acjc jc 2 h2 maths 2011 year end exam question paper 1

ANGLO-CHINESE JUNIOR COLLEGE

MATHEMATICS DEPARTMENT MATHEMATICS Higher 2 Paper 1 17 August 2011

JC 2 PRELIMINARY EXAMINATION

Time allowed: 3 hours

Additional Materials: List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Index number, Form Class, graphic and/or scientific calculator model/s on the cover page. Write your Index number and full name on all the work you hand in. Write in dark blue or black pen on your answer scripts. You may use a soft pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in the question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together.

This document consists of 5 printed pages.

[Turn Over

9740 / 01

Anglo-Chinese Junior College H2 Mathematics 9740: 2011 JC 2 Preliminary Examination Paper 1

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ANGLO-CHINESE JUNIOR COLLEGE

MATHEMATICS DEPARTMENT JC2 Preliminary Examination 2011

MATHEMATICS 9740 Higher 2 Paper 1

Calculator model: _____________________

Arrange your answers in the same numerical order.

Place this cover sheet on top of them and tie them together with the string provided.

Question No. Marks

1 /3

2 /5

3 /13

4 /8

5 /8

6 /8

7 /5

8 /9

9 /10

10 /7

11 /6

12 /8

13 /10

/ 100

Index No: Form Class: ___________ Name: _________________________


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1 Without using the graphic calculator, find the range of values of x which satisfy the

inequality 2

2

1

1

xx. [3]

2 Given that ( 1)x is a factor of 3 1,x express 2

3

2

1

x

x

as partial fractions.

Hence or otherwise, find the coefficient of 3nx in the expansion of 121 x x

in

ascending powers of x. [5]

3 (a) Find

(i) 2tan dx x x , [2]

(ii) 2

d3

xx

x x . [4]

(b) Evaluate, exactly,

(i) 1

1 22

0 sin dx x x , [4]

(ii) 1

0dx x b x where 0 1b . [3]

4 The equation of the plane 1p is given by 6 2 5x y z .

(i) Show that the line 1l with equation r =

2

1

2

1

1

9

lies on the plane 1p . [2]

(ii) Find the Cartesian equation of the plane 2p which is parallel to plane 1p and contains

the point A with coordinates 1, 1,1 . [2]

(iii) Line 2l contains point A and the point B with coordinates (9,1,1). Show that the sine

of the acute angle between 2l and 1p is 697

6972. [2]

Hence, or otherwise, find the exact perpendicular distance between 1p and 2p . [2]

5 Given that f( )y x where 1 1tan 2 tan4

y x for 0.4 0.4x , show that

2 2d1 2 1

d

yx y

x . [2]

By further differentiation of this result, find the Maclaurin’s series for y up to and including the term in 3.x [4] Denote the above Maclaurin’s series for y by g( )x . Find the range of values of x for which the value of g( )x differs from f( )x by less than 0.5.[2] [Turn Over


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6 (i) Find the roots of the equation 5 243 0z in exponential form. [3]

(ii) 5 243z can be expressed as 2 2( cos ) ( cos3 )z c z a z b z a z b , where

, ,a b c and are real. Find the exact values of , ,a b c and . [5]

7 An isosceles triangle has fixed base of length b cm. The other 2 equal sides of the triangle

are each decreasing at the constant rate of 3 cm per second. How fast is the area changing when the triangle is equilateral? Leave your answer in terms of b. [5]

8 The region R is bounded by the curve 22

1

1 4y

x

, the line

3

2x , the x-axis and the

y-axis, as shown in the diagram below.

(i) Using the substitution 1

tan2

x t , find the exact area of R . [6]

(ii) Find the volume of the solid formed when R is rotated completely about the y-axis.[3]

9 Sketch the curve given by the equation 2 2 5y ax for

0x and 0y , where a is a

positive constant. [1] The functions f and g are defined by

2f : 5x ax , 5

, 0x xa

g : 1 e , , 0.xx x x Show that 1f exists and define 1f in a similar form. [4]

Given that 2 5f ( ) = , for all , 0x x x x

a ,

(i) show that a = 1 without evaluating 2f ( )x , [2] (ii) using the result in (i), show that fg exists and find its corresponding range. [3]

[Turn Over

3

2x

22

1

1 4y

x

x

y

O


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10 Prove by Mathematical Induction that

1 2

21

1,

1

n nnr

r

x n x nxrx

x

for all n . [4]

Hence evaluate

2 3 4 143 3 4 3 5 3 15 3 . [3]

11 Detectives arrive at a crime scene at 12 p.m. and found an unfinished cup of coffee at

45 o C . In order to estimate what time the coffee was brewed, a fresh cup of coffee was made from the coffee machine in the same room, and its temperature was found to be 110 o C . After leaving the new cup of coffee in the room for 5 minutes, the temperature dropped to 80 o C .

It is known that the rate at which the temperature of the coffee falls is proportional to the

amount by which its temperature exceeds that of the room. Given that the crime scene is an air-conditioned room with temperature controlled at 25 o C , at what time was the unfinished cup of coffee brewed? Leave your answer to the nearest minute. [6]

12 A curve with equation xy f is also defined by the parametric equations

21 e , 1x y tt , t . (i) The point P on the curve has parameter p. Given that the tangent to the curve at P

passes through the point (1, 0), find the value of p. [4] (ii) Given that the graph of xy f has an inflexion point at 1t , sketch, on separate

diagrams, the graphs of xy f and fy x , showing clearly the exact

coordinates of the turning point(s) and asymptote(s), if any. [4] 13 (a) Jerry’s brother gave him some game cards for his birthday. He then starts to collect

cards for a total period of 3n weeks, counting his birthday gift as the first week of collection. Subsequently, the number of cards he buys each week is d more than the number he bought the previous week. If P is the number of cards collected in the first n weeks and Q the number of cards collected in the last n weeks, show that

Q P = 2n2d [4] (b) A fund is established with a single deposit of $2500 at the beginning of 2011 to

provide an annual bursary of $150. The fund earns interest at 3.5% per annum, paid at the end of each year.

If the first bursary is awarded at the end of 2011 after interest is earned, show that at the end of n years, the amount (in dollars) remaining in the fund is

12500 300001.035

7 7n .

When is the last year that the bursary can be awarded? [6]

- End of Paper -

acjc jc 2 h2 maths 2011 year end exam question paper 1

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