6002q1Specimen paper

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

ADDITIONAL MATHEMATICS 6002/1

PAPER 1
SPECIMEN PAPER 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.

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.

The use of a scientific calculator is expected, where appropriate.

Y ou are reminded of the need for clear presentation in your answers.

This question paper consists of 6 printed pages and 2 blank pages.
Copyright: Zimbabwe School Examinations Council, Specimen paper.

©ZIMSEC Specimen paper [Tum over

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A machine is programmed to cut lengths of string from a uniform string
coming up continuously from a factory belt. The lengths are all measured in
metres, and are as follows:

22282381208
100’ 100’ 100’ ” 100°

Calculate the total length of the string cut, giving your answer in kilometres.

The points A, B and C have position vectors
a=i-3j, b=2i- j +k and c=10j 10k respectively.
@ = Calculate the product a.(b ‘ c).

Gi) _—s Describe fully the geometrical meaning of this product.

A matrix, M, is given by

cl 3S
m=¢? 2+

vB 1 og

gc 2 é
@ ~~ ‘Find M°.

Gi) Hence or otherwise find M°.
Gii) Describe fully, the geometrical transformation represented by the
matrix M’.

The function f(x) is given by f(x) = 4x’- 4x- 3.
@ — Express f(x) in the form (ax +b) +c, where a, band care integers.

Gi) ‘Using an appropriate change of variable, or otherwise, show that

1, 4/456

=—/n

jt dx = 4
Or) 2. SNe

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@ The integral lnis given by ln = Gpin"5xdx. Use integration by parts, or otherwise, to show that 1 . m-16 In = - —cos5xsin”'! 5x +¢—— 4, 5. 6 " Sn +§ nov? (6

Gi) —w Hence evaluate

ye
(0 sin’ 5xdx. [3]

A curve, C, has a function y = k fx, where kis a positive constant and 2£x £4.
@ Find 2, giving your answer in terms of xand k [2]
IX

Gi) ~~=— The curve, C, is rotated completely about the xaxis.

Show that the area of the surface generated, S, is
pki 2\s 2 su
& = ra (16 +k pp - (8 +k Py [6]
i

Gi) Hence evaluate S:— S giving your answer correct to 3 decimal places. [2]

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It is given that a matrix

21 2 -4 6
A = @2 0 -1 shasa determinant 3.
G-23 kz

(a) @ Find the value of k

Gi) —_—sUsing this value of k find adj.A, the adjoint matrix of A.

Gii) Hence or otherwise, write down A“'.

&) _Itis given that when the point (a; b, c) is transformed by the matrix A,

its image is (- 8;- 9;

Find the values of a,

3).

bandc

A setG ={0, 1,2, 3 4}, where the elements are defined from the turning of

a minute hand of a clock as
0 = atum throug
1 = atum throug
2 = atum throug

3 = atum throug

4 = atum throug!

Defining a» bas “a turn of the element a followed by a turn of the element b’.

follows:

h 0 minutes or multiples of 60 minutes.
h 12 minutes

h 24 minutes

h 36 minutes

h 48 minutes

@ Construct the multiplication table for all the elements of the set G.

Gi) —_ Show that the set G is a group.

Gii) ‘Find the order of each of the elements.

@v) | Establish whether the set G forms an abelian group. Give a reason for

your answer.

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11

A curve, C, is given parametrically as

y =coshbtand x=sinhat where a and bare constants and tis the
parameter.

@ Find “ in terms of a, b, and t [2]
xX

&) Inthe case where b =4 and a =2, find the cartesian equation of curve

C inthe form y =f(x)and show that o = 4x, [4]
xX

(Co) Hence show that the length, L, of curve C between the points
where x = 1 and x =2is

L=V65-! wire ini ene [6]

é
é You may make use of the standard integration given as
é

V2 tx? dy = fa? +2 + * sinh xt
2 al

The function f(x) is given by f(x) = _16 43x,
(x +1)
@ = Write down the expression for fdx) and fqx). [2]

Gi) —_—- Hence find the coordinates of the tuming point(s) for the curve y= f (x),
stating their nature. [6]

Gii) Write down the equations of the asymptote(s) of the curve y= f(x) and
hence sketch the curve showing any intercepts with the axes. [5]

(a) Solve the differential equation
dy “l “1.
Gy toy (10x- 3) = (2x), given that y=0 whenx=1. [6]
IX
® Acurve,S, has its differential equation

a’ IY 9 ® +k’y = 0, where kis a constant.
dx? dx

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@ Find the general solution of the differential equation.

Gi) ~—_—- Hence find the exact particular solution of the differential
equation, given that the curve, S, passes through the points (0; 4),

(in 2; 0) and gn; 326
[8]

A line, 11, has an equation r =i- j- 3k +/ (i +4j- k,) and the line, /,, passes
through the points A (0; 6; 6) and B (8; - 6; - 34).

@ Find the vector equation of /, and show that the lines l: and lz
are perpendicular to each other. [4]

Gi) ‘Find the point of intersection of l and h. [4]

Gi) sl. Find the cartesian equation of the plane, , which contains both
i, and 1,.

2. Calculate the distance of the point P (1; - 7; 1) from the plane p. [8]

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