A.

[4]

(ii) Determine, showing all necessary working, whether C is nearer to B or to O.

[1]

(iii) Find, showing all necessary working, the exact volume obtained when the shaded region is [4] rotated through 360â&#x2014;Ś about the x-axis.

10

A curve is defined for x > 0 and is such that

4 dy = x + 2 . The point P (4, 8) lies on the curve. dx x

(i) Find the equation of the curve.

[4]

(ii) Show that the gradient of the curve has a minimum value when x = 2 and state this minimum value. [4] 11

Q x cm

R S

20

P

cm

C

1.2 rad

O The diagram shows a sector of a circle with centre O and radius 20 cm. A circle with centre C and radius x cm lies within the sector and touches it at P, Q and R. Angle POR = 1.2 radians. (i) Show that x = 7.218, correct to 3 decimal places.

[4]

(ii) Find the total area of the three parts of the sector lying outside the circle with centre C.

[2]

(iii) Find the perimeter of the region OPSR bounded by the arc PSR and the lines OP and OR. [4] 9709/12/O/N/12


1

Find the coefficient of x3 in the expansion of 2 − 12 x .

2

It is given that f (x) =

3

Solve the equation 7 cos x + 5 = 2 sin2 x, for 0◦ ≤ x ≤ 360◦ .

7

[3]

1 − x3 , for x > 0. Show that f is a decreasing function. x3

4

[3]

[4]

A

D

B

2 Ö3 cm

2 cm

C

In the diagram, D lies on the side AB of √ triangle ABC and CD is an arc of a circle with centre A and radius 2 cm. The line BC is of length 2 3 cm and is√perpendicular to AC. Find the area of the shaded region BDC, giving your answer in terms of π and 3. [4]

5

6

The first term of a geometric progression is 5 13 and the fourth term is 2 41 . Find (i) the common ratio,

[3]

(ii) the sum to infinity.

[2]

The functions f and g are defined for − 12 π ≤ x ≤ 12 π by f (x) = 12 x + 16 π , g(x) = cos x. Solve the following equations for − 12 π ≤ x ≤ 12 π . (i) gf (x) = 1, giving your answer in terms of π .

[2]

(ii) fg(x) = 1, giving your answers correct to 2 decimal places.

[4]

9709/13/O/N/12


7

y

y = 11 – x 2

A (p, q) x

O y=5–x

(i) The diagram shows part of the curve y = 11 − x2 and part of the straight line y = 5 − x meeting at [3] the point A (p, q), where p and q are positive constants. Find the values of p and q. (ii) The function f is defined for the domain x ≥ 0 by

f (x) = 

11 − x2 5−x

for 0 ≤ x ≤ p, for x > p.

Express f −1 (x) in a similar way. 8

[5]

A curve is such that 3 dy = 2(3x + 4) 2 − 6x − 8. dx

(i) Find

d2 y . dx2

[2]

(ii) Verify that the curve has a stationary point when x = −1 and determine its nature.

[2]

(iii) It is now given that the stationary point on the curve has coordinates (−1, 5). Find the equation of the curve. [5] 9

The position vectors of points A and B relative to an origin O are given by −−→ OA =

p 1! 1

and

−−→ OB =

4 2!, p

where p is a constant. (i) In the case where OAB is a straight line, state the value of p and find the unit vector in the −−→ [3] direction of OA. (ii) In the case where OA is perpendicular to AB, find the possible values of p.

[5]

(iii) In the case where p = 3, the point C is such that OABC is a parallelogram. Find the position vector of C. [2] 9709/13/O/N/12


10

A straight line has equation y = −2x + k, where k is a constant, and a curve has equation y =

2 . x−3

(i) Show that the x-coordinates of any points of intersection of the line and curve are given by the equation 2x2 − (6 + k)x + (2 + 3k) = 0. [1] (ii) Find the two values of k for which the line is a tangent to the curve.

[3]

The two tangents, given by the values of k found in part (ii), touch the curve at points A and B. (iii) Find the coordinates of A and B and the equation of the line AB.

11

[6]

y 2

y = x(x – 2)

O b

a

x

The diagram shows the curve with equation y = x(x − 2)2 . The minimum point on the curve has coordinates (a, 0) and the x-coordinate of the maximum point is b, where a and b are constants. (i) State the value of a.

[1]

(ii) Find the value of b.

[4]

(iii) Find the area of the shaded region.

[4]

(iv) The gradient,

dy , of the curve has a minimum value m. Find the value of m. dx

9709/13/O/N/12

[4]


1

Solve the inequality | x − 2 | ≥ | x + 5 |.

2

Use logarithms to solve the equation 5x = 32x−1 , giving your answer correct to 3 significant figures. [4]

3

Solve the equation

2 cos 2θ = 4 cos θ − 3,

for 0◦ ≤ θ ≤ 180◦ . 4

[3]

[4]

The parametric equations of a curve are

x = ln(1 − 2t), (i) Show that

y=

2 , t

for t < 0.

dy 1 − 2t = . dx t2

[3]

(ii) Find the exact coordinates of the only point on the curve at which the gradient is 3.

5

[3]

y

C

B (q , cos q )

R O

p

1 2

A

x

The diagram shows the curve y = cos x, for 0 ≤ x ≤ 12 π . A rectangle OABC is drawn, where B is the point on the curve with x-coordinate θ , and A and C are on the axes, as shown. The shaded region R is bounded by the curve and by the lines x = θ and y = 0. (i) Find the area of R in terms of θ .

