Cape-applied-mathematics-past-papers-2005p1c.pdf

TEST CODE

FORM TP 2005278

02105013

MAY/JUNE 2005

CARIBBEAN EXAMINATIONS COUNCIL

ADVANCED PROFICIENCY EXAMINATION APPLIED MATHEMATICS PAPER 01 - OPTION C 2 hours ( 08 JUNE 2005 (p.m.))

This examination paper consists of THREE sections: Discrete Mathematics, Probability and Distributions, and Particle Mechanics. Each section consists of 5 questions. The maxi mum mark for each secti on is 40. The max imum mark for this examination is 120. This examination consists of8 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper unti l instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwi se stated in the question, all numerical answers MUST be given exactly OR to three significant figures as appropriate.

Examination Materials: Mathe matical formul ae and tables Electronic calculator Ruler and graph paper

Copyri ght © 2004 Caribbean Examinations Council All rights reserved. 02 105013/CAPE 2005

-2SECTION A DISCRETE MATHEMATICS Answer ALL questions.

I.

(a)

In the context of graph theory, di stinguish between a 'trail ' and a 'path' .

[2 marks]

(b)

D

E

Referring to the graph above, (i)

name a path between vertices A and G

[2 marks]

(ii)

state the degree of vertex G

[1 mark]

(iii)

state the total number of vertices in the graph

[1 ma rk]

(iv)

state the total number of edges in the graph.

[1 mark] Total 7 marks

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-32.

The circuit shown below contains three gates:

a NOT gate:

an OR gate:

==D-

an AND gate:

=D-

a

b

c

Write down the Boolean expression for the logic circuit and construct the corresponding truth table. Total 8 marks

3.

p, q and r are the propositions p: q: r: (i)

you get a Grade 1 in M athematics you do all logic questions you revise your work regularly

Using p, q and rand logical connectives, state the a)

converse

b)

contrapositive

c)

inverse

of the proposition p (ii)

Express p

~

q

A

~

q

A

r.

r as an English sentence.

[6 marks] [2 marks] Total 8 marks

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-44.

For every desktop and laptop model sold by a computer store, a profit of $400 and $600 is made respectively. In any month, a maximum of90 desktops and 70 laptops are supplied to the store. Since it takes 3 hours per desktop and 4 hours per laptop to set up software, a maximum of 400 hours is set aside each month for software installations. Defining any variables used, formulate a linear programming model which could be used to determine the maximum monthly profit from the computers. [You are not required to solve the problem.] Total9 marks

5.

A linear programming problem is defined by

7x + lly

Maximise Subject to

x+y ~20 x+2y~30

x~O,y~O

(a)

Sketch a graph to show the straight lines: X

+ y

=

20,

X

+ 2y

= 30,

X

=

0

and

y

=

0. [3 marks]

Name the given lines. (b)

Shade the region represented by x + y ~ 20, x + 2y ~ 30 , x

(c)

State the coordinates of the vertices of the shaded region.

~ 0,

and y ~ 0. [2 marks]

[3 marks] Total 8 marks

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-5-

SECTIONB PROBABILITY AND DISTRIBUTIONS Answer ALL questions.

6.

Theevents AandBare suchthat P(AIB) event that A does not occur. (a)

Find P(A).

(b)

S how that P(A n B)

(c)

Hence, calculate P(B).

+·

=

P(A')

=!

and P(A uB)

= ~· where A' is the [1 mark]

= _5_ .

[5 marks]

36

[2 marks] Total 8 marks

7.

(a)

A team of 6 persons is to be c hosen from 5 men and 6 women to work on a special project. Calculate the number of ways the team can be chosen so as to include more men than women. [4 marks]

(b)

A courier service has 2 new trucks in a fleet of 6 . On a particul ar day two trucks are used to deliver parcels. Given that the trucks are chosen randomly for any particular delivery, determine the probability that exactly one new truck is used. [4 marks]

Total 8 ma rks

8.

