2019_maths_alevel_mj_paper1

Extracted Content: 2019_maths_alevel_mj_paper1

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oa Cyne
Performance Mranremen

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

Gen A
eral Certificate of Education Advanced Level

PURE MATHEMATICS 6042/1

PAPER 1

Additional materials: JUNE 2019 SESSION 3 hours
Answer paper

Graph paper

List of Formulae MF 7

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.

the accuracy required is not specified in the

Ifa numerical answer cannot be given exactly and
en to the nearest degree and in other cases it

question then in the case of an angle it should be giv
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.

on in your answers.

You are reminded of the need for clear presentati

of 6 printed pages and 2 blank pages.

This question paper consists
nations Council, J2019.

t: Zimbabwe School Exami

Copyrigh'
a [Turn over

@ZIMSEC 32019

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Evaluate fe xlnxdx.

Use the trapezium rule with 4 ordinate:

S to approximate the value

Hence find
the relative error when the trapezium rule is used t
approximate the integral, f xInxdx. ;

(a)
(b) =@&
of . xilnxdx
(ii)
x3
‘The function f (x) = x~2

(a) Express (x) inthe form A he — Boe eeee

constants.

(b) — Hence evaluate fp fax.

eee)

oa

(a) By sketching suitable graphs, show that the equation Sec x = 3- x"
has exactly one root in the interval 0 <x < in.
2

(b) Show that if a sequence of values given by the iterative formula

(c)

The rate at which a quantity x is decreasin

a 1
Xn+1 = Cos *(

a)

D where A,B, Cand D are

converges, then it converges to a root of the equation given in part (a).

Use the iterative formula in (b), with the value x» = 1, 024627 ‘a
determine the root of the equation in (a) in the interval O<x <= m
correct to 2 decimal places, showing the result of each itanitions.

and I —x.

(i)
(ii)

(b)

Form a differential equation w

Initially x = 0,2

the differential equation.

(ii)

Express 25inx — 4Cosx in the fo

and a is acute.

Hence so
0° <x < 360°.

Obtain, in
simplified terms of

If the coefficient ofxin

exact value of a.

Ive the equation 25' inx — 4Cosx =

terms of the constant @, an

hich models this situation.

rm RSin(x — @) where R

= 45, for

the series expansion for y = e*

the series expansion in (i) is — 4, fin

6042/1 52019

g is proportional to the product of x

and the rate of decrease is 1,6 units per unit time. Solve

>0

expression for the first 3

d the

(4)

[5]

[5]

[4]

[3]

[2]

(4]

nM}

8]

(4]

[3]

(2]

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4

(a) Acurve has parametric equations

(i)

(ii)

x = Cos*@,
y = 1 —2Sin?6 for 0° < @ < 2n.

pdy 4
Find —. 14)
By writing Cos3@ as Cos(2@ + @) or otherwise, prove the identity

2Sin2@Cos6 = = (Cos@ — Cos36). 13)

(b) — A dust pan is in the form of a sector ABCO of a circle of radius
r (see diagram).

@

(ii)

A

C

If the length of arc AB is =f, find angle AOB in degrees. [2]

The segment ACB is cut off along straight line AC.

Find the percentage of the area cut off to the area of the remaining
area of sector AOC. (2]

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In the diagram P is the poi i ;
. point of intersection of the curves y = e<em> andy = e3</em>,

(a) Find the x coordinate of P. [2]

(b) Hence find the

nded by the two curves

exact value of the area of the region bow
and the y axis i.e (shaded region), (4)

(i)

solid of revolution formed when the shaded
axis through 360°. [5]

(ii) volume of the
pletely about the x-

region is rotated com

oul 209 (Tarn over

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(a)

The diagram shows a plan of a building whose floor is a rectangle OABC and
whose roof is in the form of a pyramid DEFGV, with OA =4.m,OC=8 m,
OD =2 mand MV = 5 m. Mis the point where the diagonals OB and AC
intersect. Taking O as the origin and i,j and k unit vectors in the directions

OA, OC and OD respectively,

i)

(ii)

(b) The complex number u

(i)
(ii)
(iii)

write the position vectors of B and V,

find the exact value of the angle between OV and OB.

_ 4-8i

Express u in the form x + iy.

Find the magnitude and argument of u.

Sketch u on an Argand diagram.

[2]

(4)

[2]
{3}
[2]

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