Cape-applied-mathematics-past-papers-2005p1a.pdf

TEST CODE

FORM TP 2005162

02105011

MAY/JUNE 2005

CARIBBEAN EXAMINATIONS COUNCIL

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

( 08 JUNE 2005 (p.m.))

This examination paper consists of THREE sections: Discrete Mathematics, Probability and Distributions, and Statistical Inferences. 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 of7 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 otherwise stated in the question, all numerical answers MUST be given exactly OR to three significant figures as appropriate.

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

Copy1ight © 2004 Caribbean Examinations Council All rights reserved. 021050 11/CAPE 2005

-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)

1f-----------,.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]

(lii)

state the total number of vertices in the graph

[1 mark]

(iv)

state the total numbe r 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:

=0-

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 I 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 ::::> q A r . (ii)

Express p ::::> q

A

r as an English sente nce.

[6 marks] [2 marks] Total 8 marks

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-4-

4.

For every desktop and laptop model sold by a computer store, a profit of $400 and $600 is m ade respectively. In any month, a m aximum of90 desktops and 70 laptops are supplied to the store. Since it takes 3 hours per desktop and 4 hours pe r laptop to set up software, a maxi mum of 400 hours is set aside each month for software installations. Definjng any variables used, formulate a linear prograrnmjng model which could be used to determine the maxjmum monthly profit from the computers. [You are not required to solve the problem.]

Total9 marks

5.

A linear prograrnrillng problem is defined by

Maxjmjse

7x+ lly

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.

x + y ~ 20, x + 2y ~ 30, x ~ 0,

(b)

Shade the region represented by

(c)

State the coordinates of the vertices of the shaded region.

and

y ~ 0. [2 marks]

[3 marks] Total 8 marks

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-5-

SECTIONB PROBABILITY AND DISTRIBUTIONS Answer ALL questions.

6.

The events A and Bare such that ?(AlB) = event that A does not occur. (a)

Find P(A).

(b)

Show that P(A n B)

(c)

He nce, calculate P(B).

+·

P(A ' )

= ~

and P(Au B)

= ~,

where A' is the

[1 mark]

= ~ .

[5 marks] [2 marks] TotalS marks

7.

(a)

A team of 6 persons is to be chosen 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 inc!ude more men than women. [4 marks]

(b)

A courier service has 2 new trucks in a fl eet of 6. On a particular day two trucks are used to deliver parcels. Given that the trucks are chosen randomly for any particular delivery, determi ne [4 marks] the probability that exactly one new truck is used. Total 8 marks

8.

X andY are two independent random variables such thatE(X 2 ) and Var(Y) = 16. Determine

= 16, E(Y) =2, E(Y 2) =20, Var(X) =7

(a)

E(X)

[3 marks]

(b)

E(SX + 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 million 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 dollars, earned per annum by the company from the new drink: (a)

Construct the probability distribution table for X.

[2 marks]

(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. Letp denote the probability of a successful outcome and X the number of atte mpts needed to obtain a successful outcome. (a)

Name an appropriate distribution that may be used to model this situation, giving its parameter(s). [2 marks]

(b)

Calculate the value of p.

(c)

Find the probability that it takes at most two attempts to obtain the first successful outcome. [3 marks]

[2 marks]

Total 7 marks

SECTIONC STATISTICAL INFERENCES Answer ALL questions.

11.

In a study conducted to assess the extent of contaminated meat pies be ing sold in a city, 110 meat pies prepared by vendors were randomly collected and examined. A total of 91 meat pies were found to be contaminated. Calculate a 95 % confidence interval for the true proportion of contaminated meat pies in the city.

Total 6 marks

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-712.

A distributor of cereal claims that the boxes of cereal he produces have a mean weight of 400 grams with a standard deviation of20 grams. Calculate the probability that40 randomly selected boxes of his cereal will have a mean weight exceeding 405 grams. TotalS marks

13.

A study c laims that the mean number of work-hours needed to assemble a machine is 120. A sample of 15 randomly selected machines was found to have a mean assembly time of 124 hours and standard deviation 5 hours.

This information is used to test whether the mean number of work-hours is different from 120, using a 5% significance leveL (a)

State suitable null and alternative hypotheses.

[2 marks]

(b)

Identify the rejection region.

[2 marks]

(c)

Calculate the value of test statistic.

[6 marks] Total10 marks

14.

On average 35 % of students pass an introductory Mathematics course. In 2004, 45% of the 225 students pass the course. Use a 10% level of significance to test whether the proportion of students passing the course has increased. (a)

State whether a one-tailed or two-tailed test is appropriate in this situation.

[1 mark]

(b)

Calculate the test statistic.

[5 marks]

(c)

State the decision rule.

[3 marks] Total 9 marks

15.

Fifty observations of a random variable X , yielded the following summarised data: u=2030

u 2 = 85 620.

Calculate an unbiased estimate of the population mean and standard deviation.

Total 7 marks

END OF TEST

02105011/CAPE 2005