[2]

(ii) The area of the rectangle OABC is equal to the area of R. Show that

θ=

1 − sin θ . cos θ

[1]

1 − sin θn , with initial value θ1 = 0.5, to determine the value cos θn of θ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

(iii) Use the iterative formula θn+1 =

9709/21/O/N/12


6

(a) Use the trapezium rule with two intervals to estimate the value of ä

1

0

1 dx, 6 + 2ex

giving your answer correct to 2 decimal places. (b) Find ä

7

(ex − 2)2 dx. e2x

[3] [4]

The polynomial 2x3 − 4x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x + 1) the remainder is 4, and that when p(x) is divided by (x − 3) the remainder is 12. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the quotient and remainder when p(x) is divided by (x2 − 2). [3]

8

(i) By differentiating

dy 1 , show that if y = sec θ then = tan θ sec θ . cos θ dθ

[3]

(ii) Hence show that

d2 y = a sec3 θ + b sec θ , 2 dθ giving the values of a and b.

[4]

(iii) Find the exact value of ã

1π 4

0

(1 + tan2 θ − 3 sec θ tan θ ) dθ .

9709/21/O/N/12

[5]


1

Solve the inequality | 2x + 1| < | 2x − 5 |.

2

The curve with equation y =

x-coordinate of this point.

3

[3]

sin 2x has one stationary point in the interval 0 ≤ x ≤ 12 π . Find the exact e2x [4]

The polynomial x4 − 4x3 + 3x2 + 4x − 4 is denoted by p(x).

(i) Find the quotient when p(x) is divided by x2 − 3x + 2.

[3]

(ii) Hence solve the equation p(x) = 0. 4

[3]

y

O

1 2

x

p

√ The diagram shows the part of the curve y = (2 − sin x) for 0 ≤ x ≤ 12 π .

(i) Use the trapezium rule with 2 intervals to estimate the value of ã

1π 2

0

(2 − sin x) dx,

giving your answer correct to 2 decimal places.

√ (ii) The line y = x intersects the curve y = (2 − sin x) at the point P. Use the iterative formula √ xn+1 = (2 − sin xn )

[3]

to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

9709/22/O/N/12


5

ln y (1, 2.9)

(3.5, 1.4)

x

O

The variables x and y satisfy the equation y = A(b−x ), where A and b are constants. The graph of ln y against x is a straight line passing through the points (1, 2.9) and (3.5, 1.4), as shown in the diagram. [6] Find the values of A and b, correct to 2 decimal places.

6

− 12 x

(a) Find ã 4e

dx.

(b) Show that ä

3

1

7

6 dx = ln 16. 3x − 1

The equation of a curve is

(i) Show that

[2]

dy 3x − 2y = . dx 2x − 2y

[5]

3x2 − 4xy + 2y2 − 6 = 0. [4]

(ii) Find the coordinates of each of the points on the curve where the tangent is parallel to the x-axis. [5]

8

(a) Given that tan A = t and tan(A + B) = 4, find tan B in terms of t. (b) Solve the equation

2 tan(45◦ − x) = 3 tan x,

giving all solutions in the interval 0◦ ≤ x ≤ 360◦ .

9709/22/O/N/12

[3]

[6]


1

Solve the inequality | x − 2 | ≥ | x + 5 |.

2

Use logarithms to solve the equation 5x = 32x−1 , giving your answer correct to 3 significant figures. [4]

3

Solve the equation

2 cos 2θ = 4 cos θ − 3,

for 0◦ ≤ θ ≤ 180◦ . 4

[3]

[4]

The parametric equations of a curve are

x = ln(1 − 2t), (i) Show that

y=

2 , t

for t < 0.

dy 1 − 2t . = dx t2

[3]

(ii) Find the exact coordinates of the only point on the curve at which the gradient is 3.

5

[3]

y

C

B (q , cos q )

R O

p

1 2

A

x

The diagram shows the curve y = cos x, for 0 ≤ x ≤ 12 π . A rectangle OABC is drawn, where B is the point on the curve with x-coordinate θ , and A and C are on the axes, as shown. The shaded region R is bounded by the curve and by the lines x = θ and y = 0. (i) Find the area of R in terms of θ .

[2]

(ii) The area of the rectangle OABC is equal to the area of R. Show that

θ=

1 − sin θ . cos θ

[1]

1 − sin θn , with initial value θ1 = 0.5, to determine the value cos θn of θ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

(iii) Use the iterative formula θn+1 =

9709/23/O/N/12


6

(a) Use the trapezium rule with two intervals to estimate the value of ä

1

0

1 dx, 6 + 2ex

giving your answer correct to 2 decimal places. (b) Find ä

7

(ex − 2)2 dx. e2x

[3] [4]

The polynomial 2x3 − 4x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x + 1) the remainder is 4, and that when p(x) is divided by (x − 3) the remainder is 12. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, find the quotient and remainder when p(x) is divided by (x2 − 2). [3]

8

(i) By differentiating

1 dy , show that if y = sec θ then = tan θ sec θ . cos θ dθ

[3]

(ii) Hence show that

d2 y = a sec3 θ + b sec θ , 2 dθ giving the values of a and b.

[4]

(iii) Find the exact value of ã

1π 4

0

(1 + tan2 θ − 3 sec θ tan θ ) dθ .

9709/23/O/N/12

[5]


1

2

It is given that f x = 2x − 53 + x, for x ∈ >. Show that f is an increasing function.

[3]

(i) In the expression 1 − px6 , p is a non-zero constant. Find the first three terms when 1 − px6 is expanded in ascending powers of x. [2] (ii) It is given that the coefficient of x2 in the expansion of 1 − x 1 − px6 is zero. Find the value of p. [3]

3

B 8 cm

O

a rad

A

8 cm

C In the diagram, OAB is a sector of a circle with centre O and radius 8 cm. Angle BOA is ! radians. OAC is a semicircle with diameter OA. The area of the semicircle OAC is twice the area of the sector OAB.

