2016_maths_alevel_mj_paper1

Extracted Content: 2016_maths_alevel_mj_paper1

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

MATHEMATICS 9164/1
PAPER 1

JUNE 2016 SESSION 3 hours
Additional materials:
Answer paper
Graph paper
List of Formulae
Electronic 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.

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 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 and candidates are advised to attempt
questions sequentially. :

The use of an electronic 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, J2016.

©ZIMSEC 12016 [Turn over

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Find the inverse of f(x) = ax + b where b and a0 are constants. ~ [2] 3 2 Express SRS ewes 3 in the form a Be + anne where A, B, C and D (x? +1) x +i (x? +1) are constants. [3] Find the set of values of m such that the gradient of the line passing through the points (m; 4) and (1; 3 — 2m) is less than 5. [5] R

3 cm
P Q
: ay peteres a
In the diagram QPS = ra PQR = 2 PS =5 cm and QR=3 cm.
Given that tan POS = s, find the exact length of RS. [6]

() Expand (9-4)? in ascending powers of x up to and including the term
Sn 3
in x’, [4]

(ii) State the range of values of x for which the expansion is valid. [1]
iii) By using x = 5 in the expansion, find an approximation for V77
leaving your answer in the form a [2]
q

@ A geometric series has first term x and second term (x? - ei If all the

terms are positive, find the set of values of x for which the series
converges. (3]

(ii) Given that x = >, find the smallest value of n for which the sum of the
first n terms differs from the sum to infinity by less than 0,001. [4]

9164/1 J2016

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10

11

A closed cylindrical tin of radius r and height h has fixed volume V.

(i) Find the total surface area, S, of the tin in terms of a, r and V.

(ii) Show that its total surface area is smallest when its diameter and height

are equal.

(0) Show by calculation that the equation e* -cosx =2 has a root between

x=Oandx=1.

(ii) | Use Newton-Raphson method twice, starting with xo = 1, to find the root

of e* —cosx —2 =0, correct to 3 decimal places.

Solve exactly the equations:

3
(a) oe -2)= 3log;2 ~logs(x -2)

2
() Oe = Ie =3

(a) Prove the identity

tan’ @ —sin’ 6 = tan? @sin? @

(b) Solve the equation.

4sin’ Otan@ -tan@ =0 for 0 < @<2z.

(i) Express the complex number w = 8 + i
: +

(ii) Hence or otherwise, find

le |w| in the form a

2. argument of w.

The mass of an animal at time f is x grammes. The rate of increase of the mass is

v

lb

’

4-i

in the form x + iy.

directly proportional to (10 000 —x*) where 0 <x < 100.

(0) Write down the differential equation connecting x and ¢.

(ii) Solve the differential equation given that x = 10 when ¢= 0 and

that x = 50 when ¢ = 10, giving x in terms of f.

9164/1 J2016

[3]

[4]

[3]

[4]

[4]

[4]

[6]

[2]

[8]

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13

14

The function fis defined by f(x) = —2) @ + 3), x© IR.

Sketch on separate diagrams, showing clearly all the intercepts and turning
points, the graphs of:

@® y=F(x)
dy) yy =|F (>)
(ii) y = f(2x)
(iv) y= f(x-1)
W) -y=-2f(x)

i]
'
1
1
:
1
i)
'
t
1
1
‘
=)

The diagram shows region R, bounded by the curve f (x) =x+2"', the x- axis,
the lines x = 2 andx =5.
@ Calculate the exact area of the region R.

(ii) Using the trapezium rule, with 4 ordinates, find the approximate area of
region R correct to 3 decimal places.

(iii) Calculate the exact volume generated when region R is rotated completely
about the x-axis.

9164/1 J2016

[4]
[2]
[2]
[2]
[2]

[4]

[4]

[5]

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2 (a) __ Relative to the origin O, the position vectors of P, Q and R are | 3}, 4 1 3 0 | and | 1 | respectively. -1 -2 Find POR. [4] m=2 (b) The vector W =| m+3}. m6 (0) If Wis a unit vector, find the possible values of m. [2] m+3 (ii) ~=The vector V=| 2m |. -2V6 If W is normal to V, find the values of m. [4] m=4 Gii) ThevectorU=| n |. 26

Given that W is parallel to U, find the values of m and n. [4]

9164/1.J2016

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