.

~ 2 ~""'t
--1-
\
U~ S\l-\l\

Find. the set of values of k , for which the equation

{u) [2\ .-'
" '('i /0:2 _ 3x =:; k - 3 has real roots.

.: 7'

(b)
Values ofx and y are related by the equationy =:; Axb. State the type
of graph you would plot and what shape would confirm the relationship.
,

2 Express
2 f63 il~ the form a + b JZ ,
'J-J7 \.1\ ./
.where a, hand c are iruegers.
.,'

.3 Estimate the value of

f \1
COS(X2 }lx
~ng thetrapeziul11 rule with 5 equal intervals,givin~ the answer corn:ct
to 3 significant tigures. 1:1\
.-----;:i'he population or
a country increases by 3% every year. Find the time,
in years. it takes for the population to double. [-1-)

~y=e'~'i
~=---
-- t
I
i
\~ ,. I

_A~~_' ---------{~-O------------~\----P~
'The diagram above shows the ghph ot), = eX + 1. The region bel\Wcn
the curve,
_. the x - axis. the lines J: "" - 1 andx -=: 1 is rotated throuch
~ - 4 t;i\2.ht
~
angles about the x - <:IX1:)'. Find the-votume at this solid giving the unsv,,-cr
correct to 2 decimal plan::s . ./

(, Skdcl1. on the same <lX'CS, the graphs of \. -------- -,

, r = \3.\1,- 61,I and}' = LX i 21
, • •
->J.

,
Hence; or otherwise. solve the inequality
I :i I /
" 2x + .2 > \3x - 6\.

'f

-:./

,I '/
 .j' , )
7 Given that f(x);;:; 2x + (IX" + bx' - 5x + 0 has both CL+ 2) and (x -3)
as factors, determine the value of a and of b.
l51

A sequence V, is defined by
UI",= (11- 3r).

, (i) Write down the first 3 terms or the sequence.

,~
"..,. .\ II1
, '

~.
(ii) Find in terms of (1 a formula for
, ..•.,

.
1 ,\ .,,::'

I l 1'=11
f 51 -, \ "\ ~ ..f... '.
.'

~!)
@ Differentiate

, (6X
xsin + ff)
4) with respect to .v.
l~ f

~ Fvaluate exactly

10

0) the value ofxv and an estimate of the relative error in

calculnting this quantity. t ~ J
'jj
I ,\ I

I ,
(ii) the value of \' - - giving your ans\ver. \yith error hounds .)'

'x
to one significant figure. to appropriate accuracy.
....q'
f ..

1I Giventhat ./(x) = tn(2-.'1x).

(i) showthat

.) I
1 -I

(ii) find F" (x).

" "

!knee obtain a Maclaurins series expansion fur Ptt{:2 - 3.'().

neglecting terms in x~and higher degree. q 1',1
I

i -'

Siait: dearly' wh.u circumstances might justify' 11l'gkcting these terms. !I!

~ IT u rn (Pi IT

(1 l- )

-
 .-', ---.... ~-.-----~-.....~
-~~~
..--,...,.-".---.---
~-.---
-_ _._.
..
-. -

l
'12
Cf;'
If r( x) = 3 cos 2x - 2 sin x for 0° ::s;x < 360°, use the substitution u = sin x
to show that f(x) may be expressed as 3 - 2u - 6i/ . [2J
, Hence find

(i) the exact maxim~m value of f(x) ,

..",t.· '.
[31

- (ii) the values ofx for whichj(x) =].

13 Sketch on the same axes, the graphs of/

y = eJx and y = 2'( + J O. l2J
(i) Show that the equation
}r
e -2x-l0=0

has a root between - 5 ancl- 4. 1~1

(ii) State a pair of consecutive integers between which another
root lies. 111
(iii) Taking XI = - .4,.use the Newton-Raphson method once (0
find a second approximation X2, giving your answer correct to
" decimal
. -. ~ ~nlace:
..~ o , [JI
-- -.... - ------
Fans arrive at the gate of a soccer stadium at a rate inversely proportional
to the time remaining before the gate is opened. The number offans at the
gate. t minutes before it is opened is x. 10 minutes before the opening, there
are 100 fans, and the rate of arrival is 50 per minute,

Show that this situation can be modeled by the differential equation

dx 500
Lit
= [3]

lind

-~ the general solution of this equation, [21

(ii)~ he number of tans present one minute before the gate opens.
"'\

State one further assumption required forthis model to be valid. 11]

,
/

92}12/1 N:2()02

I
\ .. ~ •..
 .i¥,.,. ~,--
r

15 Points A, B, C and D have coordinates (5, 7, 3)~ (7,8,0); (b, io. 3) and
(6.9,6) respectively.

(a) . Find (i) A B and CD. and hence state clearly

two facts relating lines AB and CD.