X and Yare two independe nt random variables s uc h that E(X Var(X) =7 and Var(Y) = 16. Determine

2

)

=16, E(Y) =2, E( Y =20 , 2

)

(a)

E(X)

[3 marks]

(b)

E(5X + 3Y)

[2 marks]

(c)

Var(2X- Y)

[3 marks] Total 8 marks

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- 69.

A drink company is planning to market a new drink. The financial department of the company made an analysis of the new drink and predicted that it would earn an annual profit of$3 mill ion if the sales are high, $ 1.1 million if the sales are average and lose $2 million if the sales are low. The financial department gave the probabilities of these three situations as 0.29, p, and 0.15 respectively. If X is the profit, in millions of do llars, earned per a nnum by the company f rom the new drink: (a)

Construc t the probability distribution table for X.

[2 m a rks]

(b)

Obtain the value of p.

[2 marks]

(c)

Calculate the mean and standard deviation of X.

[5 marks] Total 9 marks

10.

On average 6 identical independent trials of an experiment are needed to obtain a successful outcome. Let p denote the probability of a successful outcome and X the number of attempts needed to obtain a successful outcome. (a)

Name an appropriate distribution that may be used to model this situation, giving its paramete(s).

[2 marks] (b)

Calculate the value of p.

(c)

Find the probability that it takes at most two attempts to obtain the fLrst successful o utcome.

[2 marks]

[3 marks] Total 7 marks

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-7-

SECTIONC PARTICLE MECHANICS Answer ALL questions.

11.

Four conc urre nt forces acting o n a particle, not in equilibrium, are represented by

where i andj are the unit vectors in the directions of x and y axes respectively. (a)

De termine the resultant, R, of these fo rces.

(b)

In orde r for the syste m to be in equilibrium, a fifth force F5 =pi + qj is added . Find the values of p and q. [4 marks]

[3 marks]

Total 7 marks

12.

A block of weight WN lies on a rough pl ane inclined at

eoto the horizontal. The coeffic ie nt of friction

between the block and plane is J.1. A horizontal force of 10 N is required to make the block just s lide up the plane. Find the weig ht W N of the block whe n J.1 =+ and 8 =40°.

Total 9 marks

13.

A train, starting from rest, travels 9 km f rom s tatio n A to statio n B. The trai n takes 80 s to accelerate uniformly to a speed of 30 ms- 1• It the n travels at thi s speed fo r t seconds before dece le rating uniforml y at

!

m s-2 for the last 0.6 km and comes to rest.

(a)

Sketch a velocity-time graph to represent this information.

[3 marks]

(b)

Calculate the total time fo r the train to travel f ro m station A to statio n B.

[5 marks] Total 8 marks

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-8-

14.

A particle of mass 3 kg rests on arough plane which is inclined at sin- 1 ~ to the horizontal . This particle is connected by a light inelastic string passing over a light smooth pulley at the top of the plane, to another particle of mass 4 kg which is hanging freely. The coefficient of friction between the 3 kg particle and the plane is

!.

(a)

Draw a clear diagram showing the forces on each particle.

(b)

Frnd

[2 marks]

(i)

the acceleration of the system when it is released from rest.

[4 marks]

(ii)

the tension in the string.

[2 marks]

[Take g = 9.8 ms-2]

Total 8 marks

15.

(a)

A bullet of mass 0 .05 kg travelling horizontally at 90 ms- 1 passes through a stationary block of wood of mass 10 kg, and emerges horizontally at 30 ms- 1• Assuming that the block is free to move on the horizontal plane, find the speed of the block after the bullet passes through it. [4 marks]

(b)

A vehicle of mass 60 tonne is travelling at a constant speed of 30 ms- 1 up a hill inclined at sin- 1

3~ to the horizontal.

The resistance to motion of the vehicle is 600 N. Find the

power at which the engine is working. [Take g

[4 marks]

= 9.8 ms-2] Total 8 marks

ENDOFTEST

02105013/CAPE 2005