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BAULKHAM HILLS HIGH SCHOOL

2015HIGHER SCHOOL CERTIFICATE

TRIAL EXAMINATION

Mathematics Extension 1

General Instructions Reading time – 5 minutes

Working time – 120 minutes

Write using black or blue pen

Board-approved calculators may be used

Total marks – 7Exam consists o

This paper consists

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Section I - 10 marksUse the multiple choice answer sheet for question 1-10

1. If O is the centre of the circle, the value of

in the following diagram is

(A) 25° (B) 40° (C) 50° (D) ) 80°

2. The point P divides the interval AB externally in the ratio 3 : 2. If A(-2,2

coordinate of the point P?(A) -13

(B) -1

(C) 4

(D) 28

50°x°

O

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8. 1√ 2 5 4

(A) sin (B)

sin

(C) sin

(D)

sin

9. The derivative of tan is:

(A)

(B)

(C) 4tan

(D)

10. The solution to |2 1| | 2| is(A) 1 (B) 1 (C) 1 1 (D) 1 1

End of Section 1

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Section II – Extended Response All necessary working should be shown in every question.

Question 11 (15 marks) - Start on the appropriate page in your answer boo

a) Solve

1

b) Find 2 2 1 using the substitution 2 1

c) Evaluate lim→sin 12

3

d)Find the constant term in the expansion .

e) (i) Show that a root of the continuous function sin2 between 0.4 and 0.5.

(ii) Hence use one application of Newton’s method with an initial

of 0.4 to find a closer approximation for the root to 2 sign

f) Solve sin2 cos for 0 2

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Question 12 (15 marks) - Start on the appropriate page in your answer book

a) Evaluate sin2

b) (i) From a group of 6 boys and 6 girls, 8 are chosen at random to

How many different groups of 8 people can be formed?

(ii) How many of these groups consist of 4 boys and 4 girls?

(iii) 4 boys and 4 girls are chosen and placed around a circle.

What is the probability that the boys and girls alternate?

c) The rate of change of the temperature (T) of an object is proportional to th between the temperature of the object and the temperature of the surround

ie

An object is heated and placed in a room of temperature 20° to cool. Aftemperature is 36

°. After 20 minutes the temperature is 30

°.

(i) Show is a solution to the differential equatio(ii) Find the value of and the value of to 3 decimal places.(iii) What was the temperature of the object when it was first place

d) Prove

135 … … … … … … … …(2n+1) = !! for 0 by mathinduction.

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a) The polynomial 6 8 has roots , and . Find :

(i) (ii) (iii)

if

has a triple root.

b) PN is the normal to the parabola 4 at the point P (2,). Theintersects the line SN which is parallel to the tangent at P. S is the focus o

(i) Sh th ti f th l PN i 2

y

N

S

ap2 P(ap, )P2,

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c) Sand is falling on the ground forming a conical pile whose semi apex ang

The volume of the pile is increasing at a rate of /. (

(i) Show that the volume of the pile is given by: (ii) Find the rate at which the height of the pile is increasing when

pile is 2 metres.

h

r

30°

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a)

A projectile is fired from the ground with an angle of projection given by

and initial velocity V.

It just clears a wall 10 high 100 away. Let acceleration due to gravity(i) Show that the equations of motion are

and

(ii) Find the initial velocity, V of the projectile. (iii) At what speed is the projectile travelling the instant it clears th

b) Copy or trace the diagram below in your exam booklet.

V

100m

10m

x°

A

C

D

E

O

Q

P

R

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c) Below is the graph of ln √

(i) Show that the equation of the inverse function is given by

(ii) Hence find the area of the shaded region above.

2 e +1e

y

x

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Name: _________

Teacher: ________

Class: ________

FORT STREET HIGH SCHOOL

2015 HIGHER

SCHOOL

CERTIFICATE

COURSE

ASSESSMENT

TASK

3:

TRIAL

HSC

Mathematics Extension 1Time

allowed:

2

hours

(plus 5 minutes reading time)

SyllabusOutcomes

Assessment Area Description and Marking Guidelines

Chooses and applies appropriate mathematical techniques i

order to solve problems effectively

HE2, HE4 Manipulates algebraic expressions to solve problems from t

areas such as inverse functions, trigonometry, polynomials

circle geometry.

HE3, HE5

HE6

Uses a variety of methods from calculus to investigate

mathematical models of real life situations, such as projectilkinematics and growth and decay

HE7 Synthesises mathematical solutions to harder problems and

communicates them in appropriate form

Total Marks 70

Section

I 10

marks

Multiple Choice, attempt all questions,

Allow about 15 minutes for this sectionSection

II 60 Marks

Attempt Questions 11‐14,

Allow about 1 hour 45 minutes for this section

General Instructions:

Section

I Tot

Q1‐Q10

Section

II Tot

Q11 /15

Q12 /15

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SECTION I (One mark each)

Answer each question by circling the letter for the correct alternative on t

Allow about 15 minutes for this section.

1

Which expression

is

a correct

factorisation

of

64 (A)

( 4 4 16

(B) ( 4 8 16

(C) ( 4 4 16

(D)

( 4 4 16

2 Which expression is equal to 3 ?

(A)

sin 3

(B)

sin 3

(C)

sin 6

(D)

sin 6

3

Which inequality

has

the

same

solutions

as

| 2 | | 3 | 5 ?

(A) 6 0

(B)

0

(C) | 2 1| 5

(D)

1

4 A Mathematics department consists of 5 female and 5 male teachers.

committees of 3 teachers can be chosen which contain at least one fe

(A) 100

(B) 120

(C) 200

(D)

2500

5

Consider the function

and its inverse function . Eva

(A) ‐3

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6

Which group of three numbers could be the roots of the polynomial e

41 42 0 ?

(A) 2, 3 , 7

(B)

1, ‐6, 7

(C) ‐1, ‐2, 21

(D) ‐1, ‐3, ‐14

7 A family of ten people is seated randomly around a circular table. Wha

that the two oldest members of the family sit together?

(A)

!!

!

(B)

!!

!

(C)

!!

!

(D)

!!

!

8) Let 1 x be a first approximation to the root of the equation cos log x

What is a better approximation to the root using Newton’s method?

(A) 1.28

(B) 1.29

(C) 130

(D) 1.31

9

What is the value of 2

3

6

sec

tan

xdx

x

? Use the substitution tanu x .

(A) 0.6009

(B) 0.6913

(C)log 3

e

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10 Let || 1.What is the general solution of 2 ?

(A) 1

, is an integer

(B)

, is an integer

(C)

,

is

an

integer

(D) 2

, is an integer

Question

11

(

15

marks)

Use

a

NEW

writing

booklet.

a) Evaluate lim→

b) Find

c)

Find

3

d) Find the acute angle between the lines 3 2 8, and 5 9.

e)

The points 2 , and 2 , lie on the parabola

(i) The equation of the chord is

. (Do NOT

If the chord passes through 0, , show that 1.

(ii)

Given the chord passes through 0, and the normals at a

point whose coordinates are

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f)

The sketch shows the graph of the curve where 2 c

The area under the curve for 0 3 is shaded.

(i)

Find the intercept.

(ii) Find the domain and range of 2 cos

.

(iii) Calculate the area of the shaded region.

Question 12 ( 15 marks) Use a NEW writing booklet

a)

Let , , be the roots of the equation 3 6 1 0.

(i)

Find 2 2 2.

(ii)

Find +

b) A particle moves in a straight line and its position in metres at anytime

by 3 cos 2 4 sin 2

(i) Express the motion in terms of cos .

