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ADDITIONAL MATHEMATICS PAPER 1 ( 4038 / 01 ) Secondary 4 Express / 5 Normal Academic 13th September 2011 2 hours

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O2011Preliminary Examination

READ THESE INSTRUCTIONS FIRST

1. Write your name, class and register number in the spaces provided.

2. Answer all questions on the foolscap paper provided.

3. Omission of essential working will result in loss of marks.

4. The number of marks is given in brackets [ ] at the end of each question or part question.

5. You should not spend too much time on any one question.

6. The total mark for this paper is 80

You are expected to use a scientific calculator to evaluate explicit numerical expressions.If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.For , use either your calculator value or 3.142.

DO NOT OPEN THIS PAPER UNTIL YOU ARE TOLD TO DO SO

This document consists of 6 printed pages.

Setter: Mrs Li

Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation

x =

Binomial expansion

where n is a positive integer and

2

Special instructions :1. Start every question on a fresh page of paper.2. Attach the green cover page to the front of your

answer scripts.

For Examiner’s Use

80

2. TRIGONOMETRYIdentities

cosec 2 A = 1 + cot 2 A

Formulae for

1 The remainder of  when divided by ( x + 1 ) is twice the remainder of  when divided by ( x – 2 ). 

(i) Find the value of k. [2]

(ii) Using the value of k found in (i), solve , leaving the non-integer solutions in the surd forms. [3]

2.Given that , find and hence solve the simultaneous

equations

[5]

3

3 Using a separate diagram for each part, represent on the number line the solutions set of

(i) x ( 2x – 1 ) > 3 [2]

(ii) x ≤ 6 – x2 [2]

Hence find the set of values of x which satisfy < [2]

4 (i) Solve the equation

[3]

(ii) Given that and ,

(a) express in terms of a and b. [3]

(b)find the value of k such that , give your answer to

4 significant figures. [3]5

A curve has the equation .

(i) Find the gradient of the curve, in the exact form, at the point when [3]

(ii) Given that y is increasing at a constant rate of 2.5 units per second.

Find the corresponding rate of change of x at the instant when ,

giving the answer correct to 4 significant figures. [2]

6 Given that

(i)find an expression for [3]

(ii)

4

hence, evaluate [3]

7 Given that the roots of are 2 and 2 where > 0 and > 0.

(i) find the values of + and , [3]

(ii) Find the equation whose roots are ( 3 – ) and ( 3 – ). [3]

8 Without using a calculator,

(i) find the value of , [2]

(ii) Hence show that [3]

9The gradient of a curve y is given by .

(i) Given that the curve passes through the point ( 2 , –1 ), find the equation of the curve. [3]

(ii) Find the x–coordinates of the stationary points of the curve. [2]

(iii)Obtain an expression for and hence, or otherwise, determine the

nature of each stationary point. [3]

10 (i)Show that

[2]

(ii) Hence, find the value of [3]

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11 The variables x and y are related by the equation , where a and b are constants. The table shows experimental values of x and y.

x 1 2 3 4 5 6

y 7.39 10.45 12.80 14.78 16.52 18.10

(i) Using graph paper, plot against and draw a straight line graph.

[3]

(ii) Use your graph to estimate the value of a and of b. [3]

(iii) Estimate the value of x when y = 15. [2]

12 (i)Sketch, on the same diagram, the graphs of and for

[3]

(ii) State, for the range of , the number of solutions of [1]

13 The diagram below shows two points A ( 3 , 4 ) and B ( 5 , 0 ) on a circle with centre P.  P also lies on the line 4y + 3x = 10.

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(i) Show that the coordinates of the point P is ( 2 , 1 ) [3]

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

(iii) Determine if the point ( – 0.5 , 3 ) is inside, outside or on the circle. Show your working clearly. [2]

End of paper 1

Answer key :

1(i) k = 3 1(ii)

2 , p = –2, q = –1

3(i)

3(ii) Hence

4(i) 4(iia)

4(iib) k = 0.7906

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4y + 3x = 10

P

A ( 3 , 4 )

B ( 5 , 0 ) x

y

0

5(i) 5(ii) 0.1178

6(i) 6(ii) 148

7(i) 7(ii)

8(i) 9(i)

9(ii) 9(iii) , max at ,

min at

10(ii)0.549 or

11(ii) a = 2 , b = – 0.5 11(iii) 4.12

12(ii) 3

13(ii) 13(iii) , outside the circle

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