Mathcad - CAPE - 2003 - Math Unit 2 - Paper 02

7/29/2019 Mathcad - CAPE - 2003 - Math Unit 2 - Paper 02

1/11

CAPE - 2003Pure Mathematics - Unit 2

Paper 02

Section A (Module 1)

1 a( ) i( ) Use the fact that ex 1

ex

ex

xto show that

d

dxe

x 1

ex

d

dxe

x

x[3 marks]

ii( ) Hence evaluate

xx2

exd [4 marks]

b( ) If x e3 t

sin t.t

t show thatd

2x

dt2

6dx

dt. 18 x 0

d2

x

dt2

6dx

dt. 18 x [8 marks]

c( ) i( ) Given that x 2yy

y > 0 express in terms of y

a( ) log2

x b( ) logx

2 [4 marks]

ii( ) Hence or otherwise solve the equation

log2

x 8 logx

2 2log2

xx

[6 marks]

a( ) i( )d

dx

1

ex

ex

1( ) ex

ex

2

d

dx

1

ex

x x

x

d

dx

1

ex

1

ex

d

dx

1

ex x

ii( ) xx2

ex

d x2

ex

xex

2 x( ) dxx2

ex

d xx

x

x2

ex

2 x ex

. x2 exd x

2e

x2 x e

x. 2 e

xK2 e

xKx

xx

x x

ex

x2

2 x 2 K

1


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b( )dx

dt3 e

3 tcos 3. t 3 e

3 tsin 3 t.

dx

dt

tt

tt 3 e

3 tcos 3. t sin 3. t( )

d

2

xdt

23 e 3 t 3 sin 3. t 3 cos 3. t( ) 9 e 3 t cos 3. t sin 3. t( )d

2

xdt

2t t t t t t

9 e3 t

sin 3. t 9 e3 t

sin 3. t 9 e3 t

cos 3. t 9 e3 t

cos 3. t 18 e3 t

cos 3 t.9 e3 t

sin 3. t 9 e3 t

sin 3. t 9 e3 t

cos 3. t 9 e3 t

cos 3. tt

t

6dx

dt. 18 e

3 tcos 3. t 18 e

3 tsin 3 t.6

dx

dt.

tt

tt

18 x 18 e3 t

sin 3 t.18 xt

td

2x

dt2

6dx

dt. 18 x 0

d2

x

dt2

6dx

dt. 18 x

c( ) i( ) a( ) log2

2y

ylog2

2y

y log2

x ylog2

x y

b( ) logx

21

log2

xlog

x2

xlog

x2

1

ylog

x2

y

ii( ) log2

x 8 logx

2 2 0log2

x 8 logx

2 2 log2

x8

log2

x2 0log

2x

8

log2

x2

y2

2 y 8 0y2

2 y 8 y 4( ) y 2( ) 0y 4( ) y 2( )

y = 4, -2 x = 16 x1

4

2 a( ) Express in partial fractions1 x

2

x x2

1.[8 marks]

b( ) Use the substitution u = cos x to evaluate

xsin3

xd [7 marks]

2


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c( ) The rate of increase of a population of insects is directly proportional to the size ofthe population at time t given in days. Initially the population is p and it doubles itssize in 3 days

i( ) Show that p p0

ekt

pkt

where p is the size of the population after t days

and k1

3ln 2.

1

3ln [7 marks]

ii( ) Find the proportional increase in population at the end of

a( ) the first day b( ) the second day [3 marks]

a( )1 x

2

x x2

1

A

x

Bx C

x2

1

Bx CBx1 x

2A x

21. Bx C( ) x( )Bx CBx

A = 1 B = -2 C = 01

x

2

x2

1

b( ) du = - sin x dx xsin2

x sin x.( ). d u1 u2

dxsin2

x sin x.( ). d u

u1

3u

2K

1

3cos

3x cos x. Kcos x. K

Alternatively: x1 cos2

x sin x. d xsin x. cos2

x sin. x. dx1 cos2

x sin x. d

xcosn

xsin x. d1

n 1cos

n 1x Kx K

n

n

I1

3cos

3x cos x. Kcos x. K

c( ) i( )dP

dtkP

dP

dtkP ln P. kt Ckt Ckt t 0 P p

0p C ln p

0.p

ln P. ln p0

. ktln P. ln p0

. kt P p0

ekt

pkt

3


4/11

t = 3 P 2 p0

p2 p

0

p0

e3 k

2 p0

p0

kk

1

3ln 2.

