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This document consists of6 printed pages (including this cover page).

[Turn over DHS 2011

Name: Index Number: Class:

DUNMAN HIGH SCHOOLPreliminary ExaminationYear 6

MATHEMATICS (Higher 2) 9740/01

Paper 1 13 September 2011

3 hours

Additional Materials: Answer PaperList of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Name, Index Number and 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 staples, 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 thecase 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 specificallystates otherwise.Where unsupported answers from a graphic calculator are not allowed in a question, you arerequired to present the mathematical steps using mathematical notations and not calculatorcommands.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, attach the question paper to the front of your answerscript.

The total number of marks for this paper is 100.

For teachers use:

Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total

Score

Max

score

5 5 5 7 7 13 10 12 12 10 14 100

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[Turn over DHS 2011 2011 DHS Year 6 H2 Math Preliminary Examination Paper 1

1 Given that2

1

( 1)(2 1),6

n

r

nr n n

=

= + + show that ( )21

(2 1)(2 3) 4 18 23 .3

n

r

nr r n n

=

+ + = + +

Hence find the exact sum of the 25th

to the 75th

term of the series. [5]

2 The position vectors of the points P, Q and R relative to the origin O, are p , q and2 1

5 5p q respectively. The point Swith position vector s relative to O is such that O

is the midpoint of the line segmentRS. Prove that 5 2 .+ =s p q 0 [2]

The point Tlies on QSextended such that QS:QT= 4:5. Show that the points P, O

and Tare collinear and find PO:OT. [3]

3 The diagram shows the graph of 2 f ( )y x= where 4 3 2f ( ) .x Ax Bx Cx Dx E= + + + +

The curve f ( )y x= has stationary points at 0x = and 3x = .

Find the exact equation of the curve f ( )y x= . Hence find the area enclosed by the

curve2

f ( )y x= in the given diagram, giving your answer to 1 decimal place. [5]

4(a) Given that f is a one-one function, determine if 1ff exists. Justify your answer. [1]

(b) The functions g and h are defined as follows:

g : ln , 1,x x x 2h : 2 2, 0.x x x x + >

(i) Given that gh exists, define gh in a similar form and find the range of gh. [2]

(ii) Sketch, on the same diagram, the graphs of g, 1g and 1g g. [3]

(iii) State the range of values ofx satisfying the equation1 1g g( ) gg ( ).x x = [1]

2 f ( )y x=

4

(3, 4)

(3, 4)

x

y

O

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5 (i) Sketch on the same diagram, the loci given by

(a) 5i 4,z = (b) 5 5i 3z = . [2]

Hence, or otherwise, find the complex numbersz that satisfy both (a) and (b).

[2]

(ii) The complex numbers, 1z and 2z satisfy the equations given in (i)(a) and

(i)(b) respectively.

Given that 0 and , 8,a a < < > find the smallest value of for which

1 2arg( 5i) arg( 5i).z a z a = = [3]

6(a) Given that 1ln tany x= , show that ( )2d

1d

yx y

x+ = .

By further differentiation of the above result, find the Maclaurins series for 1tane x ,

up to and including the term in 3.x [6]

(b) Obtain the expansion of

22

12

x

in ascending powers ofx, up to and including the

term in 4.x [2]

Using the above expansion, or otherwise,

(i) show that 2 2sin 2sec 2 2x x x x+ + + ifx is sufficiently small for 3x andhigher powers ofx to be neglected, [3]

(ii) find1

1 2r

r

r

=

by substituting a suitable value ofx. [2]

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7(a) The diagram shows the graph of f ( ),y x= which has turning points at ( 2, 4)A and

(2,3).B The horizontal and vertical asymptotes arey = 2 andx = 1 respectively.

Sketch, on separate diagrams, the graphs of

(i) f ( ),y x= [2]

(ii) 2 f ( ),y x= [3]

showing clearly all relevant asymptotes, intercepts and turning point(s), where

possible.

(b)

The graph of g( )y x= above intersects the x-axis at ( , 0) and ( , 0), where

1 and 1. > > It has a turning point (0, 1) and a vertical asymptote 1.x =

g( )y x= undergoes two transformations in sequence: a translation of 1 unit in the

positive y-direction, followed by a scaling of factor 2 parallel to the -axis.x The

resulting graph is h( ).y x=

Sketch, on separate diagrams, the graphs of h( )y x= and1

,g( )

yx

= showing clearly

all relevant asymptotes, intercepts and turning point(s), where possible. [5]

O

x = 1

1

y

Ox

( 2, 4)A

x = 1

(2,3)B

y = 2

f ( )y x=

x

y

g( )y x=

1

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8(a) Express21 2

x

x x +in partial fractions. Hence, find

2d .

1 2

xx

x x + [3]

(b) Find

(i)

2 2

e sin(e ) d ,

x x

x

[2]

(ii)1sin d ,x x [3]

(iii)2

2d .

2 3

xx

x x + [4]

9 The plane 1 has cartesian equation 5 2x y+ = . The plane 2 contains the line l

with equation

3 1

2 0

2 1

= +

r and is perpendicular to the plane with equation

5 2

2 1

0 1

= +

r . The planes 1 and 2 meet in the lineL.

(i) Find an equation for 2 in the form .d =r n [3]

(ii) Find a vector equation forL. [3]

(iii) Show that , the acute angle between l andL is given by15

cos .10

= [2]

The plane 3 has cartesian equation 2x py z q+ + = , where p and q are positive

constants.

(iv) If the three planes 1 2, and 3 do not have any point in common, show

that 1.p = If the perpendicular distance from the origin to 3 is 2 units, find

the exact value ofq. [4]

10 The terms in the arithmetic sequence { 3 1, 1,2,3,...}n n+ = are grouped into sets such

that the thr bracket contains 2rterms as shown below:

{ 4,7 },{10,13,16,19}, { 22,25, 28,31,34,37,40,43},... .

(i) Show that the total number of terms in the firstNbrackets is 2(2 1)N . [2]

(ii) Find the sum of the terms in the firstNbrackets. [2]

(iii) Show that the first term in the thN bracket is 3(2 ) 2N and find also the last

term in the thN bracket. [4]

(iv) Find the least value ofNsuch that the sum of terms in the thN bracket is more

than 1210 . [2]

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11(a) The graph of 3 2y x= + for 0 1x , is shown in the diagram below. Rectangles,

each of width1

,n

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

(i) Given that ( )2

23

1

1 ,4

n

r

nr n

=

= + show that the total area of all n rectangles,A,

is given by2

9 1 1

4 2 4n n + . [4]

(ii) Deduce the exact area under the curve for 0 1.x [1]

(iii) Find the least value ofn such that the areaA is no less than 99% of the exact

area under the curve. [2]

(b) The diagram shows the curves C1 and C2 given by the equations 21

14

yx

= and

22 cos(5 )y x= respectively. C1 cuts the x-axis at1 1

and2 2

x x= = and C2 has a

-intercepty at y = 1. The shaded region R is bounded by the curves C1, C2 and the

-axis.x

(i) State the coordinates of the points of intersection between the curves C1 and

C2, correct to 3 decimal places. [1]

(ii) Find the volume of the solid formed whenR is rotated through

(I) 4 right angles about thex-axis, [3](II) 2 right angles about they-axis. [3]

2

3

n

2

n

11nn

2n

n

3n

n

3 2y x= +

Ox

y

1

n

END OF PAPER

y

xO

y = 1

22 : 2 cos(5 )C y x=

1 2

1: 1

4C y

x=

1

2

1

2

1

R

1

2011_dhs_paper_1

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