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c) A coffee maker has the shape of a double cone 60cm high. The rad

Coffee is

flowing

from

the

top

cone

at

the

rate

of

5/ .

(i)

Show that radius ( in the bottom cone is

(ii) How fast is the level of coffee in the bottom cone rising at the insta

cone is 6 cm deep?

d) In the diagram, is tangent to both the circles at .

The points

and are on the larger cicles, and the line is a tacircle at . The line intersects the smaller circle at .

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Question

13

(

15

marks)

Use

a

NEW

writing

booklet

a) In a bag there are 6 red, 4 white and 3 black balls. Three balls are drawn sim

the probability that these are:

(i)

all red.

(ii) exactly 2 white balls.

b) In the diagram, , is a point on the unit circle 1 a

positive axis, where ‐

. The line through 0,1 and

1 at . The points 0 , and 0 , 1 are on the axis.

(i) Using the fact that ∆ and ∆ are similar,

show that

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Question

14

(15

marks)

Use

a

NEW

writing

booklet

a) From a point is due south of a tower, the angle of elevation of the top

From another point , on a bearing of 120°, from the tower, the angle o

The distance is 200 metres.

i)

Copy or trace the diagram into your writing booklet, adding the giv

diagram.

ii)

Hence find the height of the tower to the nearest metre.

b) A particle is projected horizontally from a point P, metres above , wit

per second. The equation of the motion of the particle are

= 0 and = .

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A canister containing a life raft is dropped from a helicopter to a str

helicopter is travelling at a constant velocity of 216 km/h, at a heigh

sea level, along a path that passes above the sailor.

(ii) How long will the canister take to hit the water? (Answer to one d

(Take g 10 / ).

(iii) A current is causing the sailor to drift at a speed of 3.6 km/h in th

plane is travelling. The canister is dropped from the plane when t

from the plane to the sailor is metres. What values can take i

most

50

metres

from

the

stranded

sailor?

c)

The depth of water metres on a tidal creek is given 5 4cos

, f

the time being measured in hours.

(i)

Draw a neat sketch of

5 4 cos , showing all important features.

(ii)

If the low tide one day is at 1.00 p.m., when is the earliest time tha

water can enter the creek? Give your answer in hours and minutes

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Girraween

High School

2015 Year 12 Trial Higher School Certif

Mathematics Extension 1

General Instructions

• Reading tjmc mjnutcs

• Working

t ime-

2

hours

• Write using black or blue pen

Black pen is preferred

• Boa.rd-approved calculators may

e used

• A

table

of

standard integrals is

provided at the back

of

this paper

•

In

Questions 11-14, show

relevant mathematical reasoning

and/or calculations

ll

ll

a

1,

i:

>I

J

n

q

I

Total marks - 70

Section I )

1

marks

• Attempt Questions

1-10

• Allow about 15 minutes for th

Section II )

60 marks

• Attempt Questions

11-14

• Allow about 1 hour and 45

mi

For questions 1-10 fill in the response oval corresponding to t

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on your Multiple choice answer sheet.

1. What is

the

acute angle between

the

lines

y

=

2x

-

3 and

3x

+ 5

y

nearest degree?

A)32° B

50° C)82°

D)

2.

The

number

of

different arrangements

of

the letters

of the

word

begin

and end

with

letter R is:

A)

6

(2 )'

B)

8

2

C)

6

2

3.

The

middle

tenn in

the expansion

(2x -

4)4 is

A)

81

B) 216x

2

C

384x

2

4.

,\' l {XJ

Which of the following could be the polynomial y

= P(x)?

A)

P(x)=x

3

(2-x)

B P(x) =x

(2-x)

2

D

D

'..

8

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6. Which

of the

following represents the exact value

o os xdx

0

A) re-2.J2

16

B re-2.fi.

8

C re 2.J2

16

7. Which of the following represents the derivative

of y

= cos-' (

I

A

x~x -I

B -1

~x

2

-1

C 1

~x

2

-1

i .h

8.

Let

a,,B,ybe

the roots

of

2x +

x -4x

9 = 0 .What

1st

e valu

9. I f cose

= _I and

O< e

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Questionll. 15

marks)-show all necessary working)

5

a)Solvefor x: >2

x-1

b) Find the value

of

e, such that

/3 cos

e

-

sine =

1, wher

3

) U h b

. . . , l sin2x

c se t e su st1tut10n u =

sm

-

x to eva uate .

2

dx

l+sm x

Give your answer in simplest form.

4

d)

Use

the mathematical induction to show that for all

integers n 2:: 2,

n(n

2

-1

2x1+3x2+4x3+

...............

+n n-1)=

3

e) The coefficients of x and

x-

in the expansion

of

(

are the same, where a and b are non-zero. Show that

Question 12. 15 marks)

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a

i

Find _cos-' x -I

0

J

dx IO

1

ii) Hence, evaluate

f J

1

x

2 x x

2

5

b Two points P 2ap,ap

2

) and Q 2aq,aq

2

) lie on the parabola x

The general tangent at any point on the parabola with parameter

t

is

y =

x at

2

(DO NOT prove this).

i Find the coordinates of the point of intersection T of the tang

parabola at

P

and

Q

ii) You are given that the tangents

at P

and Q intersect at an an

Show that p - q 1 + pq

iii) By evaluating the expression

x

-

4ay,

or otherwise, find the

point

T

when the tangents at

P

and

Q

meet

as

described in

c The velocity v Is of a particle moving in simple hannonic mo

the x axis is given by v

2

=

8 + 2x

x .

i Between what two points is the particle oscillating?

Question 13. 15 marks)

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a Let ABPQC be a circle such that AB= AC AP meets BC

at

X

meets

BC

at

Y

as

shown below. Let

LEAP=

a

and

LABC

=

3

B

7

\

_JP

~

x

• c

I

y

Q

i Copy the diagram in your writing booklet, marking the info

above.

ii

State why LAXC = a + fJ

.

iii Prove that L BQP =

a

.

iv Prove that L BQA = .

c

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205 m

NOTTO

SCALE

is 205 metres above the horizontal plane

BPQ .

AB is vert

of

elevation

of

from

P

is 37° and the angle

of

elevation

o

is

22

°

P is due East

of

B and

Q

is south

4 7

°

ast from B . C

distance from

P

to

Q

to the nearest metre.

d

Four

people visit a town with

four

restaurants

A B C

and

D

Each

person chooses a restaurant at random.

i

ii

Find the probability that they all choose different restaura

Find the probability that exactly two

of

them choose rest

Question 14. 15 marks).

a

The

graph of

y =I

2 sin i 2x

-1

is shown in the diagram.

y

, :\

c· I

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b)

.

S h f 1

J

x - 4

1 tale t e range o

y

= tan .

2

ii) Find

dy

for the function

dx

Jx

2

4

y=tan

1

2

c) Find the volume

of

the solid when the region enclosed entirely

by

y

= sin x and

y

= sin x over the domain Os x s r is rotated ab

2

x

axis.

d) A projectile is fired from the origin towards the wall of a fort with

Vms

at an angle

a

to the horizontal.

On its ascent, the projectile just clears one edge

of

the wall and on i

ii)

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Hence, show that the equation o the path o the projectile is

y=x 1-;}ana.

iii) The projectile is fired at 45° and the wall

o

the fort is Ometr

the

x

coordinates o the edges o the wall are the roots o the

x

- Rx IOR= 0.

iv)

f he wall o the fort is 4.5 metres thick, find the value o R.

