2018_maths_alevel_nov_paper2

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For Performance Measurement

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/2
PAPER 2

NOVEMBER 2018 SESSION 3 hours

Additional materials:
Answer paper
Graph paper

List of Formula
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.

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.

Questions are printed in the order of their mark allocations.

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 7 printed pages and 1 blank page.

Copyright: Zimbabwe School Examinations Council, N2018.

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©ZIMSEC N2018

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Section A (40 marks)

Answer all questions in this section.

— 1, When divided by

i = ax? is divisible by x
The polynomial P(x) = ax? + bx + c is divisible by ly. Find the values

x +1and x — 2 it leaves remainders of 2 and 8 respective!
of a, b, and c.

(a) Show by calculation that the equation
4Cos $x = 6-—~ hasa root between 3 and 4.

(b) Use Newton Raphson method twice to find the root correct to two
decimal places starting with x, = 3.5.

(a) Use the substitution x = Cos?6 to show that

‘

f rydx = —f2Cos@d 6.

1
(b) Hence evaluate fe Fn dx-correct to 3 decimal places.

2x?-7
(x-3) (2x+5)

(a) Express in the form A + = +

3 ax+s"

(b) Hence evaluate
2_
f. 2x¢—-7 dx

(x-3) (2x+5)

correct to 3 decimal places.

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[6]

[3]

[4]

[4]

[4]

[5]

[4]

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The diagram shows two circles touching at X. The bigger circle has centre A and radius 9 cm. the smaller circle has centre B and radius 3cm. DC is a common tangent to both circles.

Find the

@ perimeter of ABCD. [4]
(ii) angle BAD. [2]
(iii) exact area of the shaded region. [4]

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a)

on,

(b)

7 (a)

(b)

(c)

4
Section B

Answer any five questions in this section.

Each question carries 16 marks.
The line L, has equation

= ()rQ)

and the plane 7r; passes through the points A, B and C with coordinates
(2; -1; 3), (4; 2; -5) and (-1;3; —2) respectively.

Find the
(i) Cartesian equation of 7. [5]
(ii) acute angle between the plane 7, and the line L,. [3]

The plane 7 has equation

2
r.{—-1] = 6.
-1
The line Ly lies in the plane mz and is._perpendicular to L,. The line
Lz passes through the point (4; 2:1).
Find the

(i) vector equation of L», {4]

(ii) vector equation of the line of intersection of the planes 7,
and 7 . [4]

Solve the equation z? + 3z + 5 = 0, giving the answers in the form
x+iy. [5]

(i) On a sketch of an argand diagram show the locus representing
the complex numbers satisfying the equation

[zl =|z-3-5il. [2]
(ii) Find the

1. complex number represented by the point on the locus
where | z | is least. (1)

2, modulus and argument of the complex number in 1.

giving the argument correct to 2 decimal places. (3]
Use De Moivre’s theorem to express Cot 58 in terms of Cot 6. [5]
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4 8 ) (a) Prove by mathematical induction that for all positive integers, 7, 10" + 3x4"*1 4.5 is divisible by 3. [6]

mw) Under a 2 x 2 matrix M the images of A(1,0) and B(0,1) are Ai(3,5)
and B, (5,9) respectively.

Find the
(i) matrix M. [2]
(ii) image of point (2; —5) under matrix M. [2]

(c) Given the matrix

2 6 1
Po = {-1 2 -1],
4 3 1

find the inverse of matrix P. [6]

9 (a) (i) Show that the set of integers
{1, 3, 5, 7} under multiplication modulus 8 forms a group. (7]
(ii) | Write any three subgroups. [3]

(b) The set {f, g,h, k}. under the operation of composition of functions
forms a group H, where

fix 7 x
Bf
gix ——7> |
hix TT -x
1
kix ——> Te
Show the operation table for H. [6]
10 (a) i) Find
n
y Be (6)
r(rt+1)(r + 2)

r=1

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(ii) | Hence or otherwise deduce the value of

co

y issih [2]
r(r +1)(r + 2)

T=1

(b) Use the series expansion of Cos x to expand (1 + x)? Cos x up to

and including the term in x3. [4]
(c) Given that “24 2 43) = owherey = 1 at x = 0 and %=2
dx? dx dx
atx = 0.
Use Taylors’ series to find the first three terms of (1 + x)? Cos x. [4]

(i) (a) —_ Solve the inequality

R!

-2 x-6
wai < x2" (6)
(b) It is assumed that the length, /, in cm of a certain snake at time, t,
1

months after birth increases at a rate proportional to (10 — D2:
Whent = 0 L=1 and £ = 03.
(i) Show that / and ¢ satisfy the differential equation -

aH _ o1(10- 0:

at co [2]
(ii) Solve the differential equation and obtain an expression for /

in terms of f. [6]
(iii) Find the maximum length of the snake. [2]

12 (a) Theequationy = ¥x_ is used to approximate the square root of a |

number.
Find an approximation for ¥16.01. [4]

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(b) ‘The function f is defined by f(@) =1+Cosx OSketch on separate diagrams the graphs of the following functions
showing clearly all intercepts and turning points.

@ y= fx) [2]
di) y = (|f@)-3 (3]
di) y = £(8) [2]
(iv) y = £(x-2) [2]
(v) yy = —2f(x) [3]

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