2021_maths_alevel_nov_paper2

Extracted Content: 2021_maths_alevel_nov_paper2

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Fee Perfarmatee Mfoccxrement

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/2
PAPER 2

7 NOVEMBER 2021 SESSION 3 hours
Additional materials:

Answer paper

Graph paper

List of Formulae MF 7

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 in Section A and any five questions from Section B.
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 answers.

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

Copyright: Zimbabwe School Examinations Council, N2021.

©ZIMSEC N2021 [Turn over

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2 Section A (40 marks),

Answer all questions in this section.
Solve the inequality
2x —3| <x-5. [4]
(a) Prove the identity
cosec 20 + cot 20 = cot 6. 3)

(b) — Hence or otherwise solve the equation
cosec 26 + cot 26 = 0,5, for 0 < 6 < 2n, giving the answer to 2d. p [4]

(a) Find the value of x given that 3**? = +. [3]

(b) Solve the equation.
3(22<em>) — 7(2</em>)+2=0 [5]

Given that —m,n and 1 are any consecutive terms of a geometric progression and
1, n and m are the first three terms of an arithmetic progression,

(a) show that n? + 2n = 1, [5]
(b) _ hence or otherwise find the exact value of m if n is positive. [4]
(a) Show that the set {0,1,2,3}, forms a group under addition modulo 4. [7]
(b) Show that group is abelian. [2]
(c) Write down the subgroups of the group. [2]
(d) State the order of the group. 1]

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Section B (80 marks)

Answer any five questions from this section. Each question carries 16 marks.

Functions h and g are defined by

Rix ox—4, xER

3
gx xERx#tk,
(a) Find

@ the possible values of k,
(ii) the values of x for which hg(x) = 0,

Gi) = -A-1(x).

(b) Onthe same axes sketch the graphs of y = h(x) and y = h-4(x)

showing clearly the relationship between the graphs.

() Describe completely the sequence of transformation which map
1 32
the graph y Fa onto y =

A curve has the equation y = x? — xy and passes through the point P(1; >)

(a) Find the equation of the tangent to the curve at P.

(b) Hence or otherwise show that the equation of the normal to the curve at P

: F Ad, 124
intersects the curve again at ( a a ).

_ 11 , 121

(c) Find the distance between the two points (1 ; 2) and ( ree

The vector equations of the lines n and m, and the plane 7, are
4 —1
r= (=) +pl 2 |,
7 1
0 1
r= (1 + a(s) and
7 2

3
r ( 2 ) = —1 respectively where p and A are parameters,

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[3]
BI
(3)

B]
[4]

6]

[8]

[2]

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The line m intersects the plane 7, at a point P.

(a)
(b)

(c)

9 (a)

(b)

Show that the line 7 lies in the plane 7,

(i)
(ii)

i)
(ii)

(i)

ii)

(iii)

@

Gi)

Find the coordinates of point P.

Hence or otherwise find the vector equation of the plane 72
passing through point P and perpendicular to the line n.

Find the point of intersection of the line n and the plane 72

Hence or otherwise show that the vector equation of the line

through point P which lies in plane 7, and is perpendicular to line

—2 2
nis; r=(4)¥e(-1)
3 4

Find the first 3 terms of the Maclaurin series in the expansion of

In (5+ x).

Write down the first 3 terms of the Maclaurin series in the
expansion of In (5 — x).

Hence or otherwise show that when x is small In (= = 2%
Prove by mathematical induction that
a” ym mi m-n
—— =—— Xx
dxn = ) (m-n)!
for all ne Z*

er 5
Hence find “| (x°).

10 Given the complex numbers
Z = 2andz, =2+4 2i,

@)

@)

(ii)

express Zz in the form r(Cos@ + isin@),

] 4.
hence or otherwise, express the complex number ri in polar
form.

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S°

[2]
[4]

[3]

[4]

B]

[1]

[2]

[3]

[2]

[3]

(3]

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il

12

(b)

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5

Describe fully the locus represented by Re = = 0 given that zisa
complex number x + iy, [4]

By using de Moivres theorem or otherwise, find the roots of the equation,

3-1 2
2 2

Wisaivertaa=( 4 -1 0) ana =( y

(a)

(b)

()

(a)

(b)

—2 4 3

A line with the equation y = 2x — 3 is transformed under the
transformation matrix B. Show that the equation of its image is

y—6x+24=0. [5]
Evaluate A~+. [7]
Hence or otherwise solve the following simultaneous equations. [4]
38x -yt2z=1
x-y=2

—2x+4y—-3z=3

22
A learner used the value of 1 as — to calculate the volume of a sphere of
radius 8 cm.

Find correct to 3 decimal places the
(i) absolute error involved,
(ii) percentage error involved. [5]

(i) Sketch on the same axes the graphs of y = In 2x andy =x-1
and state the number of real roots for the equation In 2x = x — 1.

(ii) | Show that a root of the equation In 2x = x — 1 lies between 0.2
and 0,3,

(iii) Using 0.2 as the first approximation, use the Newton Raphson

method twice to obtain an approximation to the root correct to
3 decimal places. [11]

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