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# GCE O level A.math Specimen paper

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DC (SLM) 81102

UCLES 2013 [Turn over

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral Certificate of Education Ordinary Level

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.DO NOTWRITE IN ANY BARCODES.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles indegrees, unless a different level of accuracy is specified in the question.The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten all your work securely together.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 80.

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Paper 1 October/November 2013

2 hours

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Mathematical Formulae

1. ALGEBRA

For the equation ax2+ bx+ c= 0,

xa

b b ac

2

42

!=

- -.

Binomial Theorem

(a+ b)n= an+

(

n

1 )an1b+

(

n

2 )an2b2+ +

(

n

r)anrbr+ + bn,

where nis a positive integer and ( nr)= n!(n r)!r! .

2. TRIGONOMETRY

Identities

sin2A+ cos2A= 1

sec2A= 1 + tan2A

cosec2A= 1 + cot2A

Formulae for ABC

asinA

=b

sinB=

csin C

a2= b2+ c2 2bccosA

=1

2bcsinA

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Examiners

Use

1 The diagram shows the graph of ( )siny a bx c= + for 0 2xG G r, where a, band care positiveintegers.

4

3

2

1

O

y

x

1

2

3

2

State the value of a, of band of c. [3]

a= b= c=

2 Find the set of values of kfor which the curve ( ) ( )y k x x k1 3 12= + - + + lies below the

x-axis. [4]

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3 Show that1

1 2

cos

sin

sin

cossec

i

i

i

ii

++

+= . [4]

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For

Examiners

Use

4 The setsAandBare such that

: ,cosA x x x2

10 6 02 G G= =\$ .,

: ,tanB x x x3 0 6 02 G G= =" ,.

(i) Find n(A). [1]

(ii) Find n(B). [1]

(iii) Find the elements ofA,B. [1]

(iv) Find the elements ofA+B. [1]

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Examiners

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5 (i) Find (9 3 )sin x xd+y . [3]

(ii) Hence show thatr

( )sin x x a b9 3 d9

r+ = +r

y , where aand bare constants to be found. [3]

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6 The function ( ) 4 2x ax x bxf 3 2= + + - , where aand bare constants, is such that x2 1- is afactor. Given that the remainder when ( )xf is divided by x 2- is twice the remainder when ( )xf is divided by x 1+ , find the value of aand of b. [6]

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7 (a) (i) Find how many different 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8 and 9 if each digit may be used only once. [1]

(ii) Find how many of these 4-digit numbers are even. [1]

(b) A team of 6 people is to be selected from 8 men and 4 women. Find the number of different

teams that can be selected if

(i) there are no restrictions, [1]

(ii) the team contains all 4 women, [1]

(iii) the team contains at least 4 men. [3]

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8 (i) On the grid below, sketch the graph of ( ) ( )y x x2 3= - + for x5 4G G- , andstate the coordinates of the points where the curve meets the coordinate axes. [4]

y

x

5 4 3 2 1 1 32 4O

(ii) Find the coordinates of the stationary point on the curve ( ) ( )y x x2 3= - + . [2]

(iii) Given that kis a positive constant, state the set of values of kfor which

( ) ( )x x k2 3- + = has 2 solutions only. [1]

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9 (a) Differentiate ( )lnx x4 2 13

+ with respect tox. [3]

(b) (i) Given that yx

x

2

2=

+, show that

x

y

x

x

2

4

d

d

3=

+

+

^ h. [4]

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(ii) Hence findyx

xx

2

5 20d

3+

+

^ h. [2]

(iii) Hence evaluate2

7yx

xx

2

5 20d

3+

+

^ h. [2]

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10 Solutions to this question by accurate drawing will not be accepted.

The points ( , )A 3 2- and ( , )B 1 4 are vertices of an isosceles triangleABC, where angle B 90= .

(i) Find the length of the lineAB. [1]

(ii) Find the equation of the lineBC. [3]

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(iii) Find the coordinates of each of the two possible positions of C. [6]

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11 (a) It is given that the matrix2

4

3

1A =

J

LKK

N

POO.

(i) Find A+ 2I. [1]

(ii) Find A2. [2]

(iii) Using your answer to part (ii)find the matrix Bsuch that A2B= I. [2]

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Use

(b) Given that the matrix Cx x

x

1

12

=- +

-

x 1-

J

LKK

N

POO, show that det 0C ! . [4]

12 (a) A function f is such that ( ) 3 1x xf 2

= - for x10 8G G- .

(i) Find the range of f. [3]

(ii) Write down a suitable domain for f for which f 1- exists. [1]

Question 12(b) is printed on the next page.

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(b) Functions g and h are defined by

( ) 4 2xg ex= - for x Rd ,

( ) 5lnx xh = for x 02 .

(i) Find ( )xg 1- . [2]

(ii) Solve ( ) 18xgh = . [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the

publisher will be pleased to make amends at the earliest possible opportunity.

University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of

Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

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