End o examination

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STUDENT NUMBER

GOSFORD HIGH S HOO

2 15

TRIAL HSC EXAMINATION

EXTENSION 1 MATHEMATI

General Instructions:

• Reading time: 5minutes

• Working time: 2 hours

• Write using black or blue pen

• Board-approved calculators may

be used

• table of standard integrals is

provided

• In Questions 11-14 show relevant

mathematical reasoning and/or

calculations

Total marks: - 7

Section I 10 ma

Attempt Question

Answer on the M

sheet provided

Allow about 15 m

Section II 60 ma

Attempt Question

Start each questio

booklet

Allow about 1 ho

this section

Section

I

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Total marks 10).

Attempt Questions 1-10.

Allow about

5

minutes

for

this section.

Answer on

the

multiple choice answer sheet provided. Select the altemativ

best answers

the

question. Fill

in the

response oval completely.

1. The point A

has

coordinates -1,4)

and

the

point

B has coordinate

coordinates

of the point

which divides

AB

externally

in the

ratio

1:

A. -4, 7)

B. 4, -7)

C. 7,-4)

x

2

2

2.

The

solution to

the

inequation

:5

1

is

x

A.

x

:5

-1, x ;;;

2

c

x :5 -1

0

< x :5

2

B.

D.

- 1 : 5x

x

:5

-1

3.

A

committee

of

three is to be chosen from a group

of

five

men

and

How

many

different committees can be fo11ned

if

he

committee is

one

man

and at least one woman?

A.

220 B.

175

c 70 D.

10

4.

I f

the acute angle between the lines

x -

y 2

and

kx

-

y 5

value

of k

is

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9.

Which of

the following is

an

expression for

2cos

2

x dx

A

1 .

2

. x s m x c

2

C. x - sinZx c

D.

x sinZx c

10. One approximation

to

the solution

of

the equation

+

an·

x x

2

4

is another approximation to this solution using

one

application

ofN

A. x 1.3805

B x 1.3914

c x

1.4125

D

Section II.

Total marks 60).

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Attempt Questions 11-14.

Allow about 1 hour and 45 minutes for this section.

Answer all questions, starting each question in a separate writing booklet.

Question

15 marks) Use

a

SEPARATE writing booklet.

a) i)

ii)

d

i n d x sin 2x).

dx

Hence or otherwise find

f

x os 2x dx .

x

b) Consider the function f x)

=

2

cos-

1

.

3

i) Evaluate

f O).

ii) Draw the graph

of

y

=

f x).

iii) State the domain and range

of

y

=

f x).

c)

f ex,~

and y are the roots

of 2x

3

- 6x

x

2 = 0, find the val

uestion

2 15 marks) Use a SEPARATE writing booklet.

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a) Two points

P 2ap, ap

2

and

Q 2aq, aq

2

lie

on

the parabola

x

i)

Show that the equation of the tangent to the parabola at

y =px p

2

•

ii)

The tangent at P and the line through Q parallel to the y

at T Find the coordinates

ofT

iii)

Write down the coordinates

ofM,

the midpoint

ofPT.

iv)

Determine the locus ofM when pq -1.

b) The diagram below shows a cyclic quadrilateral MN

KL

with M

p

NOTT

PN is a tangent to the circle and LMNK 2LKNP

Copy the diagram into your writing booklet and prove that

t

Hence, show that MK bisects LLMN

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Question

3

(15 marks) Use a SEPARATE writing booklet.

(a) (i)

v

1

howthat = - ( - v

2

.

t

x

2

(ii) A particle is moving along a straight line. At time,

t

second

x metres, from a fixed point O on the line is such that t

=

x

an expression for its velocity v in terms

of

x

(iii) Hence, find an expression for the particle's acceleration a i

(b) (i) Express f3cosx

-

sinx in the form Rcos x a) where O

(ii) Hence,

or

otherwise, solve

f3cosx

- sinx

= 1.

(c How many 4-letter words consisting

of

at least one vowel and at

consonant can be made from the letters

of

the word EQUATION?

(d) The region bounded by the curve y

=

cos 2x and the x-axis betwee

Tr

x

= -

is rotated about the x-axis. Find the exact value

of

the volum

4

revolution generated.

Question 4

15 marks) Use a SEPARATE writing booklet.

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a) Use mathematical induction to prove that for all positive integers

n:

n

(r )

= n + 1) -

1

r l

b) A particle moves in a straight line so that its displacement, x metres

seconds, is given by x = 4 - 2sin

2

t.

i) Show that the motion is simple harmonic.

ii) Find the period and the centre of the motion.

iii) Show that the velocity

v

of the particle in

te1ms

of its displa

expressed as v

2

= 4(-8

+ 6x

2

.

c) i) Show that the range offligbt of a projectile fired at an angle o

v

sin2a

horizontal with velocity v is

B

where g is the accele

gravity.

The equations describing the trajectory of the projectil

. 1 2

x

= vt

cos

a

y

= vt

sm a

- - gt .

2

You ar·e

NOT

required to prove these equations)

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2015

Assessment

ask

4

Trial HSC Examination

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Mathematics Extension

Examiners - Mrs

D

Crancher Mrs

S

Gutesa Mr

S

Faulds M


o Reading Time - 5 minutes

o Working Time - 2 hours

o Write using a blue or black pen.

o Board approved calculators and

mathematical templates and instruments

maybe

used.

o Show

all

necessary working in

Questions 11 12 13 and

14

o This examination booklet consists of

13

pages including a standard integral page

and a multiple choice answer sheet.

Total marks 7

section

Total marks

(10

o Attempt

o Answer

answer s

last page

booklet.

o Allow a

section

lsection

n

Total marks

(60

o Attempt

o Answer

writing

o Start a n

question

Section I

10 marks

Attempt Questions 1 - 10

Allow

about

15 minutes for this section

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Use the multiple choice answer sheet for Questions 1 - 10.

1 The solution to the inequality x( x ) x + 1)

:: :

0 is

A)

C)

x S: -2 or OS: x

S:

1

x:S:-1 or 0:S:x:S:2

B)

D)

-2

S: x S:

0 or

x

:: : I

-1 S: x

S: 0 or

x ;: :

2

2 A committee of3

men

and 3

women

is to

be fonned

from a group of8 m

3

How

many

ways can this

be

done?

A)

C)

48

40320

c

B)

D)

1120

3003

NOTTO SCALE

In the diagram, AB is a tangent to the circle, BC= 6cm and CD = 12cm

What is the length ofAB?

5 What is the acute angle to the nearest degree that the line

2x-3 y

+ 5 = 0

(A) 27°

(B)

34°

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6

7

8

(C)

56°

(D)

63°

Which

of

the following statements is

FALSE.

(A)

cos-

1

-8) =

-cos-

1

e

(B)

sin-I

(-8)

= -sin-I f

(C)

tan-

1

-8) =

tan-I f

(D)

cos-

1

-8)

=

n

-

cos-

1

8

The ptimitive of

2x(

3x

2

- I)

ts:

1 (

)

3 (

)

(A)

5

3x·

-1

c

(B)

-

3x

- I

c

5

2x

s

2x

(

2

5

(C)

5

(3x

2

- l

c

(D)

-

3x - I c

15

The

equation(s)

of

the horizontal asymptote(s) to the curve

y

= x

+

1

-1

(A)

(C)

y=O

y= l

(B)

D)

x=±I

x=l

only

9

What

are the coordinates

of

the point that divides the interval joining th

B(4,5) externally in the ratio 2:3?