1

3ln

ii( ) t = 1 P p0

e

1

3ln 2. 1( ).

p P p0

2

1

3.p

proportional increase = 2

1

3

t = 2 P p0

2

2

3.p proportional increase = 2

2

3

Section B (Module 2)

3 a( ) Three sequences are given below

1 1, 4, 7, 10, ...

2 11

4

1

7

1

10...

3 1( )1

1( )4

1( )7

1( )10

...

Determine which of the sequences is divergent, convergent or periodic and state

which of these sequences is an arithmetic sequence

[12 marks]

b( ) i( ) Find the nth term of the series 1 (2) + 2 (5) + 3 (8) + ... [3 marks]

ii( ) Prove by mathematical induction that the sum to n terms of the series in(b) (i) above is

n2

n 1( ) [10 marks]

a( ) 1( ) un

3 n 2nn

divergent / arithmetic d 3

2( ) un

1( )n 1

3 n 2

n

nnconvergent u

ntends to. o.

3( ) un

1( )3 n 2n

nperiodic of period 2

4


5/11

b( ) i( ) nth term is n (3n -1)

ii( ) Sn

n2

n 1( )n nn

S1

2 S2

12 true for n = 1, n = 2

assume true for Sk

n = k

Sk

k2

k 1( )k kk

Sk 1

k2

k 1( ) k 1( ) 3 k 2( )k k k k k

Sk 1

k 1( ) k2

3 k 2.k k kk

Sk 1

k 1( )2

k 2( )k kk

Sk 1

is of the form for Sk

for (k+1)th term

since Sk

is true for k = 1, k = 2 Sk 1

is true for all k N

Sn

n2

n 1( )n nn

4 a( ) i( ) Show that the series

loga

b loga

bc( ) loga

bc2

... loga

bcn 1

where a, b, c > 0 n 1 is an arithmetic progression whose sum

Sn

to n terms isn

2log

ab

2c

n 1[6 marks]

ii( ) Find Sn

when n = 6 and a = b = c = 5 [2 marks]

b( ) Given the series1

2

1

24

1

27

1

210

...

i( ) show that the series is geometric [3 marks]

ii( ) find the sum of the series to n terms [3 marks]

iii( ) show that as n approaches infinity the sum of the series approaches4

7

[2 marks]

5


6/11

c( ) i( ) Write down and simplify as far as possible the first three terms in theexpansion of

1 ux( ) 2 x( )4

[7 marks]

ii( ) Given that the coefficient of x2

is 0 find

a( ) the value of u

b( ) the coefficient of x [2 marks]

a( ) i( )

loga

b loga

b loga

c loga

b 2 loga

c ... loga

b n 1( ) loga

c

un

un 1

n 1( ) loga

c loga

b n 1( ) loga

c loga

bun

un 1

na

ca

b na

ca

b

loga

c constantloga

c constant common difference = loga

c

T1

loga

ba

b

Sn

n

22 log

ab n 1( ) log

ac

na

b na

cn

Sn

n

2log

ab

2c

n 1na

b cn

n

ii( ) S6

3 log5

52

52

.. log5

52

52

. S6

3 log5

57

log S6

21

b( ) i( )1

2

1

23

1

26

1

29

... T1

1

2r

1

23

ii( ) Sn

1

21

1

23

n

11

23

n

nS

n

1

2

23

7. 1

1

23 nnn

4

71

1

23 n

iii( )

n

4

71

1

23 n

lim4

7n

4

71

1

23 n

lim

6


7/11

c( ) i( ) 1 ux( ) 24

4 2( )3

x( ) 6 2( )2

x( )2

... =...ux

1 ux( ) 16 32 x 24 x

2

... 16 32 16 u( ) x 24 32 u( ) x

2

......u u

ii( ) 24 32 u 024 32 u u3

4coeff x = 20

Section C (Module 3)

5 a( ) A bag contains 5 red balls, 7 black balls and 4 white balls. Two balls are drawn atrandom without replacement

i( ) Draw a tree diagram to represent the different ways in which the balls can bedrawn from the bag with the branches showing the probabilities of drawingthe red, black or white ball