4

5

(i) Find ! in terms of 0.

[3]

(ii) Find the perimeter of the complete figure in terms of 0.

[2]

The third term of a geometric progression is −108 and the sixth term is 32. Find (i) the common ratio,

[3]

(ii) the first term,

[1]

(iii) the sum to infinity.

[2]

(i) Show that

cos 1 1 sin 1 +  . 2 sin 1 + cos 1 sin 1 − cos 1 sin 1 − cos2 1

(ii) Hence solve the equation

sin 1 cos 1 + = 3, for 0Å ≤ 1 ≤ 360Å. sin 1 + cos 1 sin 1 − cos 1

9709/11/M/J/13

[3]

[4]


6

Relative to an origin O, the position vectors of three points, A, B and C, are given by −−→ OA = i + 2pj + qk,

−−→ OB = qj − 2pk

and

−−→ OC = − 4p2 + q2 i + 2pj + qk,

where p and q are constants.

7

−−→ −−→ (i) Show that OA is perpendicular to OC for all non-zero values of p and q.

[2]

−−→ (ii) Find the magnitude of CA in terms of p and q.

[2]

−−→ (iii) For the case where p = 3 and q = 2, find the unit vector parallel to BA.

[3]

A curve has equation y = x2 − 4x + 4 and a line has equation y = mx, where m is a constant. (i) For the case where m = 1, the curve and the line intersect at the points A and B. Find the coordinates of the mid-point of AB. [4] (ii) Find the non-zero value of m for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve. [5]

8

(i) Express 2x2 − 12x + 13 in the form a x + b2 + c, where a, b and c are constants.

[3]

(ii) The function f is defined by f x = 2x2 − 12x + 13 for x ≥ k, where k is a constant. It is given that f is a one-one function. State the smallest possible value of k. [1] The value of k is now given to be 7.

9

(iii) Find the range of f.

[1]

(iv) Find an expression for f −1 x and state the domain of f −1 .

[5]

1

A curve has equation y = f x and is such that f ′ x = 3x 2 + 3x

− 12

− 10.

1

(i) By using the substitution u = x 2 , or otherwise, find the values of x for which the curve y = f x has stationary points. [4] (ii) Find f ′′ x and hence, or otherwise, determine the nature of each stationary point.

[3]

(iii) It is given that the curve y = f x passes through the point 4, −7. Find f x.

[4]

[Question 10 is printed on the next page.]

9709/11/M/J/13


10

y

C

A (1, 1)

O

B

y = (x â&#x20AC;&#x201C; 2)4

x

The diagram shows part of the curve y = x â&#x2C6;&#x2019; 24 and the point A 1, 1 on the curve. The tangent at A cuts the x-axis at B and the normal at A cuts the y-axis at C. (i) Find the coordinates of B and C.

[6]

(ii) Find the distance AC, giving your answer in the form (iii) Find the area of the shaded region.



a , where a and b are integers. b

[2] [4]

9709/11/M/J/13


dy 6 = 2 and 2, 9 is a point on the curve. Find the equation of the curve. dx x

1

A curve is such that

2

Find the coefficient of x2 in the expansion of @ A 1 6 (i) 2x − , 2x A @ 1 6 2 . (ii) 1 + x  2x − 2x

3

[3]

[2] [3]

The straight line y = mx + 14 is a tangent to the curve y = constant m and the coordinates of P.

4

12 + 2 at the point P. Find the value of the x [5]

X C

B

10 cm

A

O

D

The diagram shows a square ABCD of side 10 cm. The mid-point of AD is O and BXC is an arc of a circle with centre O.

5

(i) Show that angle BOC is 0.9273 radians, correct to 4 decimal places.

[2]

(ii) Find the perimeter of the shaded region.

[3]

(iii) Find the area of the shaded region.

[2]

It is given that a = sin 1 − 3 cos 1 and b = 3 sin 1 + cos 1, where 0Å ≤ 1 ≤ 360Å. (i) Show that a2 + b2 has a constant value for all values of 1.

[3]

(ii) Find the values of 1 for which 2a = b.

[4]

9709/12/M/J/13


6

Relative to an origin O, the position vectors of points A and B are given by −−→ OA = i − 2j + 2k

and

−−→ OB = 3i + pj + qk,

where p and q are constants. −−→ −−→ (i) State the values of p and q for which OA is parallel to OB.

[2]

(ii) In the case where q = 2p, find the value of p for which angle BOA is 90Å.

[2]

−−→ (iii) In the case where p = 1 and q = 8, find the unit vector in the direction of AB.

[3]

7

The point R is the reflection of the point −1, 3 in the line 3y + 2x = 33. Find by calculation the coordinates of R. [7]

8

The volume of a solid circular cylinder of radius r cm is 2500 cm3 . (i) Show that the total surface area, S cm2 , of the cylinder is given by

S = 20r2 +

9

5000 . r

2

(ii) Given that r can vary, find the stationary value of S.

[4]

(iii) Determine the nature of this stationary value.

[2]

A function f is defined by f x =

5 , for x ≥ 1. 1 − 3x

(i) Find an expression for f ′ x.

[2]

(ii) Determine, with a reason, whether f is an increasing function, a decreasing function or neither. [1] (iii) Find an expression for f −1 x, and state the domain and range of f −1 . 10

[5]

(a) The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression. [4] (b) The third term of a geometric progression is four times the first term. The sum of the first six terms is k times the first term. Find the possible values of k. [4]

[Question 11 is printed on the next page.]

