Further Maths Sample Chapter

Contents 1 Number recall.............................................................4 1.1 Number: Recall and extension.......................................4 1.2 The product rule for counting........................................7 1.3 Manipulation of surds........................................................9

2 Algebra recall........................................................... 14 2.1 Recall of basic algebra....................................................14 2.2 Expanding brackets and collecting like terms.....................................................................................18 2.3 Factorising..........................................................................26

3 Geometry recall 1................................................... 34 3.1 Perimeter of compound shapes.................................34 3.2 Area of basic shapes........................................................35 3.3 Circumference and area of a circle.............................37 3.4 Volume of a cube, cuboid, prism and pyramid................................................................................39 3.5 Volume of a cone and a sphere...................................44

4 Geometry recall 2................................................... 48

10.2 Setting up equations..................................................143 10.3 S olving quadratic equations by factorisation or the quadratic formula...........................................146 10.4 Solving quadratic equations by completing the square..............................................152

11 Simultaneous equations..................................156 11.1 Simultaneous linear equations...............................156 11.2 L inear and non-linear simultaneous equations........................................................................161 11.3 U sing graphs to solve simultaneous linear equations............................................................163 11.4 U sing graphs to solve simultaneous equations, one linear and one non-linear...........165 11.5 S olving equations by the method of intersection....................................................................167 11.6 Algebraic solution of linear equations in

three unknowns...........................................................173

12 Inequalities..........................................................176 12.1 Linear inequalities.......................................................176 12.2 Quadratic inequalities................................................179 12.3 Graphical inequalities.................................................181

4.1 Special triangles and quadrilaterals..........................48 4.2 Angles in polygons..........................................................51 4.3 Circle theorems.................................................................55 4.4 Cyclic quadrilaterals........................................................58 4.5 Tangents and chords.......................................................60 4.6 Alternate segment theorem.........................................63

13 Coordinate geometry........................................188

5 Functions.................................................................. 66

14 The equation of a circle.....................................210

5.1 Function notation............................................................64 5.2 Domain and range...........................................................68 5.3 Sketching graphs of linear and quadratic functions..............................................................................70 5.4 Inverse functions..............................................................72 5.5 Sketching graphs of linear, quadratic and exponential functions ...................................................74 5.6 The significant points of a quadratic graph............80

6 Matrices..................................................................... 86 6.1 Introduction to matrices................................................86 6.2 The zero matrix and the identity matrix...................89 6.3 Transformations................................................................91 6.4 Combinations of transformations..............................94

7 Algebra...................................................................... 96 7.1 Manipulation of rational expressions.......................96 7.2 Use and manipulation of formulae and expressions................................................................99 7.3 The factor theorem........................................................101

8 Sequences...............................................................104 8.1 Number sequences........................................................104 8.2 The nth term of a sequence........................................105 8.3 The nth term of a linear sequence............................106 8.4 The nth term of a quadratic sequence....................108 8.5 The limiting value of a sequence as n → ∞...........110

9 Pythagoras’ theorem and trigonometry ........112 9.1 Pythagoras’ theorem.....................................................112 9.2 Trigonometry in ­right-​­angled triangles..................119 9.3 The sine rule and the cosine rule..............................132

13.1 Cartesian grids and straight-​­line graphs.............188 13.2 The equation of a straight line in the forms y = mx + c and y − y1 = m(x − x1)..................195 13.3 Parallel and perpendicular lines.............................202 13.4 Applications of coordinate geometry..................206 14.1 The equation of a circle centred on the origin (0, 0)........................................................210 14.2 The equation of a circle centred on any point (a, b)........................................................212 14.3 The equation of a tangent at a point on

a circle..............................................................................215

15 Indices...................................................................218 15.1 Using indices.................................................................218 15.2 Rules of indices.............................................................220 15.3 Negative indices...........................................................221 15.4 Fractional indices.........................................................223 15.5 Solving equations with indices...............................227

