2016_maths_alevel_nov_paper1

Extracted Content: 2016_maths_alevel_nov_paper1

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

“ MATHEMATICS 9164/1
PAPER 1

NOVEMBER 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 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 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, N2016.

©ZIMSEC N2016 [Turn over
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The complex number aoe is denoted by uw.

Find the

(i) modulus of u, [2]
Gi argument of w. [2]
Find the series expansion of (9 + ox)? up to and including the term inx’,
simplifying the coefficients. (4]
The equation of a circle is x° +)’ =4 and the equation of its tangent is

y=2mx tc.

Show that! 16m? -¢ +4=0. (4)
@ Given that f(x) = 2 - |x + 3], sketch the graph of y =| fx)). [2]
(ii) Hence from (i) solve the inequality |f{x)| > 1. (2]

The variable quantities x and y are related by the rule y= 7 where P is a
x

constant. Five pairs of values of x and y measured experimentally are shown in
the table.

Ty [og | 11 | 135 [140 [ 152 |

x

(i) Plot the graph of y against x 3]

\s\

(ii) Use the graph to find the value of P.
[2]

x

5500

oO x o.”
The diagram shows the position of an aeroplane P flying in a straight line at a
constant height of 5 500 metres. The angle of elevation of the plane from the
observer O is 6° when the observer is x m from a point Q, on level ground, which
is vertically below P. The speed of the aeroplane is 550 000 m/hr.
Find the rate of change of angle @ when 6 is 25°. [5]
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(i)

(ii)

ji)

(ii)

(a)

(b)

x-3
2

Express in partial fractions.

x -1
F 3x -3 Oe ‘ ;
Hence from (i)find a; Fe ie a giving the answer in terms of a single
xe
logarithm.

Show that the equation cos (45° + 8) = 2sin (45° - 6) can be reduced to
tan 8=1,

[2]

[4]

[4]

Hence or otherwise, solve the equation cos (45° + 8) = 2 sin (45° - 8) for

0°s 6 s 360°.

The present population of a country is 27 million. The population is
projected to be 38 million in 36 years time.

Calculate the annual growth rate correct to two decimal places, if this
projection is based on a geometric sequence.

15
Find the sum, S, defined by S= S(2n + |

nel

[2]

[3]

[3]

The points P, Q, R and T lie on the circle centre O and radius 7. S is on OQ produced
such that QS =r and PS = RS. Angle POS = @ radians.

@
(ii)

Calculate the shaded area PQRS.

Show that the perimeter of PTRS is given by r(x -0+V5- 4c0s0).

9164/1 N2016

[4]
[5]

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12

13

14

4

The polynomial 2x3 + ax? —bx +12 where a and b are constants is denoted by
fix). The result of differentiating f(x) with respect to x is denoted by /'(x).

It is given that (x — 1) is a factor of, fx) and when f'(x) is divided by (x — 1)
the remainder is — 5.

(i) Find the values of a and b.
(ii) | Whenaand b have these values in (i), solve the equation Ax) = 0.

Relative to the origin O the position vectors of the points A and B are

OA =31+4j +(2-b)k
OB=-i+j-k

(i) Find the unit vector in the direction of OB.
(ii) Find the value of b, if BOA = 90°.

(iii) = Write down an expression for jABi.

(iv) Hence find the value of b if iaBl is minimum.

‘A wet substance in open air loses its moisture at the rate proportional to the
moisture content. M is the moisture content after ¢ hours,
(i) Show that the differential equation connecting M and f is given by

— =kM,
dt

(ii) Solve the differential equation to obtain M in terms of ¢, given that initially

M= mo andt=0

(iii) If the substance is hung in open air, it loses half of its moisture during the

first hour.
Find the time it will lose 95% moisture, in same weather conditions.

(i) By sketching the graphs of y = tan x and y = 4x ona single diagram for

[5]
(4)

[2]
[2]

[3]

[2]

[6]

[4]

fs : ae
O<x< 3 show that the equation tan x = 4x has exactly one positive root. [3]

(ii) Show, by calculation, that the root of the equation lies between 1 and 1.5. [3]

(iii) | Show that the Newton-Raphson structure can be written in the form
x,fan?x, —tanx, +X
tan’x, -3 :

(iy) _ Use the structure in (iii) once starting with x; = 1.4, to estimate the root
correct to 4 decimal places.

(3)

[3]

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(a) Sketch ona single diagram, the graphs of y=f{x) and y= f7'(x) making
clear the relationships between the graphs. ‘ ; [2]

(b) (i) — The function gis defined by (x) = 3x? + 2x-2, for xER.
Express gfx) in terms of x in its simplest form, 22]

(ii) | Hence find the minimum value of oftx). (3]
(c) Given that g(x) = 3x" + 2x — 2 is defined for x >k,
(i) state the minimum value of k, such that g(x) can have an inverse, [3]

(ii) find, g(x).
[2]

16

The diagram shows the graph of y = e* — e** with shaded region R. Its maximum point is P,
@) Show that the x - coordinate of P is ine. [4]

(ii) Calculate the area of the shaded region R. [4]

(iii) Find the volume of the solid of revolution generated when the shaded region
R is rotated completely about the x-axis giving the answer correct to
3 significant figures. [4]

9164/1 N2016
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