9709/12/M/J/13


11

y y = Ă&#x2013;(1 + 4x)

B

A

O

C

x

 The diagram shows the curve y = 1 + 4x , which intersects the x-axis at A and the y-axis at B. The normal to the curve at B meets the x-axis at C. Find (i) the equation of BC,

[5]

(ii) the area of the shaded region.

[5]

9709/12/M/J/13


1

A curve is such that

dy  = 2x + 5 and 2, 5 is a point on the curve. Find the equation of the curve. dx [4]

2

R

Q

C O

3 cm

P

6 cm

S

The diagram shows a circle C with centre O and radius 3 cm. The radii OP and OQ are extended to S and R respectively so that ORS is a sector of a circle with centre O. Given that PS = 6 cm and that the area of the shaded region is equal to the area of circle C,

3

4

(i) show that angle POQ = 14 0 radians,

[3]

(ii) find the perimeter of the shaded region.

[2]

(i) Express the equation 2 cos2 1 = tan2 1 as a quadratic equation in cos2 1.

[2]

(ii) Solve the equation 2 cos2 1 = tan2 1 for 0 â&#x2030;¤ 1 â&#x2030;¤ 0, giving solutions in terms of 0.

[3]

(i) Find the first three terms in the expansion of 2 + ax5 in ascending powers of x.

[3]

(ii) Given that the coefficient of x2 in the expansion of 1 + 2x 2 + ax5 is 240, find the possible values of a. [3]

5

(i) Sketch, on the same diagram, the curves y = sin 2x and y = cos x â&#x2C6;&#x2019; 1 for 0 â&#x2030;¤ x â&#x2030;¤ 20.

[4]

(ii) Hence state the number of solutions, in the interval 0 â&#x2030;¤ x â&#x2030;¤ 20, of the equations

6

(a) 2 sin 2x + 1 = 0,

[1]

(b) sin 2x â&#x2C6;&#x2019; cos x + 1 = 0.

[1]

The non-zero variables x, y and u are such that u = x2 y. Given that y + 3x = 9, find the stationary value of u and determine whether this is a maximum or a minimum value. [7]

9709/13/M/J/13


7

y A (2, 14)

X B (14, 6) C (7, 2)

x

O

The diagram shows three points A 2, 14, B 14, 6 and C 7, 2. The point X lies on AB, and CX is perpendicular to AB. Find, by calculation, (i) the coordinates of X ,

[6]

(ii) the ratio AX : XB.

[2]

8

A

B O

C The diagram shows a parallelogram OABC in which ` a ` a 3 5 −−→ −−→ OA = 3 and OB = 0 . 2 −4

9

(i) Use a scalar product to find angle BOC.

[6]

−−→ (ii) Find a vector which has magnitude 35 and is parallel to the vector OC.

[2]

(a) In an arithmetic progression, the sum, Sn , of the first n terms is given by Sn = 2n2 + 8n. Find the first term and the common difference of the progression. [3] (b) The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the nth term respectively of an arithmetic progression. Find the value of n. [5] 9709/13/M/J/13


10

The function f is defined by f : x  â&#x2020;&#x2019; 2x + k, x â&#x2C6;&#x2C6; >, where k is a constant. (i) In the case where k = 3, solve the equation ff x = 25.

[2]

The function g is defined by g : x  â&#x2020;&#x2019; x2 â&#x2C6;&#x2019; 6x + 8, x â&#x2C6;&#x2C6; >. (ii) Find the set of values of k for which the equation f x = g x has no real solutions.

[3]

The function h is defined by h : x  â&#x2020;&#x2019; x2 â&#x2C6;&#x2019; 6x + 8, x > 3. (iii) Find an expression for hâ&#x2C6;&#x2019;1 x.

11

[4]

y A (1, 7) y= C

8 â&#x20AC;&#x201C;x Ă&#x2013;x

B (4, 0) O

x

8 The diagram shows part of the curve y =  â&#x2C6;&#x2019; x and points A 1, 7 and B 4, 0 which lie on the x curve. The tangent to the curve at B intersects the line x = 1 at the point C. (i) Find the coordinates of C.

[4]

(ii) Find the area of the shaded region.

[5]

9709/13/M/J/13


1

Solve the equation | 2x − 7 | = 1, giving answers correct to 2 decimal places where appropriate.

[5]

2

Solve the equation ln(3 − 2x) − 2 ln x = ln 5.

[5]

3

(i) Show that 12 sin2 x cos2 x ≡ 32 (1 − cos 4x). (ii) Hence show that ã

4

5

1π 3 1π 4

[3]

√ π 3 3 12 sin x cos x dx = + . 8 16 2

2

The polynomial ax3 − 5x2 + bx + 9, where a and b are constants, is denoted by p(x). It is given that (2x + 3) is a factor of p(x), and that when p(x) is divided by (x + 1) the remainder is 8. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x) completely.

[3]

The parametric equations of a curve are

x = e2t , (i) Show that

y = 4t et .

dy 2(t + 1) = . dx et

[4]

(ii) Find the equation of the normal to the curve at the point where t = 0. 6

[3]

[4]

(i) By sketching a suitable pair of graphs, show that the equation

cot x = 4x − 2,

where x is in radians, has only one root for 0 ≤ x ≤ 12 π . (ii) Verify by calculation that this root lies between x = 0.7 and x = 0.9.

[2] [2]

(iii) Show that this root also satisfies the equation

x=

1 + 2 tan x . 4 tan x

[1]

1 + 2 tan xn to determine this root correct to 2 decimal places. 4 tan xn Give the result of each iteration to 4 decimal places. [3]

(iv) Use the iterative formula xn+1 =

9709/21/M/J/13


7

(i) Express 5 sin 2θ + 2 cos 2θ in the form R sin(2θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the [3] exact value of R and the value of α correct to 2 decimal places.

