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SYDNEY

TECHNIC L

HIGH

SCHOOL

Celebrating 1 years of public education

Mathematics Extension

HSC ASSESSMENT TASK

JUNE 2011

General Instructions

Working

time

allowed 7 minutes

Write using black or blue pen

Approved calculators may be used

A table

of

standard integrals is provided

All necessary working should be shown

Start each question on a

new

page

Attempt

all questions

NAME:

TEACHER:

QUESTION QUESTION QUESTION

TOTAL

1

2

3

1

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Question

17 marks)

a

Find

f

x

3

lnx

dx

l

6

x 2

d

b Show that

X

5

x-4) x-1)

I

16

5

c

d

1 i and i are zeroes

of

a monic polynomial, P x),

of

degree 4

with real coefficients.

Express

P x) as

a product

oftwo

real quadratic factors.

i Solve the equation

16x

4

- 16x

2

1 = 0

using the

identity

cos48

=

8cos

4

e

8cos

2

8

+

1

ii Solve

the

equation 16x

4

- 16x

2

1 = 0

as

a quadratic in x

2

,

and hence find the exact value

of

cos2:.. Justify your answer.

12

e

The equation x

3

+

3px

+

q

=

0

has

a double root at x

=

k

3

3

2

2

3

i

Show

that

p

=

2

1

ii Hence solve x

3

- 6ix 4 - 4i

=

0 given that it has a double root. 3

2

r

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Question

2

17 marks

-

Start

a new page

a Below is a sketch ofth function

y

= g x).

There are stat ionary points at 1, 1 and

3 , 0 ).

y

X

On separate diagrams, draw neat sketches of the following, clearing showing any

important features or points.

_

i

y = f g x) dx

2

1

ii)

=g x+1)

2

iii)

yz

=

g x) 2

iv)

y =

cos-

1

g x)

2

3

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Question 2 continued)

b

The points

P

p and Q 2q . ~ l i e on the hyperbola xy

=

4.

p q

i hat is the

eccentricity

of

the hyperbola

xy =

4. 1

ii) Find the equation of the chord PQ

2

iii) Given

that the

chord

PQ

always passes through

the

point

4,

2 ) 1

show that pq = p 2

iv) Find the locus of M the midpoint

of

PQ

2

1C

c Evaluate J

0

6 sin 2x sin 4x dx

3

4

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Question

3

17 marks - Start

a new page

f

2x

a Find

x

) x

2

+10x+26

b

f

1 sinx d

Find

2

X

COS X

c The equation 2x

4

- 3x

2

- 2x

+

k

=

0 has a triple root.

Find the value of k .

d

If

a , p

andy

are the roots

of

the equation

x

3

- 6x

2

- 4x

+ 2 = 0

i Find the monic polynomial equation

with

roots

2a, {3

, 2y

ii Find the constant term in the monic polynomial with roots

1 a 1 {3 l+y

1 a

I 1-/3 I 1 y

e

Use

integration

by

parts

to

evaluate

J

0

3y

e-Y

1 -

e-Y)

2

dy

End

of paper.

Q

5

3

2

3

2

3

4

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J ndx

J dx

J

ax

dx

J

osaxdx

J inaxdx

J ec

2

axdx

ST ND RD INTEGR LS

= -

1

xn

1

n t -1; x t

0,

if

n O

1 . 0

=

-s1nax,

at:

a

1

= -acosax,

at:-0

1

= -tanax, at: 0

a

J ecax tanaxdx

1

= -secax,

a

at:-0

1

1

X 0

= tan - at:

a a

. 1 X

0

=

sm -

a > - a < x < a

a

=

ln x

~ x a

2

), x

>a> 0

NOTE: lnx=logex,

x>O

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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-

x rkJJ

Sl

vJJ

2-

s

r.....

1.11' \

()/1

r V,JJ

1:-

Z..-ort

Q VG'SI \V.N I

___ . ) _ _ ~ - ) ~ L

cl ?L

I.-

:x.

\

. l

(A- : .

u..

-=-

?t.

~ ? -

lf

r J

\ r ~

1.1

:::

jC...

I

:x,Lf{n?t.--

s

' )

-

-

?C...

c:.t

.,

(.,

-

'-t

-

J..

?'-' .f

~ . . .

-

j.. ~

+

G

-

\.4'

16

{,

\

:>C. + ,_ d..

:X.+ '2-

A

$

b)

,

-:>C. - -f) {

:x.-1-)

:1(,.

-::.

-

-+

(,t..-+

'2... : .

A

l t.-1-)..f.

1 3 { ~ - + )

.

Jc

~ l {

-

.J._

;:)(...

::

? ( .

_,

)C.::.

t

=;

~

=

-3B

.

r3

=-

.

,

A+

6

=

I

.

A :: '1..-

.

=

1... lV\ ( ')(.-

tJ

- ,

( )t - J .

.

::::

(

')._ lr-.2..

- (

s

-

(_

L...( 1 -

( '

4-)

::

?...I

r ' \ Lt

-

\l'\

..s

-

'"'

lb

-

' '>

' ; .-

In

16

( .

-

c

c..) r o o ~ S

0.( --

'

i J

l, ;_

3 - ~

3, ..