[5 marks]

ii( ) Find the probability of drawing two balls of the same colour [4 marks]

iii( ) Find the probability of drawing 1 red and 1 black ball [3 marks]

b( ) A journalist reporting on criminal cases classifies crime by age (in years) of thecriminal and whether the crime is violent or non-violent. The table below shows that150 criminal cases were classified in a particular year

Type of crime Age (in years)

Under 20 20 to 39 40 or older

Violent 27 41 14

Non-violent 12 34 22

The journalist randomly selects a criminal case for reporting

i( ) What is the probability of selecting a case involving a violent crime?

[2 marks]

ii( ) What is the probability of selecting a case where the crime was committed bysomeone less than 40 years old?

[3 marks]

7


8/11

iii( ) What is the probability of selecting a case that involved a violent crime or anoffender less than 20 years old?

[3 marks]

iv( ) Given that a violent crime is selected what is the probability the crime wascommitted by a person under 20 years old?

[2 marks]

v( ) Two violent crimes are selected for review by a judge. What is the probabilityboth are violent crimes?

[3 marks]

a( ) i( )

P (red) = 4/15

P (red) = 5/16 P (black) = 7/15P (white ) = 4/15

P (red) = 5/15P (black) = 7/16

P (black) = 6/15

P (white) = 4/15

P (red) = 5/15

P (white) = 4/16 P (black) = 7/15

P (white) = 3/15

ii( ) P (2 red) or P (2 black) or P (2 white)

5

16

4

15

7

16

6

15

4

16

3

15

37

120

5

16

4

15

7

16

6

15

4

16

3

15

iii( ) P (1 red and 1 black) = P R1

B2

. P B1

R2

. 25

16.

7

15P R

1B

2. P B

1R

2.

7

24

b( ) i( )82

1500.547

82

150ii( )

39

150

75

1500.76

39

150

75

150

iii( )82

150

39

150

27

1500.627

82

150

39

150

27

150iv( )

27

820.329

27

82v( )

82

150

81

1490.297

82

150

81

149

8


9/11

6 a( ) Show that eln y.

yeln y.

y for y real and positive [2 marks]

b( ) A disease spreads through an urban population. At time t the proportion of thepopulation who have the disease is p where 9 < p ( 1.

The rate of change of p with respect to time is proportional to the product of p and (1- p)

i( ) Form a differential equation in terms of p and t which models the situationdescribed above

[2 marks]

ii( ) Solve the differential equation [8 marks]

iii( ) Given that when t = 0 p1

10and when t = 2 p

1

5

show that 9 p1 p

32

t

9 p1 p

[10 marks]

iv( ) Find p when t = 4 [3 marks]

a( ) eln y.

celn y.

c taking logs both sides

ln y. ln e.( ). ln c.ln y. ln e.( ). c y = c eln y.

yeln y.

y

b( ) i( )dp

dtkp 1 p( ).

dp

dtkp

ii( ) p1

p 1 p( ).d tkdp

1

p 1 p( ).d k

p1

p

1

1 pd kt Cp

1

p

1

1 pd kt C

p

1 pA e

kt.

p

1 p

kte

CAe

C

p

A ekt

.

1 A ekt

.1 ekt

kt

kt

iii( ) t = 0 p1

10

1

10

A

1 A1 AA

1

9

9


10/11

t = 2 p1

5

1

5

A e2 k

.

1 A e2 k

.

1

5

k

ke

2 k 1

4 Ae

2 k

k 12

ln 94

.12

ln k ln 32

.k ln 32

.

p

1

9e

ln3

2

. t( )

11

9

ln3

2

. t( )

ln3

2

. t( )

p

1

9

3

2

t

11

9

3

2

tp

1

9

3

2

t

1

99

3

2

t

9 p p3

2

t.

3

2

t

9 p p3

2

t. 9 p

3

2

t

1 p( )9 p

9 p

1 p

3

2

t9 p

1 p

iv( ) t = 49 p

1 p

3

2

49 p

1 p

9 p

1 p

81

16

9 p

1 pp

9

25

10


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Documents

Mathcad - CAPE - 2003 - Math Unit 2 - Paper 02