Hence (ii) solve the equation

5 sin 2θ + 2 cos 2θ = 4,

giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ , (iii) determine the least value of

1 as θ varies. (10 sin 2θ + 4 cos 2θ )2

9709/21/M/J/13

[5] [2]


dy 4 . The point 3, 2 lies on the curve. Find the equation of the curve. = dx 7 â&#x2C6;&#x2019; 2x [4]

1

A curve is such that

2

Solve the inequality  x â&#x2C6;&#x2019; 8 > 2x â&#x2C6;&#x2019; 4.

3

[4]

(i) The polynomial 2x3 + ax2 â&#x2C6;&#x2019; ax â&#x2C6;&#x2019; 12, where a is a constant, is denoted by p x. It is given that x + 1 is a factor of p x. Find the value of a. [2]

(ii) When a has this value, find the remainder when p x is divided by x + 3. 4

[2]

The variables x and y satisfy the equation 5y+1 = 23x . (i) By taking logarithms, show that the graph of y against x is a straight line.

[2]

(ii) Find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the y-axis. [2]

5

The equation of a curve is

x2 â&#x2C6;&#x2019; 2x2 y + 3y = 9. (i) Show that

dy 2x â&#x2C6;&#x2019; 4xy . = dx 2x2 â&#x2C6;&#x2019; 3

[4]

(ii) Find the equation of the normal to the curve at the point where x = 2, giving your answer in the [4] form ax + by + c = 0.

6

(i) By sketching a suitable pair of graphs, show that the equation 3ex = 8 â&#x2C6;&#x2019; 2x has only one root.

[2]

(ii) Verify by calculation that this root lies between x = 0.7 and x = 0.8. (iii) Show that this root also satisfies the equation @ A 8 â&#x2C6;&#x2019; 2x x = ln . 3

1

A 8 â&#x2C6;&#x2019; 2xn to determine this root correct to 3 decimal places. 3 Give the result of each iteration to 5 decimal places. [3]

(iv) Use the iterative formula xn+1 = ln

@

[2]

9709/22/M/J/13


7

(a) Find the exact area of the region bounded by the curve y = 1 + e2x−1 , the x-axis and the lines x = 12 and x = 2. [4] (b)

y

M 1 2p

O

The diagram shows the curve y = exact x-coordinate of M .

8

x

e2x for 0 < x < 12 0, and its minimum point M . Find the sin 2x [5]

(i) Prove the identity 1  cosec x. sin x − 60Å + cos x − 30Å

3

(ii) Hence solve the equation 2 = 3 cot2 x − 2, sin x − 60Å + cos x − 30Å for 0Å < x < 360Å.

[6]

9709/22/M/J/13


1

Solve the equation | 2x − 7 | = 1, giving answers correct to 2 decimal places where appropriate.

[5]

2

Solve the equation ln(3 − 2x) − 2 ln x = ln 5.

[5]

3

(i) Show that 12 sin2 x cos2 x ≡ 32 (1 − cos 4x). (ii) Hence show that ã

4

5

1π 3 1π 4

[3]

√ π 3 3 12 sin x cos x dx = + . 8 16 2

2

The polynomial ax3 − 5x2 + bx + 9, where a and b are constants, is denoted by p(x). It is given that (2x + 3) is a factor of p(x), and that when p(x) is divided by (x + 1) the remainder is 8. (i) Find the values of a and b.

[5]

(ii) When a and b have these values, factorise p(x) completely.

[3]

The parametric equations of a curve are

x = e2t , (i) Show that

y = 4t et .

dy 2(t + 1) = . dx et

[4]

(ii) Find the equation of the normal to the curve at the point where t = 0. 6

[3 ]

[4]

(i) By sketching a suitable pair of graphs, show that the equation

cot x = 4x − 2,

where x is in radians, has only one root for 0 ≤ x ≤ 12 π . (ii) Verify by calculation that this root lies between x = 0.7 and x = 0.9.

[2] [2]

(iii) Show that this root also satisfies the equation

x=

1 + 2 tan x . 4 tan x

[1 ]

1 + 2 tan xn to determine this root correct to 2 decimal places. 4 tan xn Give the result of each iteration to 4 decimal places. [3]

(iv) Use the iterative formula xn+1 =

9709/23/M/J/13


7

(i) Express 5 sin 2θ + 2 cos 2θ in the form R sin(2θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the [3] exact value of R and the value of α correct to 2 decimal places.

Hence (ii) solve the equation

5 sin 2θ + 2 cos 2θ = 4,

giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ , (iii) determine the least value of

1 as θ varies. (10 sin 2θ + 4 cos 2θ )2

9709/23/M/J/13

[5] [2]


1

2

3

(i) Find the first three terms when 2 + 3x6 is expanded in ascending powers of x.

[3]

(ii) In the expansion of 1 + ax 2 + 3x6 , the coefficient of x2 is zero. Find the value of a.

[2]

A curve has equation y = f x. It is given that f â&#x20AC;˛ x = 

1 6 + 2 and that f 3 = 1. Find f x. [5] x + 6 x

D C

3 k

O

B E

j 6

i

4 A

The diagram shows a pyramid OABCD in which the vertical edge OD is 3 units in length. The point E is the centre of the horizontal rectangular base OABC. The sides OA and AB have lengths of 6 units â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; and 4 units respectively. The unit vectors i, j and k are parallel to OA, OC and OD respectively.

4

â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; (i) Express each of the vectors DB and DE in terms of i, j and k.

[2]

(ii) Use a scalar product to find angle BDE.

