2019_maths_alevel_mj_paper2

Extracted Content: 2019_maths_alevel_mj_paper2

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Fer Pofernmnce Mrassremcat

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS . 6042/2
PAPER 2

JUNE 2019 SESSION 3 hours

Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Scientific calculator

TIME 3 hours
INSTRUCTIONS TO CANDIDATES

Write your Name, Centre number and Candidat

¢ 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 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 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, J2019,

©ZIMSEC 32019 [Turn over

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2 Section A (40 marks).

Answer all questions in this section.

Express 2x : F .
. xP @=1) tesa? (1 partial fractions. ISI

i E V
(i) "xpress 6Cos0 — J2Sind in the form RCos(0 + a) where R>Oand
O<a < 99°, 12]

(ii) ence or otherw; < ,
for 0° <0< 360°. solve the equation V6Cos0 — V2Sing = 2

15]
Given that (1 ~— ~
(1 ax)" = 1 15x + 90x? + bx3, find the values of a;bandn. [91

The function fis defined by f(x) = in(2x — 3) forx € R,x > 2

(i) Find £~* (x) and state its domain. . me
(ii) Solve the equation
f(x) — 6f 1 (-x) +5 = 0. (6]
(i) Use the trapezium cule with 5 ordinates to find an approximate value of
S311 + Sin?(2x)] dx correct to 8 decimal places. iM
ata ma z.
(ii) Find the exact value of fol1 + Sin?(2x)] dx. 4]
(iii) Hence, find the percentage error in using the trapezium rule as an
[2]

a
approximation of Sela + sin?(2x)|dx.

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A Section B (80 marks)
—_ any five Questions

(a) from this section, Each question carsies 16 marks.
ind the im: .
4 Ay 88° Of the Tine y= Sx under the transformation matrix
2 sh 131
(b) It i * 2
'8 given that matrix M= (-: : 2)
. 4° -4 3
(i) Evaluate M2. ray
(ii) ‘ .
Find M~1, the inverse of matrix M. 16]
(iii) H .
lence, or Otherwise solve the simultaneous equations:
2x4 By 442 = 4
~3x + 2y 4 22 =14
4% ~ 4y 4.37 = 29 143
(a) Us a
“7 the Substitution u? = x — 1 to find the exact value of
f xv (x ~ ]) dx. ia
b .
) The diagram below shows the shaded region S bounded by the curve
y = 4vx , line PQ and the x— axis,
y
Find the
(i) coordinates of P, 1
(ii) equation of line PQ, 12
iii F region S,
(ii) area of reg 6042/2 32019 14]
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4

(iv) exact volume of the solid of revolution generated when region

S is rotated completely about the x — axis. [5]
The rate at which a proportion x of a certain population is infected by cholera is
directly proportional to the product of the proportion infected and that not yet
infected. At time t days the Proportion of the population infected is x and not
infected is (1 — x),
(i) Write down a differential equation relating x and ¢. (2)
(ii) Initially 20% of the population was infected and 120 days later it rose to
80%. Solve the differential equation, expressing x in terms of /. {11]
(iii) Hence, find the Proportion infected after 60 days, {2}
(iv) Describe what happens to the population as time increases. fy)
(a) The equation of a circle is given by 3x? + 3y? — 12x 4 18y + 3k = 0.
Find the value of k for which the circle has a radius 6 units. {4}
(b) Use Taylor’s series expansion in ascending powers of (x — 1) to find
x<em>inx up to and including the term in (x -1)</em> [5]
© Show that the equation x5 — 10x —5 = 0 has a root between |
and 2. [3]
(ii) Taking 1.7 as the first approximation, use the Newton-Raphson
method twice to approximate the root of the equation
x* — 10x —5 = 0 to 2 decimal places. [4]
4
The plane x has equation r.{ —1 } = 1. The points P and Q have coordinates
(—5; 1; 3) and (2; 4; —1) respectively.
(i) Show that Q lies on plane x. [2]
(i) Find the
(a) equation of the line perpendicular to x through P, [2]
(b) Cartesian equation of line PQ. {3]
(iii) Calculate the
(a) _ acute angle between x and the line PQ, I4|
[5]

(b) __ shortest distance from P to 2.

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" @) Ona Single 5 Bram sg} 6 = org(z NAde the pea: f Ss wand |z "region defined by the inequalities () Solve the ves a, [3] “quatio; 23 ©) ~S +12; ¢ Use de {6] Sinosinso® theorem tg show that 2 @)~ Theg *6Sin°0 ~ 205int 4 5Sin?9 71 he diagram gf : ho . R. PQ > on and genie 0, passing through the points P, Q and aPProximate area ofa © QPR = 6. Given that 0 is small. find the tangle PQR in terms of 0. : [4]

S

(b) Show that the set A = {1; i; —1; -i} forms a group under multiplication. [6]

(c) Prove by induction that
nm

ic += arin +1)(n4+2)

re]

for all positive integers n. {6}

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