~

J

J

Sv-

1-.

b

r.-vJ '-*

2-

t()

f L '-')

~

(

~ c . . _ -

1..:h.

-+

_

J

_

?

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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_ _ _ _ _ _ _ _ _ _ _ _ - . : : ~ ( _ ~ i _ : _ . : : > L - _ - ~ _ _ : . ~ _ : - ' _ c . . _ ' _ + _ , - , o ) . ~ - - - - - ' - = - - o______ __________

( e-.-f

:>c -:: c. se

---------------- ---- - - - - - - - - - - - - - -

2.. (

~

~

s

rfe

-

8

~

l.-

;,c..

t-

1

-

I

-:.--0

~ ~ L 1

>if

?II

uTI

+e ;.

3J

\ )

) J

-

fJ

11

SIT

tii '

tCf..)

.

niT

~

J

-

\ . . ; '

S11

~ s

771

c .

tl'if

\

-

-

1 -

J

'

/

,i_j

6

)(..

If

r

6

)(_ . -

+

f

~ )

-

rJ

...,

u.

')(..

b

CA.

\...

-- 6 ,\.+1

= 0

{6

.

j

I 6 ..- '+. t

. t

U.=

q_,_

-

::.

l

:t

J

q l.-

31.-

f

I

-

\ =

6J3

3t...

.

=

1....

E

Ll-

--------------

- - - - - - - - - - - -

~ ~ ~ ~ ~

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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)

~

0

:X.

-+

~ p : ; t c _ _

+

1

::

v

~ : X . . 1...

-

)

p

: : .0

.,_

?t.-

::

p

o

J,{e_

A

? L ~ k

-

=:J

(c...

~

-:.- p

p ~ - - k . '

rocAl

lc.J

3

+

3p,c..ij

.

V

a

-:x-

:. X 6-i

X.+

4- 4 . ~ v

2-

lc-

f-.

-::.

0

'e-:

- 0 ~

~ ~ 4 - - L f ~

[t..

l.kt:A-=-

- 6 ~

+

__::.

- L '

(

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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I

_}---.

c...

j

( \

1

\

l

7

\

-

-

- -

-

-

-

-

/

\

-

-

---

-

:7 ?t...-

l

-

1

j

-

I

I

A

1

\

\

,,

'

tl

T

T

-

I

-

;;7

I

\

I

I

\

-

1

I

\.

v

1 -

_ I

I

_5

I

I

-

I

I

I

'r

/

I

I

\

I

_I_

I

-

(

I

_I_

I

-

IJ

I

I

J

1

I

-

I

-

-

_I

1

-

J

-

-

- -

-

--

-- -

-

-

-

-

.

L

-

_j

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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..

Ill)

II\')

i'\'

l

-

K

\

\

...

7 ; ) ~

_

I

.

IV)

1\

I

v

V

1\

'

~

,

-I

" " " ~

bj

t

e. =

s:c

c

: . . - ~

\\

vY\ :...

t

p

2 -

_

r

' : '

1.p-

\..L

.. L

r

:>t.. -lj :...

- L e t ~ )

..

_j

8/10/2019 Extension 2, 4 Unit_2011_Sydney Technical High School HSC Trial Paper

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JJ

111)

s ~ \ ,

'X-

-:

lf'

J

-=-L...-

I

4-

....

1-

e ~

...

1--(

p-+ 1)

-

._

)_ . t

Pj

e

, -f 1

-

f

1

:.

p

f

c,

-

\--

v)

V ' ~ J

p o , ~ ~

(

p-+

'

p \

ni:

-

-

I

I

f t ~

,

.:) ._ f i

1

~ : : .

-

.

ne.

v

\ \

(

,

lo '>

('

~ ' : . . -

-

-

.._J

r

1

? l

. \

-

:Jr.- --

-

1i

a )

:.

3

- ; ; - -

0

-::

J3

-

-

8

-

_j

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Q v ~ t o , J

3

:t:>c...

~

eJ..

?C..

c)

-

+ l 0 ~ 1 . . 6

X..

- - - - -

~

~ ~ l

(0

-

c.( .,..._

: .

X -

+

(0

-+

'l-6

'X- - + (

0 ':)C.-+

'\...-'

~ V \ ( ) ( . . . ' \ . . + l O ? C - + ' L ~ J -

-\

lO

d.? t . .

-

( X+ s) - -4- 1

-

' '

-::...

l ' (

-

+ lOX-

L

bJ--

-lQ {eM-'

(

-x + s)

+

(._.

c

....

,..

.

b)

\

I

t

J

~ ~ : x . -

c..{ ,c.-

{,.c L

?L

..)

':..

\

s

t...(.,\..- ?L

+

soL(._ ?L-

la...._

- cJ '}L

._,

~

--

a:

.....

:1:..

+

s---

?L

'+

'- -

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J

Jd

t)

{ ~ lll

r e cl

.P 0b\.0- ,c..

~

p

_ ~ : J ~ v

' \

l..

-

( _ ~ _ ) -

Gt

~ J

t r ( ~ ) f l . ~

0

::X..,)

~

L

- -

-

L

::t..

.. . L -=- 0

-:;-

3

C6x-1-l6-=- o

X-

-

(L 'L

-

-

l ~

:> O o O O o u o o