[4]

(i) Solve the equation 4 sin2 x + 8 cos x â&#x2C6;&#x2019; 7 = 0 for 0Ă&#x2026; â&#x2030;¤ x â&#x2030;¤ 360Ă&#x2026;.

[4]

(ii) Hence find the solution of the equation 4 sin2

5

1    1 + 8 cos 12 1 â&#x2C6;&#x2019; 7 = 0 for 0Ă&#x2026; â&#x2030;¤ 1 â&#x2030;¤ 360Ă&#x2026;. 2

[2]

The function f is defined by f : x  â&#x2020;&#x2019; x2 + 1 for x â&#x2030;Ľ 0. (i) Define in a similar way the inverse function f â&#x2C6;&#x2019;1 .

[3]

. (ii) Solve the equation ff x = 185 16

[3]

9709/11/O/N/13


6

A

r

B

r O

a rad

D

E

C The diagram shows a metal plate made by fixing together two pieces, OABCD (shaded) and OAED (unshaded). The piece OABCD is a minor sector of a circle with centre O and radius 2r. The piece OAED is a major sector of a circle with centre O and radius r. Angle AOD is ! radians. Simplifying your answers where possible, find, in terms of !, 0 and r, (i) the perimeter of the metal plate,

[3]

(ii) the area of the metal plate.

[3]

It is now given that the shaded and unshaded pieces are equal in area. (iii) Find ! in terms of 0.

7

[2]

The point A has coordinates â&#x2C6;&#x2019;1, 6 and the point B has coordinates 7, 2. (i) Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx + c. [4] (ii) A point C on the perpendicular bisector has coordinates p, q. The distance OC is 2 units, where O is the origin. Write down two equations involving p and q and hence find the coordinates of the possible positions of C. [5]

8

x metres

r metres

The inside lane of a school running track consists of two straight sections each of length x metres, and two semicircular sections each of radius r metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres. (i) Show that the area, A m2 , of the region enclosed by the inside lane is given by A = 400r â&#x2C6;&#x2019; 0r2 . [4] (ii) Given that x and r can vary, show that, when A has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5] 9709/11/O/N/13


9

(a) In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Find the common difference and the first term. [5] (b) A geometric progression has first term a, common ratio r and sum to infinity 6. A second geometric progression has first term 2a, common ratio r2 and sum to infinity 7. Find the values of a and r. [5]

10

y

y = (3 â&#x20AC;&#x201C; 2x)3

(12 , 8) x

O

The diagram shows the curve y = 3 â&#x2C6;&#x2019; 2x3 and the tangent to the curve at the point

1 2

 ,8 .

(i) Find the equation of this tangent, giving your answer in the form y = mx + c.

[5]

(ii) Find the area of the shaded region.

[6]

9709/11/O/N/13


1

Given that cos x = p, where x is an acute angle in degrees, find, in terms of p, (i) sin x,

[1]

(ii) tan x,

[1]

(iii) tan 90Ă&#x2026; â&#x2C6;&#x2019; x.

[1]

2

O

8 cm

q rad O

6 cm

A

A Fig. 1

Fig. 2

Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm. The paper is cut from A to O and opened out to form the sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The angle of the sector is 1 radians. Calculate (i) the value of 1,

[4]

(ii) the area of paper needed to make the cone.

3

The equation of a curve is y = 

[2]

2 . 5x â&#x2C6;&#x2019; 6

(i) Find the gradient of the curve at the point where x = 2. (ii) Find Ă&#x201D; 

4

[3]

3

2 2 dx and hence evaluate Ă&#x201D;  dx. 5x â&#x2C6;&#x2019; 6 5x â&#x2C6;&#x2019; 6

[4]

2

Relative to an origin O, the position vectors of points A and B are given by â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OA = i + 2j

and

â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; OB = 4i + pk.

â&#x2C6;&#x2019;â&#x2C6;&#x2019;â&#x2020;&#x2019; (i) In the case where p = 6, find the unit vector in the direction of AB.

(ii) Find the values of p for which angle AOB = cosâ&#x2C6;&#x2019;1

9709/12/O/N/13

1 5

.

[3] [4]


5

y D

A (0, 8)

8y + x = 6

4 C

O

x

B (4, 0)

The diagram shows a rectangle ABCD in which point A is 0, 8 and point B is 4, 0. The diagonal AC has equation 8y + x = 64. Find, by calculation, the coordinates of C and D. [7] 6

y S (0, 12)

R

Q (x, y) T (16, 0)

O

P

x

In the diagram, S is the point 0, 12 and T is the point 16, 0. The point Q lies on ST , between S and T , and has coordinates x, y. The points P and R lie on the x-axis and y-axis respectively and OPQR is a rectangle.

7

(i) Show that the area, A, of the rectangle OPQR is given by A = 12x â&#x2C6;&#x2019; 34 x2 .

[3]

(ii) Given that x can vary, find the stationary value of A and determine its nature.

[4]

(a) An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile. (i) Given that the nth mile takes 9 minutes, find the value of n.

[2]

(ii) Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon. [2] (b) The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression. [4] [Questions 8, 9 and 10 are printed on the next page.]

9709/12/O/N/13


8

A function f is defined by f : x → 3 cos x − 2 for 0 ≤ x ≤ 20. (i) Solve the equation f x = 0.

[3]

(ii) Find the range of f.

[2]

(iii) Sketch the graph of y = f x.

[2]

(iv) State the maximum value of k for which g has an inverse.

[1]

(v) Obtain an expression for g−1 x.

[2]

A function g is defined by g : x → 3 cos x − 2 for 0 ≤ x ≤ k.