(ii) the length of BC. [2J

{b) Evaluate the scalar product AC. BD, and deduce a relationship
between AC and BD. '; I~)I

16 (a) (i) Express (In(2x + 1) - 2J1nx - I as a single togarithrn tcrm. 121

(ii) Sketch the graph 0("·

v= (in(2x+I).

(b) The tunctionris defined ~y )

5
.1' .A'-'
,- '.('".:-C - )-)2 - "' •.. ,-,;:;
.J n
, c 1'..• X2 -,
2

Find (i) the range 0(" f.

(ii)

- - --
\.. .,

;'

{
;-

\
\

\
~

\
[Tu rn O'H'{

J -
I
.J
 --~~r'~ 1
i

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

MATHEMATICS 9202/4
PAPER 4

Wednesday 6 NOVEMBER 2002 Morning 3 hours

. Additional materials:
Answer paper
Graph paper
List of Formulae

TIME 3 hours

INSTRUCTIONS TO CANDIDATES
•
/
Write your name, Centre number and candidate number in the spaces provided on the ansyvcr
paper/answer booklet.

There is no restriction on the number of questions which you may attempt.

If a numerical answer cannot be given exactly, and the accuracy required is not specified in the.
question, then in the case of an angle it should be given to the nearest degree, and in other cases
it should be given correct to 2 significant figures.

If a numerical value for g is necessary, take g = 9.81ms·-2,

INFORMATION FOR CANDIVATES

The number of marks is given in brackets [] at the end of each question or part question,

',..-- The total number of marks for this paper is 120.

Within each section of the paper, questions are printed in the order of their mark allocations and
candidates are advised, within each section, to attempt questions sequentially.

The use of an electronic calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

This question paper consists of 9 printed pages and 3 blank pages.

Copyright: Zimbabwe School Examinations Council, N2002.

@ZIMSFC N2002 [Turn over

..
 2

Section (a) : Pure Mathematics

1 Solve the equation 22x+1 - 5(2x) + 2 = 0 . [4]

2 In a certain country, 35% of all cars emit excessive amounts of pollutants. The
probability that a car emitting excessive amounts of pollutants will fail the
emission test is 0.95. The probability that a car not emitting excessive
amounts of pollutants will fail the test is 0.19.

A summary of the above information is represented by part of the tree diagram

as shown below.
fails the test

~mits

fails the test

does not
emit

Find the probability that

(i) a car will fail the test, [2]

(ii) a car emits excessive pollutants given that it fails the test. [3]

1 + t2 t dy (6]
3 If x = -- andy= --? ' find - when t = 1.
1-l2 l-r dx

9202f4 N2()02
 3

4 A class of Biology students was given a practical test. The total score for the test
was 25. The minimum mark for the grade 'above average' is 20 and the minimum
mark for the grade 'average' is 13. Marks of 12 or less are graded as below average.
The following stem-and-leaf diagram shows the results obtained by the class.

o 2 2 3 7 7 8

1 1 1 2 2 4 4 6 7 8 8 9

2 1 2

Key: 1/4 means 14 marks

(i) Sketch a diagram to show the percentages of students in the 3 grades. [2]
"-
(H) Using the classes J - 5; 6 -, 10; ... ; 2 J - 25;
group the data and draw a cumulative frequency curve. r 4]
(iii) Use your cumulative frequency curve to obtain estimates of the median
and the interquartile range. [3]

"-_/
/

9202/4 N2002
(Turn over
 4

Section (b): Statistics

5 If a random sample of size 50 is drawn from a population, with mean 20

and standard deviation 8, find the probability that the sample mean exceeds 22. [4]

6 The discrete random variable, X, has a probability distribution shown in the

table below.

x o 2 3

3 3
P(X =x) a
10 5 10

Find (i) the value of a, [1]

(ii) E(X), [2]

(iii) Var(X). [2] \'

7 W and Y are independent random variables such that W- N (l0; 100)

1
and y- N(-25; 5), find P(3W > 5" V). [5]
8 A survey was conducted on 200 'A' Level students doing two' A' Level science
subjects including Mathematics. TIns was conducted to determine the relationship
between tbe science subject taken and the Mathematics option paper preferred.
The results are summarised in the following table.

