4026Q2 SPECIMEN ADDITIONAL MATHSr

Extracted Content: 4026Q2 SPECIMEN ADDITIONAL MATHSr

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Surname Forename(s) Centre Number Candidate Number

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Ordinary L evel

ADDITIONAL MATHEMATICS 4026/2

PAPER 2

SPECIMEN PAPER 2 hours 30 minutes

Additional materials:
Mathematical tables Data Booklet
Graph paper (4 sheets)

Allow candidates 5 minutes to count pages before the examination.

TIME 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES

Write your name, Centre number and candidate number in the spaces provided on the question
paper.
Check if the booklet has all the pages and ask the invigilator for a replacement if there are
duplicate or missing pages.

Answer all questions in Section A and any five questions from Section B and Section C.
Do not answer more than four questions in Section B and not more than four questions in
Section C

Write your answers on the spaces provided on the question paper.

All working must be shown clearly.

Where necessary take g = 10 ms~’.

Electronic calculators may be used.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.
If the degree of accuracy is not specified in the question and if the answer is not exact, the answer
should be given to three significant figures. Answers in degrees should be given to one decimal

This question paper consists of 19 printed pages and 1 blank page.
Copyright: Zimbabwe School Examinations Council, Specimen Paper.

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Answer all questions in this section

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Section A [40 marks]

Mechanics and Statistics

The table shows the annual incomes, in thousands of dollars, for 120 employees at a
certain university.

(a)

(b)

Income 10-20 20-30 30-40 40-60 60-100
(thousand)
Frequency | 40 48 18 x 4
Find the value of x.
x= [2]

Calculate an estimate of the mean income.

Answer

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3 (c) Draw a histogram to illustrate this data

5

During a particular day, the time (in minutes) taken by people entering a food outlet
during lunch hour were as follows shown in the table below.

Time (min) Frequency Cumulative frequency
0<x<10 43 43

10 < x<20 51 94

20 < x <30 69 163

30 < x<40 22 x

40 < x<60 y 204

(a) Find the value of x and the value y.

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(b)

Draw a cumu.

4
ative frequency curve to represent t

he information in the table.

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(©)

(a)

5
Use the graph to estimate the lower quartile and upper quartile.
Hence calculate the interquartile range.

lower quartile = upper quartile

interquartile range = [4]

Two forces of magnitudes 6 N and P N have a resultant of magnitude 15 N.
Given that the angle between the two forces is 60°.

Find the value of P.

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(b)

6

A particle is in equilibrium under the forces 10N, 6N and QN as shown in the
diagram.

10N

Find the value of

@ 4&4,

ji) @

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7 A particle starts from rest and accelerates uniformly at 3,2 m/s? for 12 s to reach a speed of V m/s. then it decelerates at 1,8 m/s? until its speed is 10 m/s.

Calculate,

(a) the value of V,

(b) the time taken when decelerating,

Answer: [2]

(c) the total distance covered during this journey.

Answer: [6]

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(a)

(b)

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Section B: Mechanics
Answer not more than four questions in this section.

State any two contact forces.

1.

2.

[2]

Particle A of mass 2 kg rests on a smooth plane inclined at 30° to the horizon. It
is attached to a particle B of mass 4 kg, by a light inextensible string passing over a
smooth pulley P at the top of the plane.

Given that B hangs freely and the system is released from rest, find the

@ acceleration of the system and the tension in the string,

Acceleration tension [6]

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(b) (¢1)) speed with which B hits the ground,

Answer: [2]

(iii) ~_— time B takes to reach the ground

Answer: [2]

A horizontal force of magnitude 2,5 N pushes a rubber ball of mass M kg along a
horizontal surface. When the force is removed , the ball moves with uniform retardation
and passes through the points A and B with speeds 7 m/s and 4 m/s, respectively, in 6
seconds.

(a) Find the

(i) retardation of the ball,

Answer: [2]

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(a) (¢1)) distance AB,

Answer: [3]

(iii) ~~ work done by the force

Answer: [2]

(b) Assuming that there are no other resistances to the motion of the ball, find the
value of M.