9

y C

y = 8x + 2x

A B

x

O The diagram shows part of the curve y =

x-coordinates 1, 2 and 5 respectively.

8 + 2x and three points A, B and C on the curve with x

(i) A point P moves along the curve in such a way that its x-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the y-coordinate of P is changing as P passes through A. [4] (ii) Find the volume obtained when the shaded region is rotated through 360Å about the x-axis. [6] 10

A curve has equation y = 2x2 − 3x.

(i) Find the set of values of x for which y > 9.

[3]

(ii) Express 2x2 − 3x in the form a x + b2 + c, where a, b and c are constants, and state the coordinates of the vertex of the curve. [4] The functions f and g are defined for all real values of x by

where k is a constant.

f x = 2x2 − 3x

and

g x = 3x + k ,

(iii) Find the value of k for which the equation gf x = 0 has equal roots.

9709/12/O/N/13

[3]


1

Solve the inequality x2 − x − 2 > 0.

2

A curve has equation y = f x. It is given that f ′ x = x

3

The point A has coordinates 3, 1 and the point B has coordinates −21, 11. The point C is the mid-point of AB.

[3] − 32

+ 1 and that f 4 = 5. Find f x.

[4]

(i) Find the equation of the line through A that is perpendicular to y = 2x − 7.

[2]

(ii) Find the distance AC.

[3]

4

C

10

k i 8

A

O

j 6

D

B

The diagram shows a pyramid OABC in which the edge OC is vertical. The horizontal base OAB is a triangle, right-angled at O, and D is the mid-point of AB. The edges OA, OB and OC have lengths −−→ −−→ of 8 units, 6 units and 10 units respectively. The unit vectors i, j and k are parallel to OA, OB and −−→ OC respectively.

5

−−→ −−→ (i) Express each of the vectors OD and CD in terms of i, j and k.

[2]

(ii) Use a scalar product to find angle ODC.

[4]

(a) In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio. [2] (b) In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference. [4]

9709/13/O/N/13


6

A

B

11

C

D 5c m

cm

a rad

O The diagram shows sector OAB with centre O and radius 11 cm. Angle AOB = ! radians. Points C and D lie on OA and OB respectively. Arc CD has centre O and radius 5 cm. (i) The area of the shaded region ABDC is equal to k times the area of the unshaded region OCD. Find k. [3] (ii) The perimeter of the shaded region ABDC is equal to twice the perimeter of the unshaded region OCD. Find the exact value of !. [4]

7

(a) Find the possible values of x for which sin−1 x2 − 1 = 13 0, giving your answers correct to 3 decimal places. [3] (b) Solve the equation sin 21 + 13 0 =

8

1 2

for 0 ≤ 1 ≤ 0, giving 1 in terms of 0 in your answers.

[4]

 4 (i) Find the coefficient of x8 in the expansion of x + 3x2 .

[1]

 5 (ii) Find the coefficient of x8 in the expansion of x + 3x2 .

[3]

  5 (iii) Hence find the coefficient of x8 in the expansion of 1 + x + 3x2 .

[4]

9

k2 + x, where k is a positive constant. Find, in terms of k, the values of x+2 x for which the curve has stationary points and determine the nature of each stationary point. [8]

10

The function f is defined by f : x → x2 + 4x for x ≥ c, where c is a constant. It is given that f is a one-one function.

A curve has equation y =

(i) State the range of f in terms of c and find the smallest possible value of c.

[3]

The function g is defined by g : x → ax + b for x ≥ 0, where a and b are positive constants. It is given that, when c = 0, gf 1 = 11 and fg 1 = 21. (ii) Write down two equations in a and b and solve them to find the values of a and b. [Question 11 is printed on the next page.]

9709/13/O/N/13

[6]


11

y

y = Ă&#x2013;(x 4 + 4x + 4)

â&#x20AC;&#x201C;1

The diagram shows the curve y =



O

x

x4 + 4x + 4 .

(i) Find the equation of the tangent to the curve at the point 0, 2.

[4]

(ii) Show that the x-coordinates of the points of intersection of the line y = x + 2 and the curve are [4] given by the equation x + 22 = x4 + 4x + 4. Hence find these x-coordinates. (iii) The region shaded in the diagram is rotated through 360Ă&#x2026; about the x-axis. Find the volume of revolution. [4]

9709/13/O/N/13


1

Solve the inequality  x + 1 < 3x + 5.

[4]

2

y

O

P

x

The diagram shows the curve y = x4 + 2x â&#x2C6;&#x2019; 9. The curve cuts the positive x-axis at the point P. (i) Verify by calculation that the x-coordinate of P lies between 1.5 and 1.6. (ii) Show that the x-coordinate of P satisfies the equation O@ A 9 3 â&#x2C6;&#x2019;2 . x= x (iii) Use the iterative formula

xn+1 =

_P 3

[2]

1

Q 9 â&#x2C6;&#x2019;2 xn

to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

3

4

The equation of a curve is y = 12 e2x â&#x2C6;&#x2019; 5ex + 4x. Find the exact x-coordinate of each of the stationary points of the curve and determine the nature of each stationary point. [6] (i) The polynomial x3 + ax2 + bx + 8, where a and b are constants, is denoted by p x. It is given that when p x is divided by x â&#x2C6;&#x2019; 3 the remainder is 14, and that when p x is divided by x + 2 the remainder is 24. Find the values of a and b. [5]

(ii) When a and b have these values, find the quotient when p x is divided by x2 + 2x â&#x2C6;&#x2019; 8 and hence solve the equation p x = 0. [4] 5

The parametric equations of a curve are for 0 â&#x2030;¤ 1 â&#x2030;¤ 0. (i) Show that

x = cos 21 â&#x2C6;&#x2019; cos 1,

y = 4 sin2 1,

8 cos 1 dy = . dx 1 â&#x2C6;&#x2019; 4 cos 1

[4]

(ii) Find the coordinates of the point on the curve at which the gradient is â&#x2C6;&#x2019;4.