(A)

(-2,-4) (B)

(-2,11)

10

Which of the following equations

is

shown in the sketch below

y

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(A)

C)

3n

n:

I ( . )

y =

cos smx

y =

sin

i

x)

+

sin x)

IT

-

2

2TI

3

l{

-

}

2TI

3

I

TI T

(B)

D)

-

y

= sin-

1

cosx)

y =

cos·

1

x) +

cos x)

Section II

6 marks

Attempt Questions to

14

Allow about 1 hour 45 minutes for this section

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Answer each question in the appropriate writing booklet.

All necessary working should be shown in every question.

Question

15 marks)

a)

b)

c)

d)

Solve the inequality

3 l?

x(2x-1)

In what ratio does the point 14,18) divide the interval joining

X

-1,3)to

Y 4,8)?

i)

Show that the curves

=x

- x

and

=x - x

interse

at

the

point

-2,-6

ii) Dete1111ine the acute angle between the curves

=x

-

i)

and

y = x - x

at the point

of

intersection, to the neares

A class

of25

students is to

be

divided into four groups

of

3, 4, 5 and 6 students. How

many

ways can this are

Leave your answer in unsimplified fonn.

Question

12

(15 marks)

(a) Consider the function l(x) = x - I)

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(i)

Sketch y = l(x).

(ii)

Explain why l(x) does not have an inverse function fo

all x in its domain.

(iii) State a domain and range for which

l(x)

has an invers

function 1- x).

(iv)

For

x;: :

I find the equation

of

the function 1- x).

(v)

Hence , on a new set

of

axes, sketch the graph

of

y =1

b)

F

. d

f

dx

11

J9 4x

2

c) Find the exact value

of

tan(2 tan_,

i

(

d)

Find the general solution to 2 cos

x Ji.

Leave your answer in terms

of re .

(e) Differentiate (with respect to

x

Question

13

15

marks)

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(a) The points

P

(

2ap,

ap ) and Q

2aq, aq

1

lie on the parabola x

such that

OP

is perpendicular to

OQ.

y

P 2ap,

a /

0

(i) Prove that

pq

= -4.

(ii)

R is

the point such that

OPRQ is

a rectangle.

Explain why the co-ordinates

of

R

are (

2a

(p

+

q),

a

(

(iii) Show that the locus ofRisa parabola.

(b)

Find by division of polynomials, the remainder when

x +

4

i

divided by x-3.

(c)

a, f

and

y aretherootsoftheequation x

3

-3x -6x-1=0.

Question 13 continued ...

d)

(i)

Consider the curve

.f x)=sin

2

x-x l

for 0:S:x:S:ir.

Show that it has one stationary point and detennine its

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(ii)

(iii)

.f x)=sin

2

x-x l has a zero near

x

= ir .

2

Use one application

of

Newton s method t obtain ano

approximation

x

2

t

this zero.

rr/2

The graph o x = sin

2

x x + 1 is shown in the vici

By using this diagram, detennine if x

2

is a better appr

than x

1

t

the real root

of

the equation. You must just

answer.

Question

14 15 marks)

(a)

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E

F

In the diagram above,

FG

is a common tangent and FBIIGD.

(i)

Prove that FAIIGC.

(ii)

Prove that

CGF

is a cyclic quadrilateral.

(b) (i) Find:

_: .._ (x sin 3x

dx

T

6

( i) Hence, evaluate:

cos3xdx

0

(c) Use the substitution y

=

x to find

dx

~x l -x)

(d)

Use mathematical induction

to

prove the inequality:

Year 12 Mathematics Extension I T1ial 2015

Question No.

Solutions and Marking Guidelines

Outcon1es Addressed in this Question

PE3 Solves problen1s involving pem1utations and co1nbinations, inequalities and polynon1ials.

HS Ann

ies annropriate techniaues from the studv

of

geo1netry.

Outcome Solutions

PE3

(a)

3

- - ->

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H5

x(2x

I

Multiply by the square

of

the denominator

3x(2x -

1 >

x

2

(2x

-1

)

2

3x 2x-l)-x

2

2x-1)

2

> 0

x 2x-1) 3-x 2x-l)) >

0

x 2x-1) -2x

2

+x+3)

>

0

-x 2x-1) 2x-3) x+l)

> 0

I 3

:.-l

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PE3

PE3

HS

tane

-1112

l+m

1

m

2

11-5

tan

e=

I+ (11)(5)

:.B=tan-

1

5

6

6

)

:. e=

6°7' (to the nearest minute)

25

r

3 4 5 6 7

d)

(ii)

(11-1) x 3 x 4 x 5 x 6

(e)

P E)

6x5x4x3x2

6'

5

54

Year 12 2015

Mathematics Extension I

T

Question No. 12

Solutions and Marking Guidelines

Outcomes Addressed in this Ouestion

HE4 uses the relationship between functions, inverse functions and their derivat

Outcome

Question 12

a

Solutions

HE4 (i)

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HE4

HE4

HE4

HE4

\

4

(ii)

I

;

It does not have an inverse because for every

y

value there is

more than one x value.

Or

Does not pass the horizontal line test.

Or

Anything that is equivalent.

(iii)

Domain: x;:: I

Range:

y

; 0

(iv)

x=(y-1)

1

.J; = v 1

y=l+.J;

:.r (x)=I+.J;

v)

6

5

4

3

2

1

I

M

I

M

l M

2

M

solu

l M

solu

I

M

HE4 c)

now

3

:. t anB=-

4

tan

2 tan_,

i)

=

tan 2B)

2 M

solu

1 M

solu

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HE4

(d)

HE4

e)

2tanB

l - tan

2

2(i)

=-- - - - -

1-(

i

24

7

2cosx =

-./3

-./3

c o s x = -

2

I -./3

:.x=2111r±cos- -

2

Jr . .

= 2111r±- wheren1sanymteger.

6

[

I

l

-

d

1 x - l x 3

-:(tan

-) = 2(1an -)

----:,

d.

3 3 I ~

9

(

_

x)

3 )

=2 tan

- - -

3 9 x

(

tan- 5J

=6 9 x2

no v

2 M

solu

1

M

solu

3 M

solu

2 M

that

solu

l M

diff

Year 12 Trial Higher School Certificate Extension 1 Mathematics

Question No. 13 Solutions and Marking Guidelines

Outcomes Addressed in this Question

PE3 solves problems involving polynomials

and

parametric represent

PES determines derivatives which

require

the application of more tha

differentiation

H6 uses the derivative to determine the features

of

the

graph of

a fun

HE7

evaluates mathematical solutions to problems

and

communicates

form.

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Outcome Solutions

PE3 (a) (i)

OP _l_ OQ, :. mOP xm OQ = -1.

PE3

PE3

ap

2

aq

:. x = l

2ap 2aq

: p x i = - 1

2 2

:.pq=-4.

(ii) Midpoint PQ = ap+aq, ap ; aq )

As the diagonals bisect one another in a rectangle,

R

will also have the same midpoint as

PQ.

f

0 0,

0), R has midpoint ( ap +aq, ap ;aq ) , then

Ris (2a(p+q), a(p

2

+q

2

) ) .

(iii) At

R

::

;P

+

q

~;]

y=a(p

2

+q

2

)

[3]

From [2], x

2

=4a (p+q)

: x

2

=4a

2

(p

2

+q

+2pq)

Substituting [1] and [3],

x

2

= 4 a ( ;

+

2x-4) ,

: x

2

=

4a

(y-8a), which is a concave up parabola with

ve1iex ( 0,8a).

