2004 yr 12 ext 2 thsc

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S Y D N E Y B O Y S H I G H S C H O O LM O O R E P A R K , S U R R Y H I L L S

2004

TRIAL HIGHER SCHOOL

CERTIFICATE EXAMINATION

Mathematics Extension 2

General Instructions Total Marks - 120 Marks

Reading time 5 minutes. Attempt Sections A - C Working time 3 hours. Write using black or blue pen. All questions are NOT of equalvalue. Board approved calculators may

be used.

All necessary working should beshown in every question if full marksare to be awarded.

Examiner: E. Choy

Marks may NOT be awarded for messyor badly arranged work.

Hand in your answer booklets in 3sections.

Section A (Questions 1 - 3),Section B (Questions 4 - 5) andSection C (Questions 6 - 8).

Start each section in a NEW answerbooklet.

This is an assessment task only and does not necessarily reflect the content or

format of the Higher School Certificate.

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Total marks 120

Attempt Questions 1 8

All questions are of equal value

Answer each section in a SEPARATE writing booklet. Extra writing booklets are available.

SECTION A (Use a SEPARATE writing booklet)

Question 1 (15 marks) Marks

(a) Evaluate

(i)

3

20 4

dx

x

1

(ii)

1

2

0 4x dx

1

(iii)

2

1

2x x dx

1

(b) Evaluate

(i)

2 2

1 1

x

x

edx

e

2

(ii)2

0

1

4 5sin dxx

+

4

(c) (i) If4

2

0

tan nn

xdxI

=

, 0n , show that 11

2 1n n

I In

+ =

3

(ii) Hence, evaluate4

6

0

tan xdx

1

(d) Evaluate ( )2

1

ln

e

x x dx

2

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(a) (i) Sketch on the same axes the graphs 2

3y x= + and 2y x= .

(ii) Hence or otherwise:

() Solve forx, 2 3x x< + . 2

() Sketch the curve2

3

xy

x=

+.

3

(b) Let3

( )1

f xx

=

.

On separate diagrams sketch the graphs of the following:

(i) ( )y f x= 2

(ii)2 ( )y f x= 3

(iii)( )f xy e= 3

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SECTION A continued


(a) If 1 3z i= + and 2cis

6

w

=

(i) Find z . 1

(ii) argz . 1

(iii) Expressz in the form cisr . 1

(iv) Express 6 3z w in the form cisr . 1

(b) (i) Express 5 12i in the form a ib+ . 2

(ii) Hence describe the locus of the point which representsz on the

Argand diagram if

2

2 5 12 3 2z i z i + = +

(c) The origin and the points representing the complex numbers z ,

1z

and 1zz

+ are joined to form a quadrilateral.

Write down the conditions forz so that the quadrilateral will be

(i) a rhombus; 1

(ii) a square. 1

(d) (i) Find the equation and sketch the locus ofz if 2

( )Imz i z =

(ii) Find the least value of argz in (i) above. 3

END OF SECTION A

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SECTION B (Use a SEPARATE writing booklet)


(a) 3 i is a zero of 3 2( ) 4 2P z z z z m= + , where m is a real

number.

Find m.

3

(b) If , and are the roots of 3 0x px q+ + = , find a cubic

equation whose roots are 2 , 2 and 2 .

3

(c) Given a real polynomial ( )Q x , show that if is a root of

( ) 0Q x x = , then is also a root of ( ( )) 0Q Q x x = .

3

(d) Use the following identity to answer the following questions.

5 3cos5 16cos 20cos 5cos = +

(i) Solve 5 316 20 5 0x x x + = 3

(ii) Hence show that 3

3 7 9 5cos cos cos cos

10 10 10 10 16

=

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SECTION B continued


(a) Let cos sinz i = + , show that

(i) 2cosn nz z n+ = 1

(ii) 2 sinn nz z i n = 1

(b) (i) Show that for any integer kthat

( ) ( ) 28 8cos sin cos sin4 4 4 4

k kk kz i z i z

+ + =

2

(ii) Hence simplify the following products

()7 7

cos sin cos sin4 4 4 4

z i z i

+ +

1

()3 3 5 5

cos sin cos sin4 4 4 4

z i z i

+ +

1

(c) Using the results of (b) above, factorise 4 1z + into 2 realquadratic factors.

