2020_maths_alevel_mj_paper2

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The Trotter.Q774998145

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/2
PAPER 2
JUNE 2020 SESSION 3 hours

Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Non-programmable electronic 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 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.

Ifa 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 each question or part question.

The total number of marks for this paper is 120.

The use of a non-programmable electronic scientific calculator is expected, where appropriate.

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

This question paper consists of 6 printed pages and 2 blank pages.

Copyright: Zimbabwe School Examinations Council, J2020.

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2 Section A [40 marks]

Answer all questions in this section.

28
O r

In the diagram above, ABC is a semi-circle with centre O and radius r cm.
BOC = 20 radians.

Find

(a) in terms of @ and r, the perimeter of the triangle ABC. [3]

(b) the exact area of the shaded region when 6 = = tad andr=~2. [3]
‘ fs 1-sin@

(a) Prove the identity == = sech — tand. [3]

1+sing

b> Express sinx —-V3cosx in the form Rsin(x — @), where R > 0
and 0 <a <f. [2]

(ii) Hence or otherwise, solve the equation

sinx — V3cosx = —1, for 0 <x < 2n. [3]

sin(x+30°)

(a) It is given that f(x) = cos(x+30°)"

If x is small enough to neglect terms in x* and higher powers, show that

2+2V3x—-x? 5
I = Se [5]
2¥3-2x—V3x
(b) Hence find the percentage error in calculating tan305°, giving the
answer correct to one decimal place. [3]
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3

(a) A sequence U, is defined for real numbers by

_ Un
Une a

When U, =3, find
@ Up, Us and Us, 2)

(i) Hence describe the behaviour of the sequence as 7 increases. (1

(b) The first three terms of an arithmetic progression have a sum of 21 and
a product of 231.

Find the two possible values of the tenth term. [6]
The functions fand g are defined as

fs) =4-x,xeER,
g(x) = Vx,x>0.

(a) State the range of f (1]
(b) Find the domain of gf(x). [4]
(b) Sketch the graph of y = |f(x)|, indicating the intercepts with the axes. [2]

(d) State, with a reason, whether /(x) is injective or not. [2]

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4 Section B [80 marks]

Answer any five questions from this section. Each question carries 16 marks.
(3 —1 . .
(a) The matrix ( 11 ) maps points on the line y = 4x + 8 onto another

line. Find the coordinates of the only invariant point. [5]

5 2 1
(b) Giventhe matixA={2 8 -1],
3 -1 3
(i) find the inverse of A, 16]
(ii) hence or otherwise, solve the simultaneous equations

Sx+2y4+z725

2x+8y—z= 11
3x-—y+3z= 18 [4]
(iif) give a geometrical interpretation of your Snswer in (ii) above. [IW]

The equations of a line / and a plane 7 are given by

3 —1
i r=(5)+e(2).rer
-—3 4
6
n.n(m)=s.
2

The point A has position vector ( 5 )
1

+

(a) Find the value of » if

i) A lies on 7}. [2]
(i) =m and /are parallel. [2]
(b) Calculate the shortest distance from A to 7. [5]

(c) Obtain the Cartesian equation of the plane 72 that contains both Aand/. [5]

(d) ‘Find the angle between z and I. [2]

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10

{a)
(b)

(a)

(b)

(c)

(b)

5
nt
Find the exact value of {8 cos? (2x) dx. [4]
Ina chemical reaction a substance A is continuously transformed into
chemical B. The sum of the volumes of A and B is always constant and
equal to H throughout the reaction. At time ¢ hours after the start of the

reaction, the volume of B is denoted by V. The rate of the formation of
B is proportional to the volume of A present at any instant.

(iD Show that the above situation satisfies the differential equation

& = k(H — Vp. [1]
(id) Given that V = : H when t= n9, solve the differential equation,

expressing V in terms of H and [8]
(iii) Calculate the value of t when V = 0.8 H. [7]
(iv) | Sketch the graph of V against 7. I]

Shade the region represented by the complex number z satisfying the
inequalities |2z — 1 — 4i| < 2 and Re(z) < 0, on an Argand diagram. [4]

Given the polynomial f(x) = x? — 5x* + 16x — 30,
@ show that z= 1 + 3éis a root of the equation f(x) = 0, . [3]
(ii) find the other two roots of f(x) = 0. [3]

Use de Moivre’s theorem to show that

Cos68 = 32cos<em>é — 48costé + 18cos</em>6 — 1. [6]
Prove by induction that 6” — 5n + 4 is divisible by 5 for all positive
integral values of 7. 6]
w@ By sketching a pair of suitable graphs, show that the equation

xin(x +1) = 1 has exactly two real roots. [3]
(ii) ~—- Verify by calculation that a root lies between 1.2 and 1.3. [3]
(iii) Using x; — 1,25 as the first approximation to the root, apply the

Newton-Raphson method twice to find an approximate to the root.

Give the answer to 3 decimal places. 4]

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11 (a) A binary operation * is defined by

a*b= =.
Solve the equation (3* x) #1 =1. [3]
(b) Show that the set S = {0; 1; 2; 3} forms a group under addition
modulo 4. [4]
(©) Prove that the set of integers B — {1; 3; 4; 5; 9} forms a group under
multiplication modulo 11. 9]
12 (a) @ Show that r(r + 1)(r + 2)-(r- D@) & + 1) =3r(r + 1). [2]
(ii) Hence or otherwise show that
Shar +1) =F(k+DK+2) [6]
(tb) (i) Use Taylor’s theorem to obtain a series expansion for
C os (x + 4) about x = : up to and including the term
inx’, [6]
(ii) Hence evaluate cos46° leaving the answer in exact form. [2]

Taylor’s series

Fe) = fla) + f-2@- a) +P - a +O

7a.
@-aete= ee 2 (x — a)?

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