X100/701 NATIONAL QUALIFICATIONS 2011

WEDNESDAY, 18 MAY 1.00 PM – 4.00 PM

Calculators may be used in this paper.

2.

3.

Full credit will be given only where the solution contains appropriate working.

LI

X100/701

6/8610

*X100/701*

1.

Express

13 − x in partial fractions and hence obtain x + 4x − 5 2

∫x

2

13 − x dx. + 4x − 5

5

( )

2.

Use the binomial theorem to expand 1 x − 3 2

3.

(a)

Obtain

4

dy when y is defined as a function of x by the equation dx

y + e y = x2. (b)

4.

(a)

(b)

3

Given f(x) = sin x cos 3x, obtain f′ (x).

(

For what value of λ is

(

1

2

−1

3

0

2

−1 λ

6

2α − β

2

For A = 3α + 2 β −1

−1

3 2

)

3

3

singular?

)

3 , obtain values of α and β such that 2

4

(

5.

−5 − 1

A′ = −1

4

−1

3

)

3 . 2

Obtain the first four terms in the Maclaurin series of

3

1 + x , and hence write

2 down the first four terms in the Maclaurin series of 1 + x .

Hence obtain the first four terms in the Maclaurin series of

[X100/701]

3

Page two

4 (1 + x)(1 + x2 ).

2

Marks

y

6.

(0, a) x

(–1, 0)

7.

The diagram shows part of the graph of a function f(x). Sketch the graph of –1 ⏐f (x)⏐ showing the points of intersection with the axes.

4

esin x (2 + x)3 for x < 1. 1− x Calculate the gradient of the curve when x = 0.

4

A curve is defined by the equation y =

n

8.

Write down an expression for

∑r

3

r =1

(∑ ) n

2

r

1

r =1

and an expression for n

∑r r =1

9.

3

+

(∑ )

2

n

r

3

.

r =1

Given that y > –1 and x > –1, obtain the general solution of the differential equation dy = 3(1 + y) 1 + x dx

5

[Turn over

[X100/701]

Page three

Marks 10.

Identify the locus in the complex plane given by ⏐z – 1⏐ = 3. Show in a diagram the region given by ⏐z – 1⏐ ≤ 3.

11.

12.

13.

14.

(a)

Obtain the exact value of

(b)

Find

x 1 − 49x 4

π

0

4

(secx – x)(secx + x)dx.

dx.

5

3

4

Prove by induction that 8n + 3n – 2 is divisible by 5 for all integers n ≥ 2.

5

1 The first three terms of an arithmetic sequence are a, , 1 where a < 0. a Obtain the value of a and the common difference.

5

Obtain the smallest value of n for which the sum of the first n terms is greater than 1000.

4

Find the general solution of the differential equation d 2 y dy − − 2 y = e x + 12 . 2 dx dx

Find the particular solution for which y = −

[X100/701]

Page four

3 dy 1 = when and . x = 0. 2 dx 2

7

3

Marks 15.

The lines L1 and L2 are given by the equations x −1 y z+3 x− 4 y+ 3 z+ 3 = = and = = , k −1 1 1 1 2

respectively. Find:

16.

(a)

the value of k for which L1 and L2 intersect and the point of intersection;

6

(b)

the acute angle between L1 and L2.

4

Define In = (a)

1

∫ (1 + x ) 0

1

2 n

dx for n ≥ 1.

Use integration by parts to show that In =

(b)

1 + 2n 2n

x2 dx. 2 n +1 0 (1 + x )

1

3

Find the values of A and B for which A B x2 + = (1 + x2 )n (1 + x2 )n + 1 (1 + x2 )n + 1

and hence show that In +1 =

( )

1 2n − 1 + In . n +1 2n n ×2 1

(c)

Hence obtain the exact value of

∫ (1 + x ) 0

1

2 3

5

dx.

[END OF QUESTION PAPER]

[X100/701]

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