6042q1Specimen paper

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level
PURE MATHEMATICS 6042/1
PAPER 1
SPECIMEN PAPER
Additional materials:
Answer paper
Graph paper

List of Formulae
Scientific calculator

TIME 3 hours
INSTRUCTIONS TO CANDIDATES

Write your name, Centre number and candidate number in the spaces provided on the answer
paper/answer booklet.

Answer all questions.

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.

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.

The use of a scientific calculator is expected, where appropriate.

Y ou 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, Specimen paper.

©ZIMSEC Specimen paper

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When a polynomial f(x) is divided by (x— 3) the remainder is —-9 and when
divided by (2x— 1), the remainder is - 6.

Find the remainder when f(x) is divided by (x— 3) (2x- 1). [4]

. 8 . . .
Given thaty = 2x°+—, find the percentage increase in y as x increases
x

from 2 to 2.003. [4]

The diagram shows a circle centre O and radius r.
A chord PQ subtends a right-angle at the centre O of the circle. QR is a chord

such that angle POR is é.

Show that the shaded area bounded by the chords PQ and QR, and the arc PR is

1 21 Bo
2 & 5N3 GB D)

@ Sketch on the same diagram the graphs of y =|3x +4 and y =2x HI.

(ii) | Hence or otherwise, solve the inequality [3x +4 < [2x +1). [5]
2 -
Express ox ox +1 in partial fractions. [5]
(x- 2)(x? +3

0) Expand (3- 2x) * up to and including the term in x3, simplifying

the coefficients.

(ii) Hence state the set of values of x for which the expansion in (i) is valid.

[5]

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3

The variables x and y satisfy a relationship of the form
y= ax? +bx, where a and b are constants.

The values obtained in an experiment are shown in the table.

X 1 2 3 4 5
y 1.0 | 84 | 22.3 | 39.1 | 65.8

Plot the graph of x against x and determine the values of a and b.
x

Bp 29 By yO wy

The position vectors of points A, B and C are nos € n-4 sand € 3
S35 § mJ G5

respectively.

(i) Given that AB =13 units and n > 0, find the value of n.
(ii) Hence find BAC.

Find the equation of a circle whose centre is (5; 4) and touches the line joining
the points (0; 5) and (4; 1).

@ Express 6cosq - 8sing in the form Rcos (q+a).
(ii) Hence or otherwise, solve the equation

6cosq- 8sing =- 2.5 for - 180° £ g£180°.
(iii) State the minimum and maximum values of

1
6cosq- 8sing-+13

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The parametric equations of a circle are
x = OOSeCF

y =O0tF, where O<F <2p,

@ Show that 2 =socF.
dx

Gi) ‘Find the equation of the normal to the curve at the point where
p
F —

= Gin the form y =mtc.

Given that ny = xy, where y > 0,

@ 1. showthat &@ =2
dx

Gi) ~—_—- Hence or otherwise, find the Maclaurin expansion of y up to and including

the term in x2.

The diagram shows the region A bounded by the xaxis and the
curve y =sin x +/3oosx.

(a) Find the area of region A.

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5

6) @ — Express sinx + J3cosxin the form R sin (x +a).

(ii) Hence find the volume of the solid formed when A is rotated
completely about the x-axis, leaving your answer in terms of (0.

Two complex numbers z =x Hy and w = a Hib are such that
zt+iw =2 and iz tw = 243
Find
@ iL. z,
2. w,

(ii) the modulus of zy,

(iii) the argument of 2.
w

@ Use the trapezium rule with 4 equal intervals to estimate the value of

cosec’x dx, giving your answer correct to 3 decimal places.

als oO ..

NID

Gi) Evaluate exactly () cosec?xdx.

ald

Gii) Calculate the percentage error in using the trapezium rule as an
approximation to the integral.

@ By sketching two appropriate graphs, show that the equation

(x- 1) e* =x _ has two real roots.

(ii) | Show by calculation, that one of the roots lies between x =1.2
and x =1.5,

(iii) Taking x, =1.2 as a first approximation to the root of the equation,

use the Newton Raphson method twice to obtain the root correct to 4
decimal places.

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A rectangular tank has a square base with sides 3 m long and height
5m.

The tank is initially full of water. The water leaks out through the base and
sides of the tank at a rate proportional to the total area in contact with the
water. When the depth of the water is 3 m, the level of water is falling at a
rate of 0.3 m/h.

If y denotes the depth of water in the tank after ¢, hours,

() show that @ = 324 449 [5]
dt 50€3 @

(ii) solve the differential equation in (i), giving y in terms of ¢, [6]

(iii) find how long it takes for the tank to be half full. [2]

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