16 Calculus.................................................................230 16.1 The gradient of a curve..............................................230 16.2 More complex curves.................................................233 16.3 Stationary points and curve sketching................236 16.4 The equation of a tangent and normal at any point on a curve...............................237 16.5 The equation of a tangent and normal at any

point on a curve...........................................................242

17 Ratios of angles and their graphs..................246 17.1 Trigonometric ratios of angles between 90° and 360°................................................246 17.2 The circular function graphs....................................251 17.3 Special ­right-​­angled triangles.................................254 17.4 Trigonometrical expressions and equations......256

18 Proof......................................................................260

10 Solving equations..............................................140

18.1 Algebraic proof.............................................................260 18.2 Geometric proof...........................................................251

10.1 Solving linear equations............................................140

Answers......................................................................255

1 Number recall 1.1 Number: Recall and extension THIS SECTION WILL SHOW YOU HOW TO ... ✓✓understand and use the correct hierarchy of operations (BIDMAS / BODMAS) ✓✓understand and use decimals, fractions and percentages ✓✓understand and use ratio and proportion ✓✓understand and use numbers in index form and standard form ✓✓understand rounding and give answers to an appropriate degree of accuracy KEY WORDS ✓✓ratio ✓✓proportion ✓✓standard form

Most of the calculations in this chapter will require the combination of at least two basic number skills. For example, questions involving fractions will also test order of operations, a conversion of units may have several steps and a ratio problem may be combined with a percentage problem.

Questions may be answered with or without a calculator. If you use a ­non-​­calculator method, it is good practice to use a calculator to check your answers.

EXAMPLE 1 Work out 5 1 + 3 1 × 1 2 2 4 3 5 1 2 13 5 65 =5 3 ×1 = × = 12 12 4 3 4 3 11 1 2 1 5 = 11 ⇒ 1 10 5 + 3 × 1 = 10 5 +5 12 12 2 4 3 2 12

HINTS AND TIPS Remember from BIDMAS / BODMAS that multiplication is carried out before addition.

EXAMPLE 2 6. 5 + 6. 7 8. 9 – 2. 3 Put in brackets. (6.5 + 6.7) (8.9 – 2.3)

4

NUMBER RECALL

Work out

Working out the brackets first gives

HINTS AND TIPS

13. 2 132 or =2 6. 6 66

It is sensible to insert brackets before carrying out the calculation.

EXAMPLE 3 For every 3 boys in a class, there are 5 girls. There are 6 more girls than boys.

How many students are there in the class altogether?

The ratio of boys to girls is 3 : 5, giving 8 ‘parts’.

So the difference is 5 – 3 = 2 parts, which is equivalent to 6 students.

1 part is equivalent to 3 students.

There are 9 boys and 15 girls in the class, giving 24 students altogether.

EXAMPLE 4 x : y = 3 : 5 a) Write y in terms of x. b) Show that x is 30% of 2y. x 3 5x ⇒ 5x = 3y, y = a) = y 5 3 3 3 3 of 2y and = 30% b) x = y ⇒ x = 5 10 10

EXAMPLE 5 Calculate 43 - 82 ÷ 210 .

Give your answer as a mixed number.

HINTS AND TIPS

82 ÷ 210 = (23)2 ÷ 210 = 26 ÷ 210 1 1

= 2-4 = 4 = 16 2 15 1 43 = 63 16 16

Remember from ­BIDMAS / BODMAS that d ­ ivision is carried out before ­subtraction.

EXERCISE 1A

TARGETING GRADE 5

2

3

1 4

3 8

7 -2 d)

b) 3

3 2 + 4 5 3

2 e) 2 3 + 1 8

5

2

3

c) 5

5 2 -1 6 7

HINTS AND TIPS

f) 5

1 4 3 +4 -7 2 9 4

Use a calculator to check your answers.

TARGETING GRADE 6 2. Work out each of these. Give your answers as mixed numbers.

