2017_maths_alevel_nov_paper1

Extracted Content: 2017_maths_alevel_nov_paper1

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se ( D) ZIMBABWr

See

iE NCIL
SCHOOL EXAMINATIONS COU

uevel
ral Certificate of Education Advanced Le

MATHEMATICS
APER } PURE MATHEMATICS

Gen

9164/1

3 hours
NOVEMBER 2017 SESSION
Additiong} Materials:
; "sWer paper
1

stot Formulae

Graph Paper

Non-programmable electronic calculator

TIME 3 hours

INSTRUCTION

STO CANDIDATES

Write Your Nan
Paper/answer b¢

le, Centre number and Candid
ok let,

i he answer
ate number in the spaces provided on tl
Answer

all questions.

Ifa numerical answer ¢
question, then in the ¢

Cases it should be

ired is specified in the
annot be given exactly, and the accuracy ae maces oe ee Sea
ase of an angle it should be given correct to the neares g
viven correct to 2 significant fi

gures.
INFORMATION FOR CANDIDATES

i uestion or part question
Che number of marks is given in brackets [ ] at the end of each q

The total number of marks for this paper is 120.

i i vations.
Questions are printed in the order of their mark allocation

e use a non-pro, Jrammabie electronic p ry e app Op ‘tate.
] h of ig bl ] ct calcu ator 1S ex ected wher ropr

ion i swers.
You are reminded of the need for clear presentation in your an

This question paper consists of 5 printed pages and 3 blank pages.

Copyright: Zimbabwe School Examinations Council, N2017.
@©ZIMSEC N2017

[Turn over

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Siven the “omplex numbers w = 1 +2i and u= 3-4, find (@) inthe form a + ib, where a and b are real numbers (i) uty (ii) ow . (b) the argument of uw, . Functions fand are defined as f(x) =3x-1, 5 eR A(x) = 2x + 5, xER. , Find the value of x for which fh(x) = 2hf{x). " (i) Express 2x? - 347 in the form p(x+q)' +r, where p, and r are [3]

constants,

(ii) Hence, or otherwise write down the coordinates of the turning point of
y=2x? 3x47. ng}

15]

3x+8 3 F ;
Express ———_--~_ into partial fractions.
(2x +1)(x? +3)

Quantities y and x are related by the equation y*x° =c where c is a constant.

Find the percentage decrease in y if x increases by 0.5%, {5]

In a geometric progression the first term is a and the common ratio is r where
O<r<l.

If the sum of the first four terms is half the sum to infinity, find
0) the exact value of r, ii
(ii) the 9” term when a= 2. .

Find the equation of the normal to the curve y = 3e* 4 x+3 at the point
where x = 0. [6]

(i) Show that the equation e* -2x+3=0 has Only one
sketching the graphs of y=e™ and y=2x_3 on theme, oe [3]

(ii) | Taking x, =1 as your first approximation to
e* -2x+3=0, use the Newton-Raphson
correct to 3 decimal places,

the root of €quation
method twice to find the root

fal
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10

11

12

The position vectors of points A, B and C relative to the origin O are i+2 j -3k,
3i - 2 +5k and pi- pj +(p-1)k> respectively.

Find

(i) a unit vector in the direction AB,

(ii) the angle between OA and OB,

(iii) the value of p for which OB is perpendicular to OC.

3x metres, width 2x metres,

A rectangular wooden block has base length asain ld m?
vo :

height A metres, total surface area of A m’ an
(a) Express in terms of x the

(i) height, A,

(ii) _ total surface area, A.

(b) Given that x can vary, find the stationary value of the total surface
area A and determine its nature.

It is given that g(x) = 3x4 + bx? +cx? - 7x -4 has factors(x +1) and (x -1).

(i) Find the value of b and the value of c.
(ii) Factorise g(x) completely.

(i) Prove the identity cot20 = cot@ -cosec260.

(ii) | Hence, or otherwise solve the equation cot@ - cosec2@ = NS for
2

0°s 6s 360°.

[3]
Bl
(2)

(1)
2)

[5]

[5]

[3]

[4]

[5]

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4 vx+l,

the curve Y=

13 :
tg © diagram Shows the shaded region S bounded by
Ine yor -! and the fgets:

Find the exact value of the

(i) area of S, [5]
(ii) volume generated when S is rotated completely about the x-axis. [5]
14 (a) Solve the equation 2'<em>?</em> — 9(2") an 4]

(©) @ —Onithe same axes, sketch the graphs of y = x? and y=Rx-3), [3]

(ii) Hence, or otherwise, solve the inequality 2. _ 3] < x2

[3]
15 (0) Find the exact value of I 2xe* dx. :
(ii) Use the trapezium rule with 6 ordinates to Cvaluate fo ,25«
correct to 4 decimal places, 07 e*dx,
i [4]
(iii) | Hence, find correct to 3 decimal places, the Percenta
trapezium rule as an approximation to the integral, 8¢ error in using the
[2]

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16

(a)

(b)

d:
(i) — Given that y = e™ sinx and a e<em> cosx + 2e™</em> sinx, find

da’

a 21
d’y

re 1
Hence, or otherwise, obtain the Maclaurin series for

y =e™ sinx, up to the term in cae [2]

(ii)

The rate at which the temperature of a hot iron bar, 6°C, falls is
inversely proportional to its temperature at time, ¢ minutes.

Show that the above situation satisfies the differential equation

& d@ sk

a8 Me
(ii) Solve the differential equation, expressing @ in terms of f. (2]
(iii) _ If the temperature decreases from 80°C to 70°C in 20 minutes,

find its temperature after a further 20 minutes. [5]

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