2021_maths_alevel_nov_paper1

Extracted Content: 2021_maths_alevel_nov_paper1

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/1

PAPER 1

’ NOVEMBER 2021 SESSION 3 hours
Additional materials;
Answer paper
Graph paper
List of Formulae MF7
Scientific calculator [non - programmable]
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 correct 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 non-programmable scientific calculator is expected, where appropriate.

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

This question paper consists of 5 printed pages and 3 blank pages.

Copyright: Zimbabwe School Examinations Council, N2021.

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2e?<em> —7e</em>+6=0, [3]
Give the answer in exact form.

Prove the identity

sin20 _ [3]
1+cos2@ tan @.
a:
Given that y = In[cos(x? + 1)], show that = = —2x tan (x? + 1). [3]

Given that M is inversely proportional to the cube root of (n — 1) and that
when n = 9,M = 5,

find
(a) aformula connecting M and n 5 [3]
(b) the exact value of n when M = 25. [2]

(a) Express 2x? + 3x + 1 in the form

a(x + b)? +c where a, b and c are constants to be found, [3]
(b) Hence or otherwise solve the equation 2x? + 3x +1= 0. [3]
B
t
LY
A

In the diagram, A and B are two points on the circumference of
and radius r. Angle AOB is @ radians and C is the mid
y and the length of the arc BA is t.

a a circle centre O
“point of OA. The length of BC is
(a) Express y? in terms of r and 6.

(b) Hence or otherwise show that if @ is small then y? = ar?

1
SSeS,
6042/1 N2021 2 {41

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11

3

IfA= le a) prove by induction that A” = — 4 1 eal where 7

is a positive integer.

2°41

2%-1 5

(a) Solve the equation

giving the answer correct to 3. significant figures.

(b) Solve the inequality |x — 3] > 2|3x + 1.

Given that x is so small that x* and higher powers can be neglected,

(a) show that
ex x? ox?
Gx a?

(b) use the expansion to evaluate

201

101 correct to 7 decimal places.

(a) Given that f(x) = 2 — |x| and g(x) = 5x +1,
sketch on the same axes the graphs of y = f(x) and y = g(x),

(b) Find the co-ordinates of the points of intersection of f(x) and g(x).

2x « ‘ ;
, in partial fractions.

@ Expres (x+2)(x—-2)(x—-1)

. 7 2x
(b) Solve the inequality - yaaa <°

60421 N2021

[3]
(5)

(6]

[2]

B]
[5]

[4]

[4]

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(a) calculate the modulus of,

® »p [2]
Gi) gq. [2]
(b) find
(i) the argument of q, [2]
(ii) pq? in the form a + bi, [3]
(iii) Pin the form a + bi. (3]

13 The curve y = 10 — = and the line y + x = 6 intersect at two points P and Q.

Find the
(a) coordinates of P and Q, {5]
(b) _ coordinates of the midpoint of PQ, [2]

(c) _ equation of the perpendicular bisector of the line with P and Q
as end points. 3]

14 The curve y = x” + px + q, where p and q are constants, has a turning point at

(—1; -5)
Find the
(a) values of pandq, (5]

(b) equations of the tangent and the normal in the form ay + bx + ¢ = 0, at
the point where the curve cuts the y-axis. (5)

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(a)

(b)

5

A point (1; 2) is mapped onto point (7; 2) by a shear parallel to the x-axis.
Find the shear factor. [3]

A tieagle with vertices A (—3; 1), B (3; 1) and C (3; 5) is transformed by
= 1
M=() 3):

(i) Find the area of triangle ABC. (2]
(ii) Find the co-ordinates of Ai Bi and C; the images of A, B and C

under transformation M. [4]
(iii) Hence or otherwise find the area of the triangle A,B,C}. [2]

The gradient function of a curve is directly proportional to 20 — y. If y = 0,
x = 0 and the gradient is 1

(a)
(b)
(c)
(d)

Show that 2 = 0,05 (20 — y). 2B]
Solve the differential equation giving y in terms of x. [6]
Find y when x = 10 giving the answer correct to 2 decimal places. [2]
Briefly describe what happens to y as x becomes very large. {]

6042/ | N2021

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