H6, PE5

: a +

2

+r =(a+ J + r) -2 a/J + /Jr+ar)

: a +

2

+ r = 3)

2

-2(-6)

= 21.

(d) (i) f(x) =sin

2

x -x + I

f (x) =2sinxcosx-l

:. f (x)

=

sin2x-l

f (x) =

0 for stationary points.

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PE5, HE7

HE7

Solving sin

2x -

l

=

0,

sin2x =

For O:,;

x:,; n-

0

s

2x

s

2n-.

S I

. 2 7r . . 7r

o vmg, x = - , : one stationary pomt, at x = - .

2 4

Testing

x =

7r, for

f

(

x)

= in

2x-l,

4

7r

x

-

-

-

6 4 3

f (x)

J5 2

0

J5-2

2 2

As

J5-2

is negative, there is a horizontal point of

2

. fl .

7r

111 ex10n at

x

= .

4

(

)

f

n-

. '

n- n- I

n-

11

-

=sm----

=2

. 2 2 2 2

f (;

)= sinn--1 =-1.

7r

Newton s method: x, = -

- 2

7r

2 -

- __

-1

(iii)

:. x,

= 2

Year

12

Mathematics Extension 1 Trial Examination 2015

Question No. 4 Solutions and Marking Guidelines

Outcomes Addressed in this Ouestion

PE3 solves problems involving circle geometry

HE2 uses inductive reasoning in the construction

of

proofs

HE4

uses the relationship between functions, inverse functions and their der

HE6 determines integrals by reduction to a standard form throu h a iven su

Outcome Solutions M

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PE3

(a)(i)

Let

LDGH=

a

2

marks

. . LBFG = a

(corresponding angles, FBIIGD) Correct s

PE3

HE4

HE4

1

mark

No,v, LGCD = angle bet,veen a chord and Substantia

tangent is equal to the angle in solution.

the alternate segment)

Similarly,

LFAB

=

Since

LGCD

=

LFAB

=

a,

FAIIGC

(corresponding angles are equal)

(ii) Since LGCD = a

(shown above)

LGCB = 180°-

also, LBFG = a

(angles on a straight line

(sho,vn above)

L.GCB

+

L.BFG

=

180° - a

=

180°

:. BCGF

is

a cyelie quadrilateral (opposite angles supplementary)

b) (i)

(ii)

... xsin 3x = x.3cos 3x sin 3x.l

dx

= 3xcos3x+sin3x

If

~xsin3x = 3xcos3x

sin3x

dx

then 3xcos3x=~xsin3x-sin3x

dx

3

I

d .

3

I .

3

cos .x=--xs1n

x--s1n

x

3dx

3

Integrating both sides,

rr n n

- - -

6 6 6

2

marks

Correct s

1 mark

Substanti

solution.

2

marks

Correct a

correct an

1 mark

Demonst

making s

solution.

3 marks

Correct s

2 marks

Correctly

function.

t n1ark

Substanti

required

(c)

3 rnarks

HE6

Let

y=J;

Correct so

:.x=y"

2 marks

x

=

v

Uses the

dy

.

makes sub

correct so

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dx=2ydy

1 mark

I

dx

I

2ydy

Uses the g

~x(l-x)

- ~/( - /)

I

2vdv

= .v~6-y )

I

dv

=2 J(1~y')

=

2sin-

1

y+

c

but

y=J;

I d,

. . ~x(l-x)

2sin-

1

J ; c

d)

HE2

n > 2n, for all positive integral values of n 4

Prove true

for 11 = 4

3

marks

Correct s

LHS=4

RHS= 2'

2 marks

=24

=16

Prove the

24 >

16

1nakes su

correct so

:.

True for

n

= 4

1 mark

Correctly

Assmne true for

n = k

n=4.

ie. Assun1e k > 2t

k -2 >0

Prove true for 11

=

k

+

l

ic.

Prove (k+1) >2'''

Consider the difference

k+ I) - 2'" = k +I) .k - 2.2'

=k.k +k -2•-2

1

= kk -2k

+k1-2•

=

(k-1}.k +

k -2'

+k -2'

= k-1

}.k +

2(k -2')

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Section 1 (10 marks)

Attempt questions 1 ‐10. Use the multiple‐choice answer sheet provided.

1. Evaluate x

x

x 5

7sin3lim

0

(A) 3 (B) 0 (C)5

21

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2. For what values of x is ?61

4

x

x

(A) (B)

(C) (D)

3. The interval joining the points 2,3 A and y B ,9 is divided externa

point .13, xP What are the values of x and ? y

(A) ,27 x 22 y (B) ,18 x y

(C) ,6 x 12 y (D) ,27 x y

4. A circle with centre O has a tangent TU , diameter QT ,oSTU = 25 an

What is the size of RTQ ?

6. A particle moves such that when it is x metres from the origin its accele

1

2

xa e

. What is its velocity when 3 x , given that 1v when x

(A)

1

0.050 ms

(B)

1

0.070 ms

(C)

1

0.158 ms

(D)

7 Whi h f th f ll i i th t i fdx

?

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7. Which of the following is the correct expression for236 x

?

(A)-1cos

6

xc

(B)-1

cos 6 x c

(C)-1sin

6

xc

(D)-1

sin 6 x c

8. Eden, Toby and four friends arrange themselves at random in a circle.

that Eden and Toby are not together?

(A)1

120 (B)

2

5 (C)

3

5 (D)

9. If2

tan t which of the following expressions is equivalent to sin4

(A)

2

2

1

22

t

t

(B)

2

2

1

4

t

t

(C)

2

2

1

22

t

t

(D)

10. An expression for the general solution to the trigonometric equation t

any integer is:

Section II (60 marks)

Attempt all questions from 11‐14. Answer each question on a separate pag

Question 11 (15 marks)

(a) The number of animals in a local farm who will be infested with a virus

kt Ce

pn

1 where n = the number of animals infested

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p = the total number of animals

k = the growth constant

t = the time in months

C = constant

The farmer notices that initially 1 animal out of the animal population o

virus. After one month the number of animals infested with the virus in

(i)Show that after t months,

kt en

1991

200

(ii) Show that k = 1.63 (to 3 significant figures)

(iii) How many animals can the farmer expect to be infested after 3 m

(b) (i)Find

2tan

4

2 12

x

x

x

dx

d

(ii)

Hence evaluate

2

0

224 xdx

(c) A spherical metal ball is being heated such that the volume increases at

5 .min/3mm At what rate is the surface area increasing when the rad

(d) Find an expression for

3

1

x

x

e

dxe using the substitution 1 x

u e .

(b) Let 2,2 apapP and 2,2 aqaqQ be two points on the parabola 42 a x through the point ,0,a A and the tangents at P and Q meet at . R

(i) Show that .2 pqq p

(ii) Find the coordinates of R in terms of p and .q

(iii) As P and Q vary, show that R moves on a straight line.

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(iv) Find the restrictions on the x values of the locus of R.

(c) Use mathematical induction to prove that for all integers ,3n

.

1

221...........

5

21

4

21

3

21

nnn

Question 13 (15 marks) Start a new page

(a) (i) Using the auxiliary angle method express t t 2cos22sin3 in the f

A particle moves horizontally in a straight line so that its position x fro

given by:

22cos22sin3 t t x

Displacement is measured in metres and time in hours.

(ii) Find an equation to represent the acceleration of this particle and

simple harmonic motion.