Mathematics option paper

11
Subject Paper 2 Paper 3 Paper 4

Biology 32 28 21

Chemistry 25 23 50

Physics 4 11 6

(a) Calculate the expected frequency for cell corresponding to students

doing Physics and taking Mathematics option paper 3 l2J

(b) Given that L(f<J le~/" )Y = 16.9 to 3 significant figures wherez, denotes the

expected frequency, andl~ denotes the observed frequency, carry out a

test for independence at 1% significance level, state your hypothesis and
conclusion clearly. [3]

9202/4 N2002
 5

9 In a packet of oranges, the weight of an orange denoted by x, is measured. It may

be assumed that each value of x is an observation from a normal distribution.
The results of measurements of 10 randomly selected oranges are given below:

20; 15; 10; 18; ·20; 16; 14; 12; 17; 25 ..

(in arbitrary units)

Using a 1% significance level, test whether the average weight of oranges is 18. [7]

10 In a sample of 150 randomly selected vehicles, the speed, xkmh -I, was
measured on a highway. The; results were summarised as follows:

2:> 1 = 2 220000.

(a) Calculate the unbiased estimates of the mean and variance of the speed. [31

(b) 111e (J. % confidence interval for population mean, /1, was found to be
80.67 < J..I. < 159.33, calculate a correct to the nearest whole number. [4]

920114 N2002

ITurn over
 6

11 Two different companies make a similar product. In a study to find the

relationship between quantity produced (y) and the time taken (x), scatter
plots, correlation coefficients and linear regressions were used and the results
are given below.

x
+ ,
Quantity
+,
+ , Quantity
x

(y) (y)
I
+ ,
I
+ ') I
1
50 60
Time (x) Time (x)

Company 1 2

Product moment
correlation coefficient 0.93 0.06

(a) Interpret the meaning of product moment correlation coefficient

for the two companies. 121

(b) Is it possible to estimate y given that x = 90 for both companies?

Give reasons. [2]

(c) In a similar study, the results obtained were summarized as shown.

n=8 x =5 y = 12

LXY = 512
Find the regression line ofy onx. [5]

9202/4 N2002
 7

12 Sixty percent of the new cars sold in Zimbabwe in the year 2000 were made
by BEB Motor Industries (BMI). A random sample of 15 purchases in the year
2000 is selected.

(a) Find the mean and standard deviation for BMI car purchases for this
sample. [3]

(b) Find the probability that more than 3 of those selected purchased a BMl
car, giving your answer correct to 4 decimal places. [3]

(c) Find the probability that at most 2 of those selected did not purchase a
BMI car. [3]

13 Records sh0f' that there is an average ofthree accidents each day in Harare
between the times 1700 and 1800.

(a) Find the probability that there will be an accident between 1700 and
18 00 on a particular day. 12]

(b) Find the probability that there will be at least three accidents between the
times

(i) 17 30 and 18 00 on a particular day, L3 J

(ii) ] 700 and 17 30 on a particular day. [4]

9202/4 N2002
[Turn over'
 8

14 The number of accidents occurring along Masvingo-Beitbridge road was the

subject of a recent patrol. The number of accidents were monitored per week. The
following results were obtained.

Number of accidents recorded X Observed frequency

0 14

1 28

2 32

3 or more 40
\
Total 114

(a) Assuming that the number of accidents follows a Poisson distribution

with mean 3, calculate the expected frequencies. [4 J

(b) At 5% level of significance, perform a test to determine whether there is

enough evidence to confirm that the number of accidents per week follow
a Poisson distri bution with mean 3. [6J

15 (a) Friends play a game by tossing a fair coin and a fair four sided die
simultaneously. A player wins the game \ hen he obtains a tail on
the coin and a two on the die.

Find the probability that more than 5 attempts are required to win the
game. [4]

(b) When an arrow is thrown to hit a target. the probability that it hits a
target is p. A hunter throws an arrow 1000 times and observes that it
hits a target 350 times. Test, at 1% level, the hypothesis that p = 0.4
against the alternative p < OA. [6]

9202.i4 N2002
 9

16 (a) Give any two characteristics of a normal distribution. [2]

(b) A manager analysed the number of hours employees have taken as sick
leave. He found that the amount of sick leave per employee per year has
a distribution which is approximately normal with mean 52 hours and
standard deviation 14 hours.

Calculate the percentage of employees who will take

\
r'
(i) more than 60 hours of sick leave per year, [3]
"'-
(ii) no sick leave hours per year, [3]

(iij) between 30 and 40 sick leave days. [4]

(c) The company is considering setting an upper limit on the number of hours
of paid leave per year so that only 20% of employees will need to take
unpaid sick leave, Calculate the value of this upper limit. [4]

..
9202f4 N2002