HWA CHONG INSTITUTION JC2 PRELIMINARY EXAMINATION 2007

MATHEMATICS 9740/01Higher 2 Paper 1 Wednesday 12 September 2007 3 hours Additional materials: Answer paper Cover Page List of Formula (MF15)

READ THESE INSTRUCTIONS FIRST Write your name and CT class on all the work you hand in, including the Cover Page. 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 staples, paper clips, highlighters, glue or correction fluid. Do not write anything on the List of Formula (MF15). Answer all the questions. Begin each answer on a fresh page of paper. 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 a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are required 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, place the completed cover page on top of your answer scripts and fasten all your work securely together with the string provided.

This question paper consists of 5 printed pages and 1 blank page.

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1. A sequence is defined by 11 =u and nn unnu

1

2

1 +=+ for n = 1, 2, 3, …

Prove that ( )1 !n

nu

n−

= by mathematical induction. [4]

2. (a) A graph with equation ( )fy x= undergoes transformation A followed by

transformation B where A and B are described as follows:

A : a translation of 2 units in the negative direction of x -axis ,

B : a scaling parallel to the x-axis by a factor 12

.

The resulting equation is ( )( )2

18 82 1

g x xx

= + ++

.

Find the equation y = f(x), showing your workings clearly. [2]

(b) The graph of ( )hy x is shown below. = 1 -3 -1

Sketch the graph of ( )1

hy

x= . [3]

y

3. Show that 5 4 x x− + is always positive for . 0x ≥

Hence solve the inequality 25 4 0

( )( )x x x xx a x b− +

≤− −

in terms of a and b, where 0 . [5] a b< <

4. A cylinder of radius and height is inscribed in a right cone of fixed radius r h R and

height 8 units. The circular ends of the top of the cylinder is in contact with the inner

surface of the cone and the base of the cone lies in the same plane as the base of the

cylinder. Show that the curved surface area of the cylinder is 21616 r rR

π π− . Hence find

the maximum curved surface area of the cylinder, in terms of R . [6]

x

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5. Indicate clearly on an Argand diagram, the region R represented by the locus 1z z i− ≤ + . [2]

(i) Find the largest value of such that all the complex numbers w satisfying r

3w− ≤ r lie entirely inside R . [2]

(ii) The complex number lies in 1z R where 1 3 3z − = and ( )1arg 34

z i+ =π . Find

the exact value of . [2]

1z

6. A trio-package consisting of a Notebook (NB), a Home Theatre System (HTS) and a

Television set (TV) cost a total of $7545. Two companies offer the following discounts

if a customer buys the trio-package:

Discounts given for each item in the package Company

NB HTS TV Total price

A 10% 5% 5% $7068.35

B 20% 25% 10% $6341.50

During the Great Singapore Sale, ABS Bank credit card holders are entitled to a 60%,

50% and 30% discount for the NB, the HTS and the TV respectively for the trio-

package. How much would an ABS Bank credit card holder have to pay in total for the

trio-package? [6]

7. Express )2)(1(

107++

+rrr

r in partial fractions. [2]

Hence, find ∑= ++

+n

r rrrr

1 )2)(1(107 , giving your answer in the form f( )M n− , where M is a

constant.

Deduce the exact value of 22

21 30( 1)( 2r

rr r r

∞

=

+)+ +∑ . [6]

8. Let 11+= xey . Show that

( )322

2

2

12+

=⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

xy

dxdy

dxydy .

Hence, find the Maclaurin’s expansion for y , up to and including the term in .

Deduce the Maclaurin’s expansion for

2x

1+−

= xx

ey up to and including the term in . [8] 2x

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9. An arithmetic progression is grouped into sets containing 1, 3, 9, … integers as shown below, so that the number of integers in the set is .

{ }nuthn 13 −n

{ } { } { } ...,494541373329252117,13,9,5,1

(i) In either order, find expressions in terms of , for n (a) the first integer in the thn set, [3] (b) the last integer in the thn set. [2]

(ii) Show that the sum of the terms in the nth set is given by ( )23 3n nA B−⎡ ⎤+⎣ ⎦ ,

where A and B are integers to be determined. [4]

10. The parametric equations of a curve are C

t and sinty e t= , where tπ π− ≤ ≤ . costx e=

(i) Sketch the curve C , showing clearly the axial intercepts. [2]

(ii) Find the equation of the tangent to the curve at the point where P4

t π= . [4]

(iii) Given that O is the origin and is the point on the curve C such that Q ˆ2

POQ π=

with making an obtuse angle with the positive x-axis, find the area of the triangle . [4]

OQPOQ

11. The diagram shows the shaded region R bounded the lines 2y x= , 32

y = , the x - axis

and the curve2

2

3 1xyx−

= . y

O

2

2

3 1xyx−

= y = 2x

x

32

13

34

R

23

(a) Show that the area of R is given by 2 2 3

1 2 3

3 1 dxa xx−

− ∫ a where is a constant

to be determined. By using the substitution 1 sec3

x θ= , find the exact value of

the area of R . [7]

(b) Find the volume of the solid generated when R is rotated through 2π about the x -axis, giving your answer correct to 3 decimal places. [3]

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12. The functions f and g are defined by

( )( )3

4f : , , 3, 13 1

g : ,x

x x x xx x

x e x− −

→ ∈ ≠+ −

→ ∈

− ≠

(i) Sketch, on separate clearly labelled diagrams, the graphs of and ( )fy x=

( )gy x= . Hence, or otherwise, show that the inverse function of g does not exist. [3]

(ii) The function is defined by h ( )h : g ,x x x→ ∈ A ,

where A is the maximal subset of the interval ( )0,∞ such that the inverse

function exists. 1h−

State the set A and give the rule and domain of 1h− . [4] (iii) Determine which of the following is a function: (a) f g (b) -1f h In each case, if the function exists, give its domain and range. [3]

13. The curve C has equation 2 4x axy

x b+ +

=+

.

It is given that C has a vertical asymptote 1x = − and a stationary point at . 2x =

(i) Determine the values of a and b . [3] (ii) Find the equation of the other asymptote of . [1] C

(iii) Prove, using an algebraic method, that cannot lie between two values (to be

determined). [4] C

(iv) Draw a sketch of C, showing clearly any axial intercepts, asymptotes and

stationary points. [3]

(v) Deduce the number of real roots of the equation ( )( ) ( )222 24 1 4x x x x− + = − + 4 . [2]

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