Cape-applied-mathematics-past-papers-2005p1b.pdf

TESTCODE 02105012

FORM TP 2005277

MAY/JUNE 2005

CARIBBEAN EXAMJNATIONS COUNCIL

ADVANCED PROFICIENCY EXAMINATION APPLIED MATHEMATICS PAPER 01 - OPTION B 2 hours

( 08 JUNE 2005 (p.m.))

This examination paper consists of THREE sections: Discrete Mathematics, Particle Mechanics and Rigid Bodies, Elasticity, Circular and Harmonic Motion. Each section consists of 5 questions. The maximum mark for each section is 40. The maximum mark for this examination is 120. This examination consists of8 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwise stated in the question, all numerical answers MUST be give n exactly OR to three significant figures as appropriate.

Examination Materials : Mathematical formulae and tables Electronic calculator Ruler and graph paper

Copyright © 2004 Caribbean Examinations Council All rights reserved. 02105012/CAPE2005

-2-

SECTION A DISCRETE MATHEMATICS Answer ALL questions.

1.

(a)

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

[2 marks]

(b) A

~--------------~ c

G D

H 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 mark]

(iv)

state the total number of edges in the graph.

[1 mark] Total 7 marks

GO ON TO THE NEXT PAGE

021050 12/CAPE 2005

- 32.

The circuit shown below contains three gates:

a NOT gate:

an OR gate :

==D-

an AND gate:

=0--

a

b

c

Write down the Boolean expression for the logic circ uit and construct the corresponding truth table.

Total S marks

3.

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

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

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

converse

b)

contrapositive

c)

inverse

of the proposition p

Cu)

Express p

~

q

1\

~

q

1\

r.

r as an English sentence.

[6 marks] [2 marks] Total 8 marks

GO ON TO THE NEXT PAGE

02105012/CAPE2005

-4-

4.

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 mode l 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+ ll y

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 ~ 0,

a nd y ~ 0.

[2 marks] (c)

State the coordinates of the vertices of the shaded region .

[3 marks] Total 8 marks

GO ON TO THE NEXT PAGE 02105012/CAPE 2005

-5-

SECTIONB PARTICLE MECHANICS Answer ALL questions.

6.

Four concurrent forces acting on a particle, not in equilibrium, are represented by

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

[3 marks]

(a)

D etermine the resultant, R, of these forces.

(b)

In order for the system to be in equilibrium, a fifth force F 5 =pi+ qj is added. Find the values of p and q. [4 marks]

Total 7 marks

7.

A block of weight WN lies on a rough plane inclined at 8° to the horizontal. The coefficient of friction between the block and plane is J.l. A horizontal force of 10 N is required to make the block just slide up the plane. Find the weight WN of the block when J.l=} and 8 = 40°.

Total 9 marks

8.

A train, starting from rest, travels 9 km from station A to station B. The train takes 80s to accelerate uniformly to a speed of30 ms- 1• It then travels at this speed fort seconds before decelerating uniformly at

~ ms-2 for the last 0.6 km and comes to rest.

(a)

Sketch a velocity-ti me graph to represent this information.

[3 marks]

(b)

Calculate the total time for the train to travel from station A to station B.

[5 marks] Total 8 marks

GO ON TO THE NEXT PAGE

02105012/CAPE 2005

-69.

A particle of mass 3 kg rests on a rough plane which is inclined at sin-1

~ to the horizontal. This particle

is connected by a 1ight 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)

Find

[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

10.

(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 hori zontally 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 ~ 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] TotalS marks

GO ON TO THE NEXT PAGE 021050 12/CAPE 2005

-7-

SECTIONC RIGID BODIES, ELASTICITY, CIRCULAR AND H ARMONIC MOTION Answer ALL questions.

11.

A rig id uniform rod, AB, of le ngth 120 em and mass 2.4 kg, is freely hinged to a vertical wall at A . The rod is maintained in a horizontal pos ition by a fixed smooth support at C, whereAC =80 em. A force of 16 N is applied at Bin the direction shown in the diagram. 80 em

16N

(a)

Copy the diag ram, and show the forces acting at A and at C.

[2 ma rks]

(b)

Determine the magnitude of the reaction at A.

[7 marks]

[Take g

= 10 m s-2] Total 9 marks

12.

A fixed point A is a distance 8 l vertically above a fi xed point B. A particle of mass m is attached at 0 between A a nd B by two e lastic strings OA and OB so that A, 0 and Bare in a vertical line. The string OA has a m odulus of e lastic ity of 3 mg and a natural leng th of 2l. The string OB has a modulus of e lastic ity of 6 mg and a natural length of3l. IfOA =y, (a)

Write down, in terms ofy, land mg, expressio ns for the tensions in OA and OB.

[4 marks] (b)

Show that y =4l when the particle is in equilibrium.

[4 marks] Total 8 marks

GO ON TO THE NEXT PAGE 02 1050 12/CAPE 2005

•

<

-8-

13.

A light inelastic string of length a is fastened at one end to a fi xed point 0 with the string hanging vertically downwards. A particle P of mass m, that is attached at the other end, is given a horizontal velocity u. When OP makes an angle 8with the downward vertical, the particle has a velocity v and the tension in the string is T. (a)

(b)

Find expressions for

e

(i)

v in terms of u, a, g and

(ii)

Tin te rms of m, u, a, g and

[5 marks]

e.

[2 marks]

Show that the particle will perform motion in complete vertical circles if u 2 ~5ga.

[3 marks] Total10 marks

14.

A particle moving with simple harmonic motion has a period of 4nseconds, and its maximum speed is 1.5 ms- 1• Calculate (a)

the amplitude

[4 marks]

(b)

the maximum acceleration.

[2 marks]

Total6 marks

15.

The centre of mass of a solid right circular cone of homogeneous material is at a distance -1 h from the 4 base, where his the height of the cone. A body consists of a solid right circular cone of base radius r, and height 4 h, together with a solid right circular cylinder of the same base radius and the same height. The two are glued together so that the base of the cone coincides with one end of the cylinder. Find, in terms of h, the distance of the centre of mass of the body from the vertex of the cone.

Total 7 marks

ENDOFTEST

02105012/CAPE2005