5 3 1 2 7 ÷2 b) 5 ÷ 16 c) 8 4 7 1 2 3 1 e) 5 4 + 4 ÷ 7 1 f) 2 - 1 ÷ 2 2 3 4 4 9 2 2 1 1 1 + 3 2 1 1 3 g) 3 + 3 ÷ - h) 2 6 i) 1 1 4 3 2 5 1 – 1– 2 6 4

1 1 a) 4 ×2 3 2 2 3 2 d) 2 + 1 × 3 8 5

(

)( )

()

HINTS AND TIPS Use a calculator to check your answers.

5

1 1 a) 4 +2

NUMBER: RECALL AND EXTENSION

1. Work out each of these. Give your answers as mixed numbers.

3. The ratio of A : B = 5 : 4 and the ratio of B : C = 3 : 7.

Work out the ratio of A : C. Give your answer in its simplest form.

4. z is 10% more than y. y is 20% more than x.

By what percentage is z more than x?

5. At a show, t­ wo-​­fifths of the people are men and the rest are women.

30% of the men are over 60 years old. 40% of the women are over 60 years old.

What proportion of all the people at the show are over 60?

6. Work out the number of minutes in January.

Give your answer in standard form.

7. In a school, there are equal numbers of boys and girls. At the end of the school year, 20% of the boys leave and 25% of the girls leave. There are now 450 girls in the school.

Work out the number of boys now in the school.

8. How many times is 2.56 × 1027 bigger than 5 × 105?

Give your answer in standard form.

9. A fuel tank is ­three-​­quarters full . Then 10% of the fuel is taken out.

There are now 54 litres in the tank.

How much will the tank hold when full?

10. The price of a loaf is increased by 14%. This week a shop sells 12% fewer loaves than last week.

Does the shop take more or less money from selling loaves this week?

You must show your working.

11. These three numbers are written in standard form.

8.1 × 105 5.7 × 106 2.3 × 107 a) Work out the smallest answer when two of these numbers are multiplied together. b) Work out the largest answer when two of these numbers are multiplied together.

Give your answers in standard form.

12. At the start of the year, a train is half full. Each week the number of

6

NUMBER RECALL

people catching this train increases by 1%.

How many weeks will it be before the train is full?

13. a : b = 1 : 4 a) W rite b in terms of a. a) Work out the ratio 3a + b : a + 5b, giving your answer in its simplest form.

14. Given that x% of 2y = 25, work out the value of y% of 3x.

14.3 The equation of a tangent at a point on a circle THIS SECTION WILL SHOW YOU HOW TO … ✓✓work out the gradient from the centre to a point on the circle ✓✓work out the gradient of a tangent to a circle ✓✓work out the equation of a tangent at a point on a circle.

KEY WORDS ✓✓tangent

Work out the equation of the tangent to the circle x2 + y2 = 25 at the point (3, 4).

The centre of the circle is at (0, 0).

4–0 4 = .

The gradient of the radius joining (0, 0) to (3, 4) is 3 – 0 3 3 The gradient of tangent is – (using the negative reciprocal) 4 3 The equation of the tangent at (3, 4) is y – 4 = – (x – 3) 4 which simplifies to 4y – 16 = –3x + 9

4y + 3x = 25

y

(3, 4)

O

x

215

EXAMPLE 7

14.3 The equation of a tangent at a point on a circle

A tangent to a circle is a straight line that touches the circle at a single point. The tangent is perpendicular to the radius of the circle at that point Given the equation of a circle it is possible to work out the equation of the tangent by following these steps.

Step 1: Write down the coordinates of the centre of the circle.

Step 2: Work out the gradient of the radius joining the centre to the difference in the y-coordinates

point on the curve, using gradient = difference in the x-coordinates Step 3: Work out the gradient of the tangent using the negative reciprocal of the gradient of the radius.

Step 4: Work out the equation of the tangent, using y –y1 = m(x – x1).

EXAMPLE 8 Show that the equation of the tangent to the circle (x – 1)2 + (y – 2)2 = 116 at the point (11, –2) is 2y – 5x – 59 = 0 The centre of the circle is at (1, 2).