(iii) Given that the particle is at the origin at noon, between what time

the particle be more than one metre to the right of the origin for t

at t = 0 be noon). Give your times correct to the nearest minute.

(b) Consider the function .1cos2

1 1 x y

Question 14 (15 marks) Start a new page

(a) In a BMX dirt bike competition the take‐off point O for each competito

the downslope. The angle between the downslope and the horizontal i

from O with velocity V sm / at an angle to the horizontal, where 0

on the downslope at some point ,Q a distance D metres from .O

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(i) Show that the Cartesian equation of the flight path of the biker is

.sec

2

tan2

2

2

V

gx x y

(ii) Show that

.sincos3cos42

g

V D

(iii) Show that

.2sin32cos42

g

V

d

dD

(b)

(iv)

(i)

Show that D has a maximum value and find the value of for w

Considering the identity ,111 2 nnn x x x where n is a posshow that for integer values of ,r

r nr

k r

n

k

nr

k

k C C C 11 2

2

0

provided 0

.

The flight path of the biker is give

cosVt x and2

1 2gt y

where t is the time in seconds af(DO NOT PROVE THIS)

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NORTHERN BEACHES SECONDARY CO

MANLY SELECTIVE CA

HIGHER SCHOOL CERTIFIC

Trial Examination

2015

Mathematics Extension


Reading time – 5 minutes

Working time 2 hours

Section I Multiple Choic

10 marks

Attempt all question

Manly Selective Campus

2015 HSC Mathematics Extension 1 Trial

Multiple Choice: Answer questions on provided answer s

Q1. The diagram shows a circle with centre O. The line PT is

circle at the point T . ∠TOP = 4 x° and ∠TPO = x°.

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What is the value of x?

(A)

9

(B) 18

(C) 36

(D) 72

Q2.

Which of the following is a simplified expression for

(A) sin x

(B) cos x

(C)

tan x

(D) cot x

x° P

T

4 x°

O



Q4.

What is the obtuse angle between lines ?

(A) 15o

(B) 75o

(C)

105o

(D) 165o

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Q5. What is the value of ?

(A)

(B)

(C)

(D)

Q6.

In how many ways can 5 people be selected from a group of 6

arranged in a line so that the two oldest people in the selected

either end of the line? (NB. No two people are the same age.)

(A)

720

(B)

144

(C)

72

(D)

36

Q7. The remainder of the division is equal to

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Question 11: Start A New Booklet

(a) Evaluate .

(b) (i) Verify that (αβ+αγ+ βγ)² = α²β²+α²γ²+ β²γ² + 2αβγ(α+

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( ) y ( β γ βγ) β γ β γ βγ(

(ii)

Hence, or otherwise, if α, β and γ are the roots of

evaluate

(c) (i) Determine the vertical asymptotes for

(ii) Hence sketch the curve

(d) Find the general solution of the equation



Question 12 Start A New Booklet

(a) Use the substitution to show

(b) A particle moves with acceleration . Initially, the partic

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to the right of the origin and its velocity is 4m/s.

Find the displacement of the particle when it is at rest.

(c) ABCD is a cyclic quadrilateral. The tangents from Q touch the circl

The diagonal DB is parallel to the tangent AQ, and QA produced in

produced at P.

Let < QAB = α.

(i) Prove that Δ BAD is isosceles, giving reasons.

(ii) Find

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The tangent to the parabola at cuts the x-axis at T an

P cuts the y-axis at N .

The equation to the tangent is given by

(i) Show the coordinates of N are

(ii)

Let M be the midpoint of NT. Find the Cartesian equation

(b)

Use mathematical induction to prove that ( )2

1 1n n+ + − is divisibl

for all integers 1n ≥ .

(c)

A school band is to be formed with a brass section containing

percussion section containing 4 students.



Question 13 continued

(d)

At time t years the number N of individuals is given by

constants a > 0 , b > 0. The initial population size is 20 and the limi

size is 100.

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(i) Show that

(ii) Find the values of a and b.



Question 14

(a)

Warehouse A has 100 computers and the probability that of select

computer which is defective is 0.02.

Warehouse B has 100 computers, two of which only, are defectiv

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Joe buys three computers from Warehouse A and three computers

Warehouse B. What is the probability that exactly one of the com bought is defective?

(b) Two towers T 1 and T 2 have heights h metres and 2h metres resp

second tower is due south of the first tower. The bearing of tow

surveyor is 292°. The bearing of the tower T 2 from the surveyor

angle of elevation from the surveyor to the top of tower T 1 is angle of elevation from the surveyor to the top of tower T 2 is 60°.

d



Question 14 continued.

(c) A ball is projected vertically from the ground with a speed of 49 m

the ball at time t is given by2

4.9 49 . y t t = − + (Do NOT show this.)

At the same time, a second ball is projected from the ground in

angle of projection θ . Its horizontal displacement is given by x

h i h i i b2

4 9 98 it t θ+ ( O h hi )

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height is given by 2

4.9 98 sin y t t θ = − + . (Do NOT show this.)

(i)

Find the maximum height of the ball that was projected ve

(ii) Find the value of θ at which the second ball should be pro

the first ball when the first ball reaches its maximum heigh

(iii) Find the horizontal distance between the two balls wh

projected into the air. Give your answer in exact form.

(e)

Consider the binomial expansion

Show that, if n is even:

2015 MSC HSC X1 Trial Solutions

Q1

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Q2

Q3

Q4


Q5

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Q6

Q7


Q9

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Q9

Q10Cubic of form


Q11

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4

-

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3-

w

r

c

2

f

t

1

f


Q12

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3 m

2 m

sim

1 m

su

lim


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c-i

c-ii


c-iii

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d-i

The solution is close to zero therefore reasonable approximation.

d-ii


Q13

a)i)

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ii)


b)

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c)i)

ii)

d)i)


ii)

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Question 14

a P(exactly one computer defective)

=P(1 defective from A, 0 from B) +P(0 from A, 1 from B)

3

2

c

1

u

r

b

3

2


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i 2

1

s

ii

2

1

a

e


d 4

i

3w

l

2

t

e

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1t

NORMANHURST BOYS HIGH SCHOOL

N E W S O U T H W A L E S

2015

BOSTES Number: _____

CLASS (Please circle): 12M

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HIGHER SCHOOL CERTIFICATE

TRIAL EXAMINATION

Mathematics ExtenGeneral Instructions

Reading time - 5 minutes

Working time - 2 hours

Write using black or blue pen Board-approved calculators may be

used

A table of standard integrals is provided

at the back of this paper

In Questions 11-14, show relevant

mathematical reasoning and/or

calculations

Total marks - 70

Section I Pages

10 marks

Attempt Questions

Answer on the Mult

provided

Allow about 15 min

Section II Pages

60 marks

Attempt Questions 1

Section I

10 marks

Attempt Questions 1 – 10

Allow about 15 minutes for this section

Use the multiple-choice answer sheet for Questions 1-10

1

NOT TOSCALE

y

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The diagram above represents a sketch of the gradient function of th

Which of the following is a true statement? The curve ( ) y f x has

(A) a minimum turning point occurs at 4 x

(B) a horizontal point of inflexion occurs at 2 x

(C) a horizontal point of inflexion occurs at 4 x

(D) a maximum turning point occurs at 0.

2 Solve3

1

SCALE

y = f ' ( x)

x-4 -2 2 4

3 If

,

, evaluate 2.