M= [3]

(c) Find the time it takes the ball to come to rest from A.

Answer: [2]

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Particles A, of mass 0,3kg and B, of mass 0,2kg are attached to the ends of a light
inextensible string. The string passes over a smooth pulley P. Initially the system is held at
rest with the string taut, B on the floor and A at a height of 0,5m above the floor (see
diagram). The system is then released.

0.5m

B
SSSSSLLSSS SSS LSC

(a) Find the acceleration of the system and the tension in the string.

Answer: [6]

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(b)

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At the instant when particle A hits the floor, the string becomes detached from
particle A.

Find the

@ greatest height reached by particle B,

Answer: [2]

(¢1)) speed with which particle A hits the floor,

Answer: [2]

(iii) time taken by particle A to hit the floor.

Answer: [2]

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13 (a) A particle of mass 5 kg is attached to two strings and hangs freely in equilibrium

under gravity with the two strings making angles of 35° and 47° with horizontal.
Calculate to three significant figures, the tension in each string.

Tension = [6]

(b) A water pump raises 60 kg of water per second through a height of 20 m.
The water emerges at the surface with a speed of 30 m/s.

@ Find the kinetic energy and the potential energy given to the water each
second.

Answer: [4]

(¢1)) Hence, find the effective power developed by the pump.

Answer: [2]

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A car of mass 20 000 kg is travelling at a constant speed of 25 m/s. The engine of the car
is working at 70 kw.

(a) Calculate the resistance to motion.

Answer: [2]

(b) Given that the power output and resistance to motion remain constant,
calculate the

@ maximum speed of the car up a plane inclined at an angle @ to the,

: . 1
horizontal where sin @ = Too”

Answer: [3]

(ii) —_ acceleration up the slope at the instant when speed of the car is 10 m/s.

Answer: [7]

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Answer not more than four questions in this section.

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Section C: Statistics

10 (a) The table shows the probability distribution for the discrete random variable X .

xX 1 2 3 4 5 6
P(X =x) | k 2k 3k 4k 5k 6k
@ Find the value of k
ke
(i) Show that E(X) = 42.
Answer.
(iii) Calculate Var (X).
Var(X)

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16 10 (b) =‘ Ina basketball tournament the scores of 128 players on the first day of the tournament were recorded. The results are as shown in the table below.

score 3 4 R) 6 7
frequency 21 83 17 5 2

Find the mean and variance of the scores.

Mean = Variance = [5]

11 A bag contains 6 red balls and 3 blue balls. The balls are identical except for colour.
Three balls are drawn from the bag one after the other and not replaced.

(a) _‘ Draw a tree diagram to illustrate probabilities for all the possible outcomes.

[4]
(b) Use the tree diagram to find the probability that
@ two balls are red and one is blue, in any order,
Answer: [2]

di at least one ball is blue,

Answer: [2]

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(iii) the balls are of the same colour,

Answer: [2]

(iv) _ the first ball is red, given that at least one ball is blue.

Amswer; [2]
12 (a) A random variable X ~ Bin (10; p), where p <0,5.
Given that variance of X is 1,875 find,

@ the value of p,

PS [4]
(i) = -E(X),

EX)=_ [1]
(iii) = P(X > 3).

Answer: [3]

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18 12 (b) It is given that X ~N(15;07) If P(X <10) =0, 1. calculate the value of co.

o= [4]

13 The mass of a packet of soup coming off a production line is normally distributed with
mean 50 gand standard deviation 8 g. A random sample of 25 packets from the production
line were selected.

(a) _ State the distribution of X

Amswer, ee [2]
(b) = Find P (XK > 54)

Amswer, ee [3]
(c) Find P (48<X <53)

Answer: [4]

(d) Find the symmetric 95% confidence interval about the mean.

Answer: [3]

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14 A continuous random variable, X, has probability density function given by,
ax+b, 0<x<2

0 ; otherwise

where a and bare positive constants.
Given that P(X <1) =5.

(a) find the value of a and b.

(b) show that E(X) = >

() calculate V ar (X).

Var(X): [5]

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