2

(d) Using (a) and (c) above, prove the identity 2

2cos 2 2cos 1 =

(e) The complex numbers z x iy= + , 1z x iy= + and 22

z

z

= are

represented by the points P, 1P and 2P in the Argand diagram

respectively.

(i) Show that O, 1P and 2P are collinear where O is the origin. 3

(ii) Show that 1 2 2OP OP = 2

END OF SECTION B

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SECTION C (Use a SEPARATE writing booklet)


(a) A particle of mass m is projected vertically upwards with a

velocity ofu1

ms

, with air resistance proportional to itsvelocity.

(i) Show that after a time tseconds, the height above the ground is 4

( )1 2 1ktg ku gt

x ek k

+= ,

where kis a constant and g is the acceleration due to gravity.

(ii) At the same time another particle of mass m is released from rest,

from a height h metres vertically above the first particle. Youmay assume that at time tseconds, its distance from the ground isgiven by:

4

( )2 2 1ktg gtx h e

k k

= +

Show that the two particles will meet at time Twhere

1ln

uT

k u kh

=

(b) A vehicle of mass m moves in a straight line subject to a

resistance 2P Qv+ , where v is the speed and P and Q are

constants with 0Q > .

(i) Form an equation of motion for the acceleration of the vehicle. 1

(ii) Hence show that if 0P = , the distance required to slow down

from speed

3

2

U

to speed Uis

3

ln 2

m

Q

.

3

(iii) Also show that if 0P > , the distance required to stop from speedUis given by

3

( )2ln 1D kU = +

where kand are constants

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SECTION C continued


The diagram above shows a solid with a trapezoidal base EDTH

of length b metres.The front endHTSR is a square with side length a metres.

The back is the pentagonABCDEwhich consists of the rectangle

ACDEwith length 2a metres and width a metres, surmounted bythe equilateral triangleABC.

Consider a slice of the solid, parallel to the front and the back,with face formed by both the trapezium KLMNand the rectangle

KNQP, which has thickness x and is at a distancex metres

fromHT.

Question 7 continued on page 9

A

B

C

DE

K

h

L M

N

PQ

R S

TH

a

a

2a

x

x

W

F

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Question 7 continued Marks

(i) Show that the height,BW, of the equilateral triangleABCis 3a

metres.

2

Top View Base of Solid

(ii) Given that the perpendicular height of the trapezium KLMNis h

metres ie VU = h, use the similar trianglesBWFand VUF, in the

Top View, to find h in terms ofa, b andx.

3

(iii) Given that the trianglesBLMandBRS are similar, show that

( )a b xLM

b

=

3

(iv) Using the cross section of the base, find the length ofPQ in

terms ofa, b andx.

3

(v) Find the volume of the solid. 4

Question 8 starts on page 10

DE

H T

P Q

b

x

a

2a

B

L V M

R F S

W

x

b

U

a

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SECTION C continued


(a) If 0a > , 0b > and a b t+ = show that 3

1 1 4a b t

+

(b) There are n ( 1n > ) different boxes each of which can hold up to2n + books. Find the probability that:

(i) No box is empty when n different books are put into the boxes at

random.

1

(ii) Exactly one box is empty when n different books are put into theboxes at random.

2

(iii) No box is empty when 1n + different books are put into theboxes at random.

2

(iv) No box is empty when 2n + different books are put into theboxes at random.

2

(c) PQRS is a cyclic quadrilateral such that the sides PQ, QR, RSand SP touch a circle atA, B, CandD respectively.

Prove that:

(i) ACis perpendicular toBD. 2

(ii) Let the midpoints ofAB, BC, CD andDA beE, F, G andH

respectively.

Show thatE, F, G andHlie on a circle.

3

End of paper

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STANDARD INTEGRALS

11 , 1; 0,if 0

11

ln , 0

1, 0

1cos sin , 0

n n

ax ax

x dx x n x n

n

dx x xx

e dx e aa

axdx ax aa

+

=

=

=

2

1

2 2

1

2 2

1sin cos , 0

1sec tan ,

1sec tan sec , 0

1 1tan , 0

1sin , 0,

axdx ax aa

axdx axa

ax ax dx ax aa

xdx a

a x a a

xdx a a x a

aa x

=

=

=

= +

= > < >

= + +

+

= >

2004 yr 12 ext 2 thsc

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