(A)

(B)

(C)

(D)

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4 Find the acute angle between the tangents to the graphs y x and y

(A) 27°

(B) 30°

(C) 45°

(D) 63°

5 The polynomial 2 5 2 has 2 as a factFind the value of k .

(A) –7

(B) 7

(C) –12

(D) 12

7 Find the domain and range of 3 cos (A) Domain: 0 3 Range: 0

(B) Domain: 1 1 Range: 0

(C) Domain: Range: 0 3

(D) Domain:

Range: 0 3

8 A stone is thrown at an angle of α to the horizontal. The position of t

seconds is given by cos x Vt and 21

sin2

y Vt gt where

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2

acceleration due to gravity and v m/s is the initial velocity of projectio

What is the maximum height reached by the stone?

(A)

sinV

g

(B)

sing

V

(C)

2 2sin

2

V

g

(D)

2

2

sin

2

g

V

9 The volume of a sphere of radius 8 mm is increasing at a constant rat

Determine the rate of increase of the surface area of the sphere.

(A) 0.06 mm2/s

(B) 0.60 mm2/s

(C) 1.25 mm2/s

Section II

60 marks

Attempt Questions 11 ‒ 14

Allow about 1 hour and 45 minutes for this section

Answer each question in a SEPARATE writing booklet.

Your responses should include relevant mathematical reasoning and/or ca

Question 11 (15 marks) Use a SEPARATE writing booklet.

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(a) (i) Evaluate

.

(ii) Use Simpson’s rule with 3 function values to approximate

(iii) Use your results to parts (i) and (ii) to obtain an approximati

Give your answer correct to 3 decimal places.

(b) Using one application of Newton’s method with = 2 as the first

approximation, find the second approximation to the root of the equ 2 3. Correct answer to 3 decimal places.

(c) The polynomial3 2( )P x x bx cx d has roots 0, 3 and –3.

(i) What are the values of b, c and d ?

(ii) Without using calculus, sketch the graph of ( ) y P x .

(iii) Hence or otherwise, solve the inequality2 9

0 x

x


(a) The point 28,19 divides the interval AB externally in the ratioFind the value of k if A is the point 4,3 and B is the point 2,

(b) (i) Show thatsin cos

sin( )4 2

x x x

(ii) Hence or otherwise, solvesin cos 3 x x

for 0 2 x

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22

(c) 2, and 2, are two points on the parabola2

x

(i) Show that the equation of the normal to the parabola at P is

32 x py ap ap .

(ii) Find the co-ordinates of R, the point of intersection of the no

P and Q, in terms of p and q.

(iii) If 2, find the cartesian equation of the locus of R.

(d) Evaluate

√ 1 3 using the substitution 1 3

(e) After t years the number of animals, N, in a national park decrease

according to the equation:

0.09( 100)dN

N dt


(a)

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The diagram shows a cylindrical barrel of length l and radius r . The

is at one end of the barrel, at the very bottom of the rim. The point B

very top of the barrel, half-way along its length. The length of AB i

(i) Show that the volume of the barrel is

.

(ii) Find l in terms of d if the barrel has maximum volume for the

(b) Two circles are intersecting at P and Q. The diameter of one of the

PR.

(c)

Question 13 (continued)

A rocket is fired from a pontoon on the sea. The rocket is aimed at ahigh cliff, 240m from the pontoon. The angle of projection of the ro

45º and its initial velocity is 40√ 2 .

(i) Taking the point of projection as the origin O, derive expres

the horizontal component x and vertical component y of the

of the rocket at time t seconds.

(Assume the acceleration due to gravity is 10

(ii) Show that the path of the rocket is given by the equation

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(iii) Find the time taken for the rocket to land on top of the cliff.

(iv) Find the exact velocity of the rocket when it reaches this poi

(Hint: velocity includes magnitude and direction)


(a) Use mathematical induction to prove that 3 7 is divisible b

for all integers 1.

(b) The velocity of a particle moving in a straight line is given by

10 where x metres is the displacement from a fixed point O and v is the

in metres per second. Initially the particle is at O.

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(i) Show that the acceleration of the particle is given by

10

(ii) Express x in terms of time t .

(iii) What is the limiting position of the particle?

(c) A particle moves in a straight line and its displacement x metres fro point O at any time t seconds is given by the equation

24cos 1 x t .

(i) Prove that the particle is undergoing simple harmonic motio

(ii) State the period of the motion.

(iii) Sketch the graph 24cos 1 x t for 0 t .

Clearly show the times when the particle passes through O.

(iv) Find the time when the velocity of the particle is increasing

STANDARD INTEGRALS

n

x dx 1

, -1; 0 if 01

n+1 x n x , n <

n

1 dx x ln , > 0 x x

axe dx 1

, 0axe aa

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cos ax dx 1 sin , 0ax aa

sin ax dx 1

- cos , 0ax aa

2sec ax dx 1 tan , 0ax aa

sec tanax ax dx 1

sec , 0ax aa

2 2

1dx

a x

-11 tan , 0 x

aa a

2 2

1dx

a x

-1sin , 0, - < < x

a a x aa

2 2

1dx

x a 2 2ln , 0 x x a x a

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2015 HSC ASSESSMENT TASK

MathematicsExtension 1

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Reading time – 5 minutes

Working time – 2 hours

Write on one side of the paper

(with lines) in the booklet provided

Write using blue or black pen

Board approved calculators may

be used

All necessary working should be

shown in every question

Each new question is to be started

on a new page. Attempt all questions

Class Teacher:(Please tick or hig

Mr Berry Mr Ireland Mr Lin Mr Weiss Ms Ziaziaris Mr Zuber

(To be used by the exam markers only.)

Q ti

Student Number:

Section I

10 marks

Attempt Questions


Use the multiple choice answer sheet for Questions

_______________________________________1. What is the value of

(A) 0

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(B)

(C) 1

(D)

2. is a linear function with gradient , find the gradient of

(A) 4

(B)

(C)

(D)

3.

Which of the following best describes the above function?

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Which of the following best describes the above function?

(A)

(B)

(C)

(D)

4. What are the coordinates of the point that divides the interval joining th

B( externally in the ratio 1:3?

(A)

(B)

(C)

(D)

6. The polynomial has roots , and . Wha

(A) 2

(B)

(C) 4

(D)

7. The line TA is a tangent to the circle at A and TB is a secant meeting the

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Given that TA = 4, CB = 6 and TC = x , what is the value of x ?

(A) 2

(B) 4

(C) 6

(D)8

8. Given that , find an expression for

(A) 2

(B) 4

(C) 8

(D) 16

10. An approximate solution to the equation is . U

Newton’s method, a more accurate approximation is given by:

(A)

(B)

(C)

(D)

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Section II

60 Marks

Attempt Questions

Allow about 1 hour and 45 minutes for this section

Answer each question on a NEW page. Extra writing booklets are available.

In Questions , your responses should include relevant mathematical reaso____________________________________________________

Question 11 (15 Marks) Start a NEW page.

(a) When the polynomial is divided by th

is the value of a?

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(b) (i)

(ii)

(iii)

(iv)

(c) Find the acute angle between the lines and

(d) Evaluate


(a) (i) Without using calculus, sketch the graph of

(ii) Hence solve

(b) Using the substitution find the exact value of:

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(c) (i) A chef takes an onion tart out of the fridge at into a room where th

. The rate at which the onion tart warms follows Newton’s law, tha

where k is a positive value, time t is measured in minutes and temperat

degrees Celsius.

Show that is a solution to and find the v

(ii) The temperature of the onion tart reaches in 45 minutes. Find the

(iii) Find the temperature of the onion tart 90 minutes after being removed

(d) (i)

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ABC is a triangle inscribed in a circle.

