CAPE - Pure Mathematics - Unit 22006

Paper 02

Section A - Module 1

1 a( ) If f (x) = x3 ln2 x show that

i( ) f1 x( ) x2 ln x. 3 ln x. 2( ).f1 x( ) x x x 5 marks

ii( ) f11 x( ) 6 x ln2. x 10 x ln. x. 2 xf11 x( ) x x x x x 5 marks

b( ) The enrolment pattern of membership of a country club follows an exponential logistic function N

N800

1 k e rt.kk ε R, r ε R

where N is the number of members enroled t years after the formation of the club. The initial membership was 50 persons and after one year there are 200 persons enroled in the club.

i( ) What is the LARGEST number reached by the membership of the club?

2 marks

ii( ) Calculate the EXACT value of k and r 6 marks

iii( ) How many members will there be in the club 3 years after its formation?

2 marks

a( ) i( )ddx

x3 ln2 x x3 2 ln. x.

xln2 x 3. x2d

dxx3 ln2 x x

xx

x x

2 x2 ln x. ln x. ln. x. 3. x2 x2 ln x. 3 ln x. 2( ).2 x2 ln x. ln x. ln. x. 3. x2 x x x

ii( )

ddx

x2 ln x. 3 ln x. 2( ). 2 x ln. x. 3 ln x. 2( ). x2 1x

3 ln x. 2( ). ln x.3x

.ddx

x2 ln x. 3 ln x. 2( ). x x x xx

x xx

2 x ln. x. 3 ln x. 2( ). x 3 ln x. 2( ). x 3. ln x. 6 x ln2. x 10 x ln. x. 2 x2 x ln. x. 3 ln x. 2( ). x 3 ln x. 2( ). x 3. ln x. x x x x x

1

b( ) i( )∞t

800

1 k e rt.lim 800

∞t

800

1 k e rt.lim

ii( ) t = 0 508001 k

50k

k80050

1 k 15

200800

1 15 e r200

rr ln 5.r ln 5.

iii( ) N800

1 15 e ln 5( ). 3( ). lnN

800

115125

yields5000

7

approx 714 members

2 a( ) i( ) Express 1 x

x 1( ) x2 1.in partial fractions 6 marks

ii( ) Hence find x1 x

x 1( ) x2 1.d 3 marks

b( ) Given that In0

1xxn exdn

n where n ε N

i( ) Evaluate I1 4 marks

ii( ) Show that In e n In 1.n Inn 4 marks

iii( ) Hence or otherwise evaluate I3 writing your answer in terms of e

3 marks

2

a( ) i( )1 x

x 1( ) x2 1expands in partial fractions to

1x 1( )

x

x2 1

ii( ) x1

x 1d

x

x2 1ln x 1( ). 1

2ln x2 1. ln A.

12

ln x2 1. Ax x

ln Ax 1

x2 1

..

b( ) i( )0

1xx ex. d x ex. 1

0

.0

1xex 1( ) d

0

1xx ex. d x x

I1 e e 1 I1 1

ii( ) In xn ex 1

0

.0

1xn xn 1. exdxn x n n

n In e n In 1.n nn

iii( ) I3 e 3 I2 I2 e 2 I1

I3 e 3 e 2 1( )( ) I3 6 2 e

3

Section B - Module 2

3 a( ) i( ) Show that the terms of

1

m

r

ln 3r.

=

are in arithmetic progression

3 marks

ii( ) Find the first 20 terms of this series 4 marks

iii( ) Hence show that

1

2 m

r

ln 3r.

=

2 m2 m ln 3.m2 m 3 marks

b( ) The sequence of positive terms {xn} is defined by xn 1 xn2 1

4xnn x1

12

<

i( ) Show by mathematical induction or otherwise that xn12

<

for all positive integers n 7 marks

ii( ) By considering xn 1 xn or otherwise show that xn xn 1< 3 marks

a( ) i( ) ln 3. 2 ln 3. 3 ln. 3. ... m ln. 3. T1 ln 3.T1 ln 3. d ln 3.d ln 3.

ii( ) Sm ln 3. 1 2 3 ... m( ). ... Smm2

m 1( ) ln 3.m 1

S20202

20 1( ) ln 3.202

20 1( ) ln S20 210 ln. 3.S20 210 ln. 3.

iii( ) S2 m2 m2

2 m 1( ) ln 3.m 1 S2 m 2 m2 m ln 3.m2 m

4

b( ) i( ) x112

< x2 u12 1

4u x2

12

<

assume that xn12

< for n = k then xk12

< for k ε N

xk 1 xk2 1

412

2 14

<x

by the assumption xk 112

< whenever xk12

<

ii( ) xn 1 xn xn2 1

4xnxn 1 xn xn xn xn 1 xn xn

12

2 14

14

xn 1 xn xn

xn 1 xn 0>

from (1) xn12

< hence xn 1 xn>

4 a( ) Sketch the functions y = sin x and y = x2 on the same axes 5 marks

b( ) Deduce that the function f (x) = sin x - x2 has EXACTLY two real roots 3 marks

c( ) Find the interval in which the non-zero root α of f (x) lies 4 marks

d( ) Starting with a first approximation of α at x1 = 0.7 use one iteration of the Newton-Raphson method to obtain a better approximation of α to 3 dec places

