2014_maths_alevel_mj_paper2

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

MATHEMATICS 9164/2

PAPER 2

JUNE 2014 SESSION

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 answer
paper/ariswer booklet.

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

Ifa 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 = 9.81 ms”.

INFORMATION FOR CANDIDATES

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 6 printed pages and 2 blank pages.

Copyright: Zimbabwe School Examinations Council, J2014.
©ZIMSEC 12014 [Turn over

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(b)

2

Section (a): Pure Mathematics

The least value of the function x° + px +q is 3 and this occurs when
x= 2.

Find the values of p and q.

Find the values of k for which the equation x? + (3k -7)x +2k+6=0
has real roots.

. . : d”
Prove by induction that if y = xe*, then oa =(x+n)e” for all natural values

of a.

(a)

(b)

@)

Gi)

(ii)

(a)

(b)

If y= tao -=

dy
} show that x = —cosecx,
he

+4. . :
Express 4 in partial fractions.

2
x=

2
4
Hence find the exact value of ‘ a + dx.
Jay 4

. i a
Prove the identity sins _ £0890 =4cos26.
sin@ cos@

By substituting @ = 18° into the identity in part (i) show that

4sin18°cos36° = 1. . ‘

Using the result of part (ii), deduce that sin18° is the root of the
equation 4x —8x° -1=0.

Pal
2

Express <— in the form a + bi.
ee
: ztl a
Sketch on an Argand diagram the locus defined by arg] —~7=—|=—.
1+-+3i) 4

Use de Moivre’s theorem to show that 4cos’ @ = cos3@ + 3cos@.

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(b)

The triangle ABC of area 11 square units is mapped into triangle
A'B'C! by M = _—
1 5)

Find the area of triangle A'B'C!.

ean x —-2 2\x
The point ( is transformed by | Jo | }
y y -2 3hAy

Find the equations of lines which are mapped onto themselves.

~1 -2 2 3-3 3
Giventhat A’ =|2 5 -4], B’=/-1 1. 1 | and that
1 1 -l 2 4 -2
x 12
AB] y| =} 0 |, find x, y and z.
z 0

The point P has position vector i+ 37 + 3k, the line / has an equation
r=it5j-k+¢(i-j+k) and the plane x has an equation
r=5it+5k +A(2i+ j) + u(2i+ fj + 4k).

(i)
(ii)
(iii)
(iv)

Find the point of intersection of / and x.
Find the length of the perpendicular from P to x.
Find the length of the perpendicular from P to /.

Find the acute angle between / and x, correct to the nearest 0.1°.

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Section (b): Mechanics

A particle A rests in equilibrium on a smooth horizontal table.

Forces of magnitude 4 N, 6 N and Y newtons act on A (see diagram).

(i) Find the values of Y and @. [4]
(ii) State one assumption made in answering the question. [1]

Particles A and B of masses 5 kg and 8 kg respectively are connected by means
of a light inextensible string which passes over a smooth pulley. Particle A lies

on a smooth plane inclined at an angle tan” (3) to the horizontal and particle

B is hanging freely. :

When the system is released from rest, calculate

@ the tension in the string, [3]

(ii) _ the acceleration of the particles for the part of the motion before B hits
the ground. (2]

A particle decelerates uniformly from a velocity of 25 ms“! to 15 ms in

2 seconds. It maintains a velocity of 15 ms”! for 4 seconds and then accelerates
uniformly to a velocity of 27 ms™ in 3 seconds.
(i) Sketch a (¢, v) graph for the whole journey. 3]

Gi) Given that it travels a total distance of 193 m, find the value of 4. BI

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A particle is projected from a point O on a level ground with initial velocity
of 10 ms” at an angle of elevation 6.

(i) Given that the particle passes through a point which is 3 m horizontally
away from O and 2 m above the level of O, calculate the value of &. [4]

(ii) Calculate the distance from the point of projection to the point where the
particle is 8 m below the ground. [4]

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13

14

15

Section (c): Statistics

A bag contains 24 counters of which 6 are red, 8 are green and 10 are yellow.
Three counters are taken from the bag at random without replacement.

56
@ Show that the probability that 2 of the counters taken are green is 353°

(ii) Given that 2 of the counters are green, find the probability that the first
counter taken is red.

The probability that a boy hits a target is 0.8. Assuming that shots are
independent of each other and suppose that during each practice period, the boy
fires shots until he hits the target.

(i) Find the mean and standard deviation of the number of shots fired per
practice period.

(i) Find the probability that the boy will need to take at least five shots to
hit the target.

The following are television prices in dollars taken in 40 different shops.

40 130 170 240 360 520 170 130
240 360 520 120 220 170 330 480
160 290 200 120 480 160 219 330
70 1406 180 =260 370 90 150 200
280 450 80 140 190 420 270 120

@ Construct a stem and leaf diagram for the data.
(ii) “Find the

1. median,

2. quartiles.
(iii) = Draw a box and whisker plot.
The probability that a seed grown under specified conditions will germinate
and produce a plant is 0.8. The minimum number of seeds, x, are to be planted
under these conditions to ensure a probability of at least 0.9 that 60 or more
seeds will germinate and produce a plant.
@ Using a suitable approximation show that n? - 149.2n +5531.620.
(¢ 1) ne Solve the inequality in part (i).

2. Hence or otherwise find the minimum value of n, the number of
seeds to be planted.

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