MAN is the tangent at

Ato the circ

CD and BE are altitudes of the triangle.

Copy the diagram into your answer booklet.

(ii) Give a reason why BCED is a cyclic quadrilateral

(iii) Hence show that DE is parallel to MAN.

Question 13 (15 Marks) Start a NEW page

(a) Is the graph of identical to ? Give a reason for yo

(b) (i) A particle is moving in a straight line. At time t seconds it has displacem

fixed point O on the line, velocity and acceleration given

the particle is 5m to the right of O and moving towards O with a speed o

Explain whether the particle is initially speeding up or slowing down.

(ii) Find an expression for in terms of .

(iii) Find where the particle changes direction.

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(c) (i) Express in the form

(ii) Hence, or otherwise, solve for

(d) (i)

A square ABCD of side 1 unit is gradually ‘pushed over’ to become a rho

decreases at a constant rate of 0.1 radian per second.

At what rate is the area of rhombus ABCD decreasing when ?


(a) Prove that is a multiple of 10 for all positive integers

(b) (i) Show that

(ii) Hence, using a similar expression, find a primitive for

(iii) The curves and intersect at

The curve also intersects with the x axis at Q.

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Find the area enclosed by the x -axis and the arcs OP and PQ.

(c) (i) A parabola has parametric equations

Sketch the parabola showing its orientation and vertex.

(ii) Point is the point on the parabola where

Point is the point on the parabola where

Find the equation of the locus of the midpoint of and state its geom

(iii) A line with gradient m passes through and cuts the parabola at dis

Find the range of possible values form.

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NORTH SYDNEY GIRLS HIGH SCHOOL

2015 TRIAL HSC EXAMINATION

Mathematics Extensio

General Instructions Total marks – 70

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Reading Time – 5 minutes

Working Time – 2 hours

Write using black or blue penBlack pen is preferred

Board approved calculators may be used

A table of standard integrals is provided at

the back of this paper

In Questions 11 – 14, show relevant

mathematical reasoning and/or calculations

NAME:______________________________ TEACHER:______

STUDENT NUMBER:

Section I Pa

10 marks

Attempt Quest Allow about 15

Section II Pa

60 Marks

Attempt Quest

Allow about 1 section

Section I

10 marks

Attempt Questions 1−10


Use the multiple-choice answer sheet for Questions 1 – 10.

1 What is the value of3 2

3

2 3 5 7lim

4 x

x x x

x x

(A)1

2

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(B) 2

(C)1

2

(D) 2

2 Which of the following is equivalent to 3 sin cos ?

(A) 2sin6

3 A curve is defined by 2 x t and loge y t .

Which of the following is the value ofdy

dx at the point 2, 0( ) ?

(A) 1

4

(B) 1

2

(C) 1

(D) 2

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4 What is the value of0

2sin2lim

3tan3 x

x

x?

(A)2

3

(B)3

2

(C)4

9

(D) 1

5 What is the value of sin , given that ACD in the diagram

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(A)2 2

33 5

(B)2 5

35

(C)2 2

35

(D) 4 133 5

6 What is the correct expression for 2

?

4

dx

x

x

7 The graph below represents the depth of water in a channel (in m

over time (in hours).

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Which of the following is NOT true?

(A) The centre of motion is at 8 m

(B) The period of oscillation is 8 hours

(C) The amplitude is 8 m

(D) The rate of change in the depth of water is the fastest whe

8 Which of the following are the roots of the equation 3 24 x x x

(A) 1, 3, 2

9 What is the value of 1cos sin where 2

?

(A)

(B) 2

(C)2

(D)2

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10 In solving1 2

1

x

x x

within the natural domain, three students

following inequalities.

Student I: 2

1 2 x x

Student II: 3

1 2 1 x x x

Student III: 3

1 2 1 x x x x

Which students will obtain the correct solution to the original in

(A) Student I only

(B) Student II only

Section II

Total marks − 60

Attempt Questions 11−14

Allow about 1 hour 45 minutes for this section.

Answer each question in a SEPARATE writing booklet. Extra writing bo

In Questions 11 to 14, your responses should include relevant mathemati

calculations.


(a) Differentiate 1cosx ex with respect to x.

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(a) Differentiate cos x ex with respect to x.

(b) Find sin23 x

ó

õô dx

(c) The point P divides the interval joining A 1, 5( ) to B 2, 3( ) extein the ratio 4 : 3 . Find the coordinates of P .

(d) Find the size of the acute angle between the line 2 y x and the2 y x at the point of intersection 2,4 .

Give your answer to the nearest degree.

1dx


(a) Angela is preparing food for her baby and needs to use cooled

The equation y = Aekt describes how the water cools, where t

minutes, A and k are constants and y is the difference betwtemperature and the room temperature at time t , both measur

Celsius.

The temperature of the water when it boils is 100C and the roo

is a constant 23C.

(i)

Find the value of A.

(ii) The water cools to 88C after 5 minutes. Find the value o

three significant figures.

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(iii) Angela can prepare the food when the water has cooled to

How much longer must she wait?

(b) A particle’s displacement satisfies the equation2 5 4t x x , w

measured in cm and t is in seconds. Initially, the particle is 4 cmright of the origin.

(i) Show that the velocity is given by1

2 5v

x

.

(ii) Find an expression for the acceleration, a in terms of x.

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(c) Consider the parabola 2 4 x y .

22 , P p p and

22 ,Q q q lie on the parabola.

22 ,Q q q

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(i) Find the equation of the chord PQ.

(ii) Show that if PQ is a focal chord then 1 pq .

(iii) T 2t ,t 2( ) , 0t and R 2r ,r 2( ) are two other points on the pdistinct from P and Q.

If TR is also a focal chord and P , T , Q and R are concyclic

show that p2 + q2 = t 2 + r 2.

22 , P p p


(a) A particle is undergoing simple harmonic motion such that its

displacement x centimetres from the origin after t seconds is giv

2 4 sin 23

x t

.

(i) Between which two positions is the particle oscillating?

(ii) At what time does the particle first move through the origi

positive direction?

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(b) Use the principle of mathematical induction to prove 3 7 4n n

integers 3n .

Question 14 continues on page 13


(c) Consider the region enclosed by the circle 2 2 2 x a y a and

shown in the diagram below, where 0 2b a

.

ab x

y

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(i) Show that the volume of the spherical cap formed by rotat

region around the x-axis is given by

2

33

bV a b

cubic units

(ii) A spherical goldfish bowl of radius 10 cm is being filled w

at a constant rate of 75 cm3 per minute.


(d) Consider the function 1

f x x x

whose graph is shown belo

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(i) By restricting the domain of the original function to 0 x

find the equation of 1 f x .

(ii) Hence, without solving directly, find the value(s) of x

for which1

16 x

x

. Leave your answer in exact form.

Mathematics Extension 1 Trial HSC 2015 – Suggested Solutions

Section I

1. D

Degree of numerator and denominator is the same. The

limit is the ratio of the leading coefficients ie2

2

1

.

2. B

Using auxiliary angle method, this is o the form

sin R where2

23 ( 1) 2 R .

1tan

63

.

3. B

Use parametric differentiationdy dy dx

6. B using standard i

7. C

The amplitude is the distan

to the extreme of motion w

8. B

Sum of roots = 4 and pro

9. C

1 1

1

1

2

cos sin cos sin

cos cos

cos cos

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Use parametric differentiation.dx dt dt

.

1dy

dt t

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