The gradient of the radius joining (1, 2) to (11, –2) 2 – –2 4 2 = =– = 1 – 11 –10 5 5 The gradient of tangent is (using the negative reciprocal) 2 5 The equation of the tangent at (11, –2) is y – –2 = (x – 11).

2 which simplifies to 2y + 4 = 5x – 55.

So 2y – 5x – 59 = 0.

y

(1, 2) x

O

(11, –2)

EXERCISE 14C

TARGETING GRADE 8 1. Point A(4, 2) lies on the circumference of the circle x2 + y2 = 20. a) Find the gradient of the line segment joining A to the origin (0, 0). b) Find the gradient of the tangent to the circle at A. c) Find the equation of the tangent to the circle at A in the form y = mx + c.

2. Show that the tangent to the circle x2 + y2 = 73 at the point (–8, 3) is given by the equation 3y = 8x + 73.

3. Find the equation of the tangent in the form y = mx + c for these situations. a) Circle x2 + y2 = 34, tangent at (3, –5) b) Circle x2 + y2 = 40, tangent at (–2, –6) 2 2 2 c) Circle x + y = a , tangent at (p, q) 1 4. A circle centred at the origin has a tangent with equation y = – x + c at the point (3, 9).

3 a) Find the value of c. b) Find the equation of the circle.

5. P is a point (2, 3) on the circle x2 + y2 = 13.

Work out the equation of the tangent to the circle at P.

6. P is a point (–1, 4) on the circle x2 + y2 = 17.

Work out the equation of the tangent to the circle at P.

7. P is a point (3, 1) on the circle x2 + y2 = 10.

216

THE EQUATION OF A CIRCLE

Work out the equation of the tangent to the circle at P.

TARGETING GRADE 9 8. P is a point (6, 2) on the circle (x – 3)2 + (y – 5)2 = 18.

Work out the equation of the tangent to the circle at P.

9. P is a point (2, 0) on the circle (x + 4)2 + (y + 2)2 = 40.

Work out the equation of the tangent to the circle at P.

10.

P is a point (5, –2) on the circle (x – 3)2 + (y + 1)2 = 5.

Work out the equation of the tangent to the circle at P.

­Exam-​­style questions 1. The equation of a circle is x2 + y2 = 25. a) Write down the coordinates of the centre of the circle. b) Work out the radius of the circle.

3 c) Where does the line y = x cross the circle?

4

(1 mark) (1 mark) (3 marks)

2. a) A circle has its centre at the origin and passes through the point (6, -4).

Work out the equation of the circle.

(2 marks) b) Is the point (5, 5) inside or outside the circle?

Give a reason for your answer.

(2 marks) 3. The equation of a circle is (x - 2)2 + (y + 3)2 = c.

The line x = 5 is a tangent to the circle.

Work out the value of c.

(3 marks)

4. The diagram shows a circle and a straight line. y 3

6

0

x

P –3

a) Work out the equation of the circle. b) Work out the equation of the line. c) Find the coordinates of the point P, where the line and the circle cross.

( )

(2 marks) (2 marks) (3 marks)

6. A circle passes through the points (6, 0), (6, 10) and (0, 10).

Work out the equation of the circle.

(4 marks)

7. P is a point (4, 2 ) on the circle x2 + y2 = 20.

Work out the equation of the tangent to the circle at P.

(4 marks)

8. P is a point (5, 3) on the circle (x – 1)2 + (y + 2)2 = 41.

Work out the equation of the tangent to the circle at P.

(4 marks)

9. A circle has the equation x + y = 50. a) Find the coordinates of the two points, A and B, where the circle intersects the line with equation 2x + y = 15.

The tangents of the circle at A and B intersect at point C. b) Find the coordinates of point C. 2

2

(4 marks) (6 marks)

217

(2 marks) (2 marks)

EXAM-STYLE QUESTIONS

8

.

5. The circle x2 + y2 = 50 is translated with the vector –7 a) Find the equation of the new circle. b) Show that the new circle crosses one axis but not the other one.