Mathcad - CAPE - 2000 - Math Unit 2 - Paper 01

7/29/2019 Mathcad - CAPE - 2000 - Math Unit 2 - Paper 01

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CAPE - 2000Pure Mathematics - Unit 2

Paper 01

Section A (Module 1)

1 Solve for x the equation e2 x 3 ex 2 0e2 x 3 ex 2 [6 marks]

ex 1 ex 2 0ex 1 ex 2 x = 0, x = ln 2

2 Find the derivative of the function f when

a( ) f x( ). x2 tan 3 x.f x( ). x x [3 marks]

b( ) f (x) = x ln x - x x > 0 [3 marks]

a( ) 2 x tan. 3. x 3 x 2 sec2 3 x

b( ) x1

x. ln x. 1 ln x

3 Expressx2

x 1( ) x 2 1.in partial fractions [6 marks]

x2

x 1( ) x2 1.expands in partial fractions to 1

2 x 1( ).( )12

x 1( )

x2 1

.

4 By means of the substitution u2 x 3u2 x show that

x2 x 1

x 3( )

3

2

d u414

u2dx

2 x 1

x 3( )

3

2

d u

2 du dx2 du dx u2 2 u 2 7

u2d u4

14

u2du

2 2 u 2 7

u2d u

1


2/7

5 The rate of decline of an insect population due to the application of a certain type of insecticide can be modelled by means of the differential equation

dx

dt

600

1 6 t

dx

dt t

where x is the number of insects alive after t hours after the application of the insecticide.

If there were 1000 insects initially calculate

a( ) the number of insects alive after 24 hours [4 marks]

b( ) how long the population of insects will survive [2 marks]

a( )

1000

xx1 d

0

24

t600

1 6 td

1000

xx1 d

x 100 ln 1 6 t( ). 24

0

. 1000t x ~ 502

b( )1000

0x1 d

0

t

t600

1 6 td

1000

0x1 d

t

100 ln 1 6 t( ). 1000100 ln 1 6 t( ).

te10 1

6t ~ 3671 hours

Section B (Module 2)

6 a( ) Write down the first four terms of the series

1

20

r1( )

r 1r2

=[3 marks]

b( ) Express the infinite series 3 + 5 + 7 + 9 + ... in sigma notation [3 marks]

a( ) 1 4 9 16 ...

2


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b( )

1

r

2 r 1( )

=

or

0

r

2 r 3( )

=

or

2

r

2 r 1( )

=

7 Find the coefficient of x6 in the expansion of 1 3 x( ) 1 2 x( )9

as a series of ascending powers of x [6 marks]

1 3 x( ) ...4032 x 5 5376 x 6 ... 5376 12096( ) x 61 3 x( ) ...4032 x 5 5376 x 6 ... x 6720 x 6

8 Find the sum of the convergent geometric series

S 21

2

1

8

1

32......

S2

11

4

yields8

3

9 a( ) Show that the equation 2 x3 6 x 1 02 x3 6 x 1 has a root between 0 and 1

[3 marks]

b( ) A first approximation to is 0.5. Find a second approximation to the root to onedecimal place

[4 marks]

a( ) f 0( ). 1f 0( ). f 1( ). 7f 1( ). continuous, by IVT root in [0, 1]

b( ) 0.5 2 0.5( )3

6 0.5( ) 16 0.5( ) 2 6

yields 0.2

3


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10 The true length and breadth of a rectangular lawn are known to be 50 metres and 35metres respectively. If the true length is measured as 50.3 metres and the breadth as 35.5metres calculate

a( ) the error in the area of the lawn [3 marks]

b( ) the percentage error in the area correct to one decimal place [2 marks]

a( ) area min 50.25( ) 35.45( )min yields area min 1781.3625min

area max 50.35( ) 35.55( )max yields area max 1789.9425max

area true 50( ) 35( )true yields area true 1750true

error area1781.3625 1750( ) 1789.9425 1750( )

2area

yields 35.6525 m 2

b( ) error percent35.6525

1750100( )percent yields 2.0 %

Section C (Module 3)

11 a( ) At a football match there are four ticket offices and each can accommodate 75people standing in line. An official observes the number of people standing in thelines at random times

Determine the sample space for this experiment [2 marks]

b( ) One morning there were 39 cars and 15 trucks lined up on a certain highway at apolice spot-check

What is the probability that the police will search a car and then a truck? [4 marks]

a( ) let ticket offices be {1, 2, 3, 4}

S = (1, 0), (1, 1), (1, 2), ... , (1, 75)(2, 0), (2, 1), (2, 2), ... , (2, 75)(3, 0), (3, 1), (3, 2), ... , (3, 75)(4, 0), (4, 1), (4, 2), ... , (4, 75)

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b( ) P car 1 and truck 2.. 39

54

38

53P car 1 and truck 2

.. yields247

4770.518=

12 a( ) Three beads are chosen at random with replacement from a box containing 4 red 5green and 3 blue beads. What is the probability that the three beads chosen arenot of the same colour?

[3 marks]

b( ) There are seven participants for a certain contest. How many differentcombinations of winner and runner-up are possible?

[2 marks]

a( ) P same colour.( ).4

12

3 5

12

3 3

12

3

P same colour.( ). yields1

8

P not same. colour.( ). 11

8P not same. colour.( ). yields

7

8

b( ) one winner and one runner-up from seven =

7 C. 2.7 !

7 2( ) ! 2 !7 C. 2. yields 21

13 Use the tree diagram given below to find the probabilities

a( ) P ( A ^ B ) [3 marks]

b( ) P ( A U B ) [4 marks]

P(B |A) = 1/4

P(A) = 1/3_

P(B |A) = 3/4__ P(B |A) = 1/5

P(A) = 2/3_ _

P(B |A) = 4/5

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a( ) P ( A ^ B ) = P(A) x P(B/A) = 1/12

b( ) P A. P B given. A.. P A U. B.( ).P A. P B given. A.. P U B

P (A U B) = 12

3

4

5yields

7

15

14 The population P (t) (in thousands) of a city is given by

P t( ). 1000 1 0.04( ) t 0.003( ) t2P t( ).

with t measured in years and t = 0 corresponding to the year 1980. Find

a( ) the rate of change of P (t) in the year 1985 [3 marks]

b( ) the average rate of change of P (t) from 1983 to 1993 [3 marks]

a( )dP

dt 1980 19851000 0.04( ) 2 0.003( ) 5( )( )

dP

dtyields 70

b( )dP

dt 1983 1993

1000 0.04( ) 2 0.003( ) 10( )( )

10

dP

dtyields 10

15 An insurance company estimates that a certain model car after x years depreciates at therate of 8x % of its value $c

a( ) Determine in terms of c the estimated value V of this model car after 2 years

[3 marks]

b( ) A businessman bought a new car of this model fifteen months ago for $40 000.What will be the value of his car at this time as calculated by the insurancecompany?

[3 marks]

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a( )dV

dx0.08 x c.

dV

dxx c

c

VV1 d

0

2x0.08 x c. d

c

VV1 d c

V 0.04 x 2 c 2

0

.x c V = $ 0.84c

b( )40000

VV1 d

1.25

2x0.08 x 40000( ) d

40000

VV1 d

V 1600 x2

21.25

.40000x V = $36 100

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Mathcad - CAPE - 2000 - Math Unit 2 - Paper 01