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JURONG JUNIOR COLLEGEJ2 Preliminary Examinations

MATHEMATICS 9740/01Higher 2 13 September 2011

Paper 1 3 hours

Additional Materials: Answer PaperGraph Paper

Cover PageList of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your name and civics class on all the work you hand in.Write in dark blue or black pen on both sides of the paper.You may use a soft pencil for any diagrams or graphs.Do not use highlighters, glue or correction fluid.

Answerall the questions.

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.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 a question, you are required topresent 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, with the cover page in front.

This document consists of6 printed pages

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1 A new intern in a design firm, Isaac, was tasked to buy lunch for his team of 6 members

(including himself) everyday. The staff canteen offers 3 dishes: Chicken Rice, Fried Noodles

and Nasi Lemak. The purchases he made over 3 days are summarised in the table below.

Find the price for each dish and hence find how much he would have spent if he bought 2

packets of Chicken Rice, 3 packets of Fried Noodles and 1 packet of Nasi Lemak on Day 4. [5]

2 Solve the inequality 5 2 1 3x x x+ < + . [5]

3 A sequence of real numbers 1 2 3, , ,...u u u is defined by 1u = 8 and 1 8 8n nu u n+ = + + for all 1n .

Prove by induction that, for every positive integern, ( )22 1 1nu n= + . [4]

Hence, show that nu is a multiple of 8 for every positive integern. [2]

4 (i) Express3

(3 2)(3 1)r r +in partial fractions. [1]

(ii) Hence, show that

( ) ( )11 1 1

13 2 3 1 3 3 1

n

r r r n=

=

+ + . [3]

(iii) Deduce the exact value of the sum( ) ( ) ( )1 1 1

28 31 31 34 34 37...+ + + . [3]

Day 1 Day 2 Day 3 Day 4

Chicken Rice 2 2 3 2

Fried Noodles 2 0 2 3

Nasi Lemak 2 4 1 1

Total cost $13 $12 $14.50 ?

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5 By means of the substitution 2d

(1 )d

= +y

z xx

, express the differential equation

( )2

2

2

d d1 2 1

d d + =

y yx x

x xas a differential equation involvingzandx. [2]

Hence find the solution of the differential equation ( )2

2

2

d d1 2 1 ,

d d

yx x

x x

+ =

given that

1= y andd

2d

y

x= when 0.x = [5]

6 (a) An arithmetic progression has first term a and common difference d, where a and dare

non-zero. If the first, fifth and tenth terms of the arithmetic progression are in a

consecutive geometric progression, show that 16a d= . Given that the sum of the third

and sixth terms of the arithmetic progression is 13, find the values ofa and d. [4]

(b) The sum of the first n terms of a sequence is given by 3(1 3 )n . By finding the nth

term of the sequence, or otherwise, show that this is a geometric progression, and state

the values of the first term and common ratio. [4]

7 The complex numberzsatisfies the following relations

| 1 i | | 3 i | z and ( )3

0 arg 5 3i4

+ z

.

(i) Illustrate both of these relations on a single Argand diagram. [5]

(ii) Find the exact values ofzwith the greatest and least values of argz. [3]

8 The displacement of a particle, tseconds after leaving a fixed point O, is modeled by the curve

Cwith parametric equations 1 e , e .t tx y t = + =

(i) Find the exact equation of the tangent to Cwhich is parallel to the line 3 3y x+ = . [6]

(ii) Given that the particle travels along Csuch that the rate of change of itsx-coordinate is

2 units/s, find the rate of change of itsy-coordinate atx = 3. [3]

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3

n

11n

n

2n

n

3n

n

x

e= xy

O 2

n

1

n

1

4

R

x

y

e= xy

e=yx

O

9 (a) The graph of e ,xy = for 0 1x , is shown in the diagram below.

Rectangles,each of width

1

n where n is an integer, are drawn under the curve.

(i) Show that the total area of all the n rectangles is

( )1

e 1

e 1

nn. [3]

(ii) By considering the area of the region bounded by the curve e= xy , 1= and the

x andy-axes, show that1 1

e 1> +nn

. [3]

(b) The diagram below shows the shaded region R bounded by the curves e= xy ,e

=y ,

the line 4=y and the y-axis. Find, to 2 decimal places, the volume of the solid

generated whenR is rotated through 2 radians about they-axis. [4]

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10 The equations of two planes 1 and 2 and the equation of the line l1 are as follows:

1: 2x 5y + 2z= 2,

2: 6x 15y + z= ,

l1: r = 2i + 4j +(3i 2k), .

(i) Find the position vector ofA, the point of intersection of1 and l1. [2]

(ii) Another line l2, which lies in 1, passes throughA and is perpendicular to line l1.

Find a vector equation ofl2. [3]

(iii) The plane 3 contains lines l1 and l2. Find the acute angle between 1 and 3. [2]

(iv) Find the values ofand such that 1 and 2 do not intersect. [2]

11 (i) Expand ( )1

2 3

+ as a series in ascending powers ofx, up to and including the term in

3. State the values ofx for which this expansion is valid. State also the coefficient of

.n [4]

(ii) Given thate

,2 3

=+

x

yx

show thatd

(2 3 ) (1 3 ) 0.d

+ + =y

x x y [2]

By further differentiation of this result, find the Maclaurins series fory in ascending powers of

x up to and including the term in 3.x [4]

(iii) Verify the correctness of the series found in (ii) by using the standard series expansion

for ex and the result obtained in (i). [2]

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12 (a) The curve D has the equation( )

2

,1

x ay

+=

+where a is a constant such that 1 3a<

and 1x .

(i) Find the equations of the asymptotes ofD. [2]

(ii) Show that the stationary points ofD are ( ),0a and ( )2,4 4a a . [2]

(iii) Draw a sketch ofD, which should include the asymptotes, turning points and

points of intersection with the axes. [3]

(iv) Hence state the set of values ofkfor which the liney = kdoes not intersectD.[1]

(b) The curve G given below has equation y = f(x). Sketch, on separate diagrams, the

graphs of

(i) ( )f 3 x= , [3]

(ii)( )1

fy

x= . [3]

y

2

(3, 2)

2x

y = 1

0

y = f(x)