AH_Mathematics_all_2011

X100/701 NATIONAL QUALIFICATIONS 2011

WEDNESDAY, 18 MAY 1.00 PM – 4.00 PM

MATHEMATICS ADVANCED HIGHER

Read carefully 1.

Calculators may be used in this paper.

2.

Candidates should answer all questions.

3.

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

LI

X100/701

6/8610

*X100/701*

©

Marks Answer all the questions.

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

and simplify your answer.

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

expressing your answer in the form y = f(x).

5

[Turn over

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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 = −

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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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