2020_maths_alevel_nov_paper2

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General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/2
PAPER 2

NOVEMBER 2020 SESSION 3 hours
Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Scientific calculator (Non-programmablc)

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,

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.

If a numerical value for g is necessary, take g = 9.81 ms”.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ } at the end of cach 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,

= ee

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

Copyright: Zimbabwe School Examinations Council, N2020,

©ZIMSLC N2020 {Turn over

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Answer all questions in this section.
(a) A sequence is defined as U, = 1 + (-1)".
(i) List the first four terms of the sequence,

(ii) | Describe the behaviour of the sequence.

(b) The sum,§S,, of an arithmetic progression is given by S, = n(3n + 2).

Find the first term and the common difference.

The complex number Z, and Z, are such that Z; = —1—- ¥3i and
_k =3+ 4i,

(a) Express in the form a + bi

(i) Z zs Z, ’
(Ot

(b) Calculate

(i) = |Zal,
(ii) the argument of Z,.
1 n

; ° Re SS oe,
Prove hy induction Lr=t r(r+1) nti

A, B, C and D are the points with position vectors 2i+ 2j +k, i+ 4j + 2k,
i+ 3j +k and 2i + Ak, where A is a constant, relative to the origin, O.

(a) Find the
(i) unit vector in the direction of OA,
(ii) position vector of the mid-point of A and B,

(iii) | position vector OE given that 4AB = CE.

(b) Given that the angle between AD and BC is 90°, find the value of A.

6042/2 N2020

[1]
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[2]

[2]

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(2]
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[3]
(3]

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(a) = The outcome table shows the entries of numbers modulo 4 under the operation
of addition.
(i) Identify the entries a and b. [4]
(ii) = Write down the identity element. (1)
(b) — Find f In(x +3) dx. [5]
6042/2 N2020

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(b)

(2)

(b)

(a)

(b)

(c)

(i)

(ii)
(i)
(ii)

(iii)

(i)

(ii)

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Section B [80 marks]
Answer any five questions from this section.
Each question carries 16 marks.
Given that f(x) = e~*sinx, use Maclaurin’s theorem to find the
series expansion of fix) up to and including the term inx. [4]
Hence or otherwise evaluate (0.4) to 3 decimal places. [2]

Sketch on the same axes the graphs of y = Inx and y = 2—X. [3]

Verify that equation Inx = 2 — x has one real root in the interval
Les <2. [3]

Starting with x» = 1.8 use the Newton-Raphson method twice to

calculate the root of Inx + x — 2 = 0 giving the answer correct to
3 decimal places. [4]

Express 3 — 3V3i in exponential form, re, where ris the modulus
of the complex number and 6 is the argument. [3]

a

Ilence or otherwise find all the roots of the equation

Z* — 3 + 3V3i = 0 in exponential form giving the answers
correct to three significant figures, [6]

Use de Moivre’s theorem or otherwise to show that

Stan@—-10tan?@+tan°@

tans? = 1-10tan*@+5tan‘’6d ~ [7]

Write down the matrices for the following transformations,

(i) areflection in the y — axis, [1]
5
(ii) | astretch by a factor . parallel to x-axis. [1]
Solve the equation
1 1 1
y ytl y-1/=0, [3]
y-1 2y yr |
| | . 2-3 4
(i) Find the inverse of the matrix A =[ 3 2 -1) [7]
a ae
(ii) Hence or otherwise determine the Solution of the equations:
2x —3y+4z= 22
3x +2y—-z= 10
X+y+z= 14 [4]

™A22 N2020

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11

12

Given that y =

3x+1"

dy d?
(a) find —,—> and © — ~ showing | in each case that the derivatives can

dx’ dx?
(-—3)"n!
be written in the fort acne [6)
(b) Prove the suggested n" derivative by induction. [10]

The points A, B and C have coordinates (1; 2; 3); (2; -1; 5) and (1; 3; -1
respectively relative to the origin O.

1 4
Line / passes through A and has equation r = (2) +A @)
3 1

2 1
Line mm passes through point B and has equation r = (-1) +f (-1)

5 1
(a) Show that AB is perpendicular to line /. (3]
(b) — Find the coordinates of the point of intersection of line / and line m. [5]
(c) Find the angle between line m and line /. [4]
(d) Find the equation of the plane passing through points A, B and C. [4]
(a) Find f cotxdx. [3]

(b) Find the exact value of eer om —, ax i in the form a + In (; ) where a, b and c

are constants, [6]
(c) | Evaluate the exact value of R e*cos3xdx. [7]
6
(a) (i) Express 3cos26 — sin26 in the form Rcos(2@ + @), [2]
(ii) | Hence or otherwise solve the equation 3cos2@ — sin2@ = 2 for
0<8@ < 360°, [4]
(b) Prove that sin3x = 3sinx — 4sin°x. [4]

(c) Hence or otherwise solve the equation sin3x = sin*x for 0 < x < 2n
give the answers in terms of fr, [6]

6042/2 N2020

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