2014_maths_alevel_nov_paper1

Extracted Content: 2014_maths_alevel_nov_paper1

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For Pesformance Measurement

ZIMBABWE SCHOOL EXAMINATIONS COUN CIL

General Certificate of Education Advanced Level

MATHEMATICS 9164/1

A
PAPER 1
NOVEMBER 2014 SESSION 3 hours
Additional materials: |
Answer paper
{ Graph paper
KL bist of Formulae S/S

“\TIME 3 hours
INSTRUCTIONS TO CANDIDATES

; Write your name, Centre number and candidate number in the spaces provided on the answer
paper/answer booklet,

There is no restriction on the number of questions which you may attempt.

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 { blank page: 5

Copyright: Zimbabwe School Examinations Council,

©ZIMSEC [Turn over
ane —

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pg
real numbers. ji? [3]
i of
Solve the inequality
10 -
ae sagt th [4] .

Given that F = kS*, where k is a constant, find the percentage change in Fif §

increases by 5%. [4]
(i) Use the trapezium rule with four trapezia of equal width to estimate
the value of.
2 be
f 21s dx,
by
giving your answer correct to 4 decimal places. [4]

(ii) Explain whether the trapezium rule has underestimated or overestimated
the value of this integral, {1]

P is a fixed point on the circumference of a circle with centre O and radius 5 cm.
Another point Q moves round the circumference of the same circle at a
constant speed of 3 cms“.

The angle POQ is @radians and the length of arc PQ is x cm.

Find the rate of increase of

(i) Gin radians per second, [3]
(ii) the area of sector POQ. [3]
3)
Express rm tal in partial fractions. [6]
(x - 2)(x + 5)

9164/1

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7 The table shows the values of a variable p obtained experimentally from
the values of q.

= a —_ bs |

The variables p and gq are related by the relation log p= blog q + log a.

By plotting the graph of log p against log g, determine the approximate values
of a and b. [6]

8 (a) It is given that f(x) = px? +5x-4,

Find the set of values of P for which /(«) is positive for all real values
of x. . [2]

(b) The remainder when 4x4 — 5x2 — 13x +3 is divided by x +} is equal
to the square of the remainder when 2x? - 3 is divided by x+ 5,

- Calculate the possible values of 5, correct to 2 decimal places. [4]
9
S Q
R
M
P

The diagram shows a triangular prism MNPRQS with MP = NP = 5 units. O is the
midpoint of MN.

The unit vectors i and & are taken along OM and OP Tespectively, and / is
taken parallel to MS,

Given that OR = O7+4k,

find
@ the unit vector in the direction PS, [3]
(ii) the angle SPQ. [3]

9164
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I

12

13

14

4

The complex number zsatisfies the equation

Find

@)

(ii)

z in the form x + iy,

1
the modulus and argument of —.
Zz

In a geometric series, the ratio of the sum of the first four terms to the sum of
the first two terms is 17: 16.

@)

(ii)

Calculate the value of the common ratio r given that r> 0.

Given that the third term is 2=, find the value of the sum to infinity.

It is given that e~”(1~4x) =(7+43x).

@)

(ii)

)
(ii)

(a)

(b)

Show that y = ~S[m(7 +3x)~In(1-44x)].

Hence obtain the Maclaurin expansion of y up to and including the term
inx’,

Use the binomial expansion to simplify (x +2)’ - (x -2)’,
Hence use your answer in (i) to find the exact value of

(V5 +2)’ - (V5 -2)’,

Prove that sec2@ +tan26 = sinesoost
cos@ —sin@

By using the substitution U = sinx or otherwise, find the exact value
= cos

of [2 — a .

9 34+c08° x

9164/-

[3]

[4]

[4]

[3]

[2]

[5]
[6]

[2]

[4]

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16

17

Gi)

(iii)

(a)

{b)

(a)

(b)

By sketching two appropriate graphs, show that the equation
sinx +1-4x? =0 where -275x< 22, has two real roots. [2]

Verify by calculation that the positive root in (i) lies between x =0.6
and x =0.7. [3]

Taking x, = 0.6 as the first approximation of x in (i), use the
Newton-Raphson Method twice to find the root correct to 5 decimal
places. {4]

Find the perpendicular distance of the point P(-3; 2) from the line whose
equation is 3x -Sy =7. {6]

Find the equation of a circle which passes through the point (5; 4) and

touches the y-axis at (0; 2). [6]
The gradient function of a curve is given b YL i and
” ‘ a (3x +4)

M(2; 6) is a point on the curve.
Find the equation of the curve. . [4]
The rate of increase in the number of people, x, who own a laptop is

proportional to the product of x and N ~ x, where Nis the total
population.

Initially x = aN and after one week x = <n ‘

@ Form a differential equation from the above information and show
x
= 37,

that after ¢ weeks,
N-x

Gi) Find the time, f, in (i) when x = oN .

(iii) Sketch the graph of x against ¢ for the equation in (i).
[11]

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