2023_maths_alevel_nj_paper1

Extracted Content: 2023_maths_alevel_nj_paper1

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level
PURE MATHEMATICS 6042/1
PAPER 1
JUNE 2023 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.

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 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 a
Sp. . af:

We
4

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

Copyright: Zimbabwe School Examinations Council, J2023.

©ZIMSEC J2023 [Turn over

PATA

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(b) State the behaviour of the sequence. (lj

A and B are two points on the circumference of a circle centre O and radius r,
The minor arc AB subtends an angle of 26 radians at O.
If the area of the minor segment is one third of the area of the minor sector AOB,

show that 20 = 3sin 6 cos 0, [4]
A vatiable p is inversely proportional to the square of 2q -+ 1, Given that p= :

when g = 2, Find the value of g when p = : [4]
(a) Solve the inequality (1.05)"~* < 60. : [3]
(b) Hence state the largest integral value of 7, for the inequality in (a) [1

Solve the following simultaneous equations:

aed

xy ='1

axty = 3 [4]
Differentiate with respect to x

(a) = e** sin 3x, [2]
(b) In(t + x?), [2]

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TIAVGLTGUUANNI EH aL

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A curve C has parametric equations

x= 2t +2 : y= 2t—=, where t is a parameter.

(a)
(b)
(a)

(b)

(a)
(b)

Find x? and y? in terms of t.

Hence by evaluating x* — y? find the cartesian equation of C.
y gs y q

1

Express in the form x”,

4
x \3

Solve <2 =2*
ve = TF .
© 27x16

Divide a? — b® by a- b.
Given that x* + x? +%+1= (x? + A)(x? -1)+Bx+C,

determine the numerical values of A, B and C.

Functions h(x) and g(x) are defined by:

A(x) = 3x-+7 forxe€ R

g(x) = x#2forx ER

(a)

(b)

Prove by Induction i

Write down in terms of x
@® h*@),

(i) g7*(@) stating the values of x for which g~4(x) is not
defined.

Sketch the graph of A(x) and h-1(x) on the same axes making
clear the relationship between the two graphs.
.
n
1 _ 2" -1)
> P p-t

r=1

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4]
[2]
[2]

[3]

[2]

[4]

[5]

13]

[7]

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13

14

1s

4

1 2 =°5
Given that the matrix A = (2 5 3)

Ap 7

find the
(a) value of p for which the determinant of A = 54, [3]
(b) inverse of matrix A using the value of p obtained in (a) above. [5]
(a) —_ Use Taylor series to expand : as series of ascending powers of (x ~ a),
up to and including the term in (x — a)*. [6]
(b) Use the expansion to evaluate a correct to 3 significant figures, [2]
(a) Express 4 cos @ — 3 sin @ in the form R sin(6 — a), where R > 0 and
0° < ~ < 360°, giving @ to the nearest degree. (3]
(b) — Solve the equation 4 cos 6 — 3sin 9 = 3, for 0° < 6 < 360°. [3]
eo ay 1
(c) Find the least value of Toosdoa simb48" [2]
(a) — Given that z, = 344i, andz, =1+4i,
find
@ 4-22, [2]
(i) = argument of z, — 2. [2]
(b) =i) Solve the equation z? — 4z + 53 = 0, expressing the roots in the
forma + bi, wherea, bE R [3]
Gi) ‘State the sum and product of the roots of z? —-4z7+53 = 0. [2] ‘
‘ye

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5

16 The points A, B and C have position vector 2i — k, 3 +2j — k and 5i — 3j + Ak
respectively.

(a) Calculate the angle ABC
correct to the nearest 0.1° [4]

(b) — Given that N is a point on vector AG such that AN : NC = 1:2,

find the
r()) position vector of N, {3]
Gi) unit vector in the direction of NC. [3]

17 The straight line has equation 2y —x-—5 = 0.
Another straight line # passes through the point P(—2, 6) and is perpendicular

to 2.

Find the

(a) equation of m giving the answer in the form ax + by + ¢ = 0, [3]
(b) co-ordinates of the point of intersection of # and m. [3]
(c) perpendicular distance from P to the line £. [2]

(a) _It is given that the points Q(7; 6) and R (—7; —1) lie on line 2. .
Find the area of the triangle PQR. [3]

18 (a) Sketch on the same axis the graphs of y = e~* and y = sin x where
Osxsu. [2]
(b) Hence state the number of roots of the equation e~* — Sin x = 0 {l]

(c) Show that the smallest root of e~* — Sin x = 0 lies between x = 0 and

nak ms 13]

(c) Starting with x) = 0,8, use the Newton -Raphson method twice to
estimate the smallest root of e~* — Sin x = 0, giving the answer correct
to three decimal places. [4]

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