9709/21/O/N/13

[4]


6

(a) Find e2x + 6 dx, e2x

[3]

(ii) Ó 3 cos2 x dx.

[3]

(i) Ô

(b) Use the trapezium rule with 2 intervals to estimate the value of Ô

2

1

6 dx, ln x + 2

giving your answer correct to 2 decimal places.

7

[3]

(i) Express 3 cos 1 + sin 1 in the form R cos 1 − !, where R > 0 and 0Å < ! < 90Å, giving the exact [3] value of R and the value of ! correct to 2 decimal places. (ii) Hence solve the equation

3 cos 2x + sin 2x = 2,

giving all solutions in the interval 0Å ≤ x ≤ 360Å.

9709/21/O/N/13

[5]


1

(i) Find Ô

2 dx. 4x − 1

(ii) Hence find Ô

7

1

[2]

2 dx, expressing your answer in the form ln a, where a is an integer. 4x − 1

[3]

2

e3x−1 has one stationary point. Find the coordinates of this stationary point. The curve y = 2x

[5]

3

Solve the equation 2 cot2 1 − 5 cosec 1 = 10, giving all solutions in the interval 0Å ≤ 1 ≤ 360Å.

[6]

4

(i) The polynomial ax3 + bx2 − 25x − 6, where a and b are constants, is denoted by p x. It is given [5] that x − 3 and x + 2 are factors of p x. Find the values of a and b.

(ii) When a and b have these values, factorise p x completely. 5

The parametric equations of a curve are

x = 1 + ït,

6

[2]

y = 3 ln t.

(i) Find the exact value of the gradient of the curve at the point P where y = 6.

[5]

(ii) Show that the tangent to the curve at P passes through the point 1, 0.

[3]

(a) Find Ó sin x − cos x2 dx.

(b)

[4]

(i) Use the trapezium rule with 2 intervals to estimate the value of Ó

10 2

10 4

cosec x dx,

giving your answer correct to 3 decimal places.

[3]

(ii) Using a sketch of the graph of y = cosec x for 0 < x ≤ 12 0, explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i). [2]

9709/22/O/N/13


7

y

R x

a

O

The diagram shows part of the curve y = 8x + 12 ex . The shaded region R is bounded by the curve and by the lines x = 0, y = 0 and x = a, where a is positive. The area of R is equal to 12 . (i) Find an equation satisfied by a, and show that the equation can be written in the form A O@ 2 − ea a= . 8 (ii) Verify by calculation that the equation a =

O@

2 − ea 8

A

has a root between 0.2 and 0.3.

5 [2]

a Q 2−e n (iii) Use the iterative formula an+1 = to determine this root correct to 2 decimal places. 8 Give the result of each iteration to 4 decimal places. [3]

_P

9709/22/O/N/13


1

Solve the inequality  x + 1 < 3x + 5.

[4]

2

y

O

P

x

The diagram shows the curve y = x4 + 2x â&#x2C6;&#x2019; 9. The curve cuts the positive x-axis at the point P. (i) Verify by calculation that the x-coordinate of P lies between 1.5 and 1.6. (ii) Show that the x-coordinate of P satisfies the equation O@ A 9 3 â&#x2C6;&#x2019;2 . x= x (iii) Use the iterative formula

xn+1 =

_P 3

[2]

1

Q 9 â&#x2C6;&#x2019;2 xn

to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

3

4

The equation of a curve is y = 12 e2x â&#x2C6;&#x2019; 5ex + 4x. Find the exact x-coordinate of each of the stationary points of the curve and determine the nature of each stationary point. [6] (i) The polynomial x3 + ax2 + bx + 8, where a and b are constants, is denoted by p x. It is given that when p x is divided by x â&#x2C6;&#x2019; 3 the remainder is 14, and that when p x is divided by x + 2 the remainder is 24. Find the values of a and b. [5]

(ii) When a and b have these values, find the quotient when p x is divided by x2 + 2x â&#x2C6;&#x2019; 8 and hence solve the equation p x = 0. [4] 5

The parametric equations of a curve are for 0 â&#x2030;¤ 1 â&#x2030;¤ 0. (i) Show that

x = cos 21 â&#x2C6;&#x2019; cos 1,

y = 4 sin2 1,

8 cos 1 dy = . dx 1 â&#x2C6;&#x2019; 4 cos 1

[4]

(ii) Find the coordinates of the point on the curve at which the gradient is â&#x2C6;&#x2019;4.

9709/23/O/N/13

[4]


6

(a) Find e2x + 6 dx, e2x

[3]

(ii) Ó 3 cos2 x dx.

[3]

(i) Ô

(b) Use the trapezium rule with 2 intervals to estimate the value of Ô

2

1

6 dx, ln x + 2

giving your answer correct to 2 decimal places.

7

[3]

(i) Express 3 cos 1 + sin 1 in the form R cos 1 − !, where R > 0 and 0Å < ! < 90Å, giving the exact [3] value of R and the value of ! correct to 2 decimal places. (ii) Hence solve the equation

3 cos 2x + sin 2x = 2,

giving all solutions in the interval 0Å ≤ x ≤ 360Å.

9709/23/O/N/13

[5]

A Level Maths Subsidiary 2003 to 2013 P1 & P2  

A Level Maths Subsidiary 2003 to 2013 P1 & P2 To Pass Your Cambridge Examination

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