8 marks

5

a( )

sin x( )

x2

x

2 1 0 1 2

1

1

2

b( ) since there are only 2 intersections then only 2 real roots exist

c( ) f 0.5( ). 0.229f 0.5( ). f 1( ). 0.159f 1( ).

by the intermediate value theorem 0.5 < α < 1)

d( ) x2 0.7sin 0.7. 0.72

cos 0.7. 2 0.7( ).x2 0.7sin 0.7. 0.72

cos 0.7. 2 0.7( ). x2 0.943

6

Section C - Module 3

5 a( ) i( ) How many numbers lying between 3 000 and 6 000 can be formed from the digits 1, 2, 3, 4, 5, 6, if no digit is used more than once in forming the number?

5 marks

ii( ) Determine the probability that a number in 5 (a) (i) above is even 5 marks

b( ) In an experiment p is the probability of success and q is the probability of failure in a single trial. For n trials the probability of x successes and (n - x) failures is represented by

n C x (p)x q n-x n > 0

Apply this model to the following problem.

The probability that John will hit the target at a firing practice is56

He fires 9 shots.

Calculate the probability that he will hit the target

i( ) AT LEAST 8 times 7 marks

ii( ) NO MORE than seven times 3 marks

a( ) i( ) first digit is 3, 4 or 5 select first digit in 3 ways

select second digit in 5 ways

select third digit in 4 ways

select fourth digit in 3 ways

no of ways = 3 x 5 x 4 x 3 = 180 ways

ii( ) for an even number it must end with 2, 4 or 6

if 4 is the first digit - no of ways = 2 hence 1 x 4 x 3 x 2 = 24 ways

if 4 is not first digit - no of ways = 2 hence 2 x 4 x 3 x 3 = 72 ways

total = 96 ways

P even number.( ). 96180

P even number.( ). 2445

0.5332445

7

b( ) i( )

P X 8( ). P X 9( ). 9( )56

8 16

56

9P X 8( ). P X 9( ). 0.5427

ii( ) P X 7( ). 1 0.5427P X 7( ). 0.4573

6 a( ) If A

1

1

1

2

2

2

1

1

3

and B

2

1

0

1

1

1

1

0

1

i( ) find AB 3 marks

ii( ) deduce A 1 3 marks

b( ) A nursery sells three brands of grass-seed mix, P, Q and R. Each brand is made from three types of grass, C, Z and B. The number of kilograms of each type of grass in a bag of each brand is summarised in the table below.

Grass-seed mix Type of grass (kilograms)

C grass Z grass B grass

Brand P 2 2 6

Brand Q 4 2 4

Brand R 0 6 4

Blend c z b

A blend is produced by mixing p bags of Brand P, q bags of Brand Q and r bags of Brand R

i( ) Write down an expression in terms of p, q and r, for the number of kilograms of Z-grass in the blend

1 mark

8

(ii) Let c, z and b represent the number of kilograms of C-grass, Z-grass and B-grass respectively in the blend. Write down a set of THREE equations in p, q r, to represent the number of kilograms of EACH type of grass in the blend.

3 marks

iii( ) Rewrite the set of THREE equations in (b) (ii) above in the matrix form MX = D where M is a 3 by 3 matrix, X and D are column matrices

3 marks

iv( ) Given that M 1 exists write X in terms of M 1 and D 3 marks

v( ) Given that M 10.2

0.35

0.05

0.2

0.1

0.2

0.3

0.15

0.05

M 1

calculate how many bags of EACH brand, P, Q, and R, are required to produce a blend containing 30 kilograms of C-grass, 30 kilograms of Z-grass and 50 kilograms of B-grass.

4 marks

a( ) i( )

1

1

1

2

2

2

1

1

3

2

1

0

1

1

1

1

0

1

. yields

4

0

0

0

4

0

0

0

4

ii( ) AB 4 I. B 4 A 1. A1 14

2

1

0

1

1

1

1

0

1

.A1

1

1

1

2

2

2

1

1

3

1

yields

12

14

0

14

14

14

14

0

14

9

b( ) i( ) 2p + 2q + 6r = z

ii( ) 2p + 4q = c 2p + 2q + 6r = z 6p + 4q + 4r = b

iii( )

2

2

6

4

2

4

0

6

4

p

q

r

.c

z

b

2

2

6

4

2

4

0

6

4

p

q

r

.c

z

b

iv( )

p

q

r

M 1c

z

b

.M

c

z

b

v( )

p

q

r

0.2

0.35

0.05

0.2

0.1

0.2

0.3

0.15

0.05

30

30

50

. p = 3 q = 6 r = 2

10