2015_3u_papers

BAULKHAM HILLS HIGH SCHOOL

2015

HIGHER 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  Show all necessary working in Questions 11-14  Marks may be deducted for careless or badly arranged work

Total marks – 70 Exam consists of 11 pages. This paper consists of TWO sections.

Section 1 – Page 2-4 (10 marks) Questions 1-10  Attempt Questions 1-10 Allow about 15 minutes for this section. Section II – Pages 5-10 (60 marks)  Attempt questions 11-14 Allow about 1 hour and 45 minutes for this section. Table of Standard Integrals is on page 11

1

Section I - 10 marks Use 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:

50°

x°

O

(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) and B(8,-3) what is the coordinate of the point P? (A) -13 (B) -1 (C) 4 (D) 28

3.

How many distinct arrangements of the letters of the word if the s are separated. (A) 720 (B) 1800 (C) 2160 (D) 2520

2

are possible in a straight line

4.

The polynomial (A) -28

6

2

2. What is the value of

has a factor of

?

(B) -20 (C) 20 (D) 28 5.

The acute angle between the lines 4

2 and

2

1 to the nearest degree is :

(A) 12° (B) 13° (C) 40° (D) 41° 6.

The equation of an inverse trig function drawn below is : y

–

3 2

1 2

1 2

–

7.

(A)

sin

(B)

sin 2

(C)

3sin

(D)

3sin 2

x

3 2

A particle moving in simple harmonic motion with displacement 9 16 . What is its amplitude (A) and its period (T) ? (A) A=3 T=

(B) A=3 T= (C) A=4 T =

(D) A=4 T =

3

and velocity , is given by

1

8. √25 (A)

4 sin

(B) sin (C) sin (D) sin

9.

is:

The derivative of tan (A)

(B) (C) 4 tan (D)

1|

10. The solution to |2

1

(B)

(D)

2| is

1

(A)

(C)

|

1

1 1

1 End of Section 1

4

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 booklet

Marks

3 a)

Solve

b)

Find

c)

d)

e)

f)

1

2 2 1

Evaluate lim →

sin 3

using the substitution

2

3

1

1 2

2

2 Find the constant term in the expansion

.

sin2

(i)

Show that a root of the continuous function between 0.4 and 0.5.

(ii)

Hence use one application of Newton’s method with an initial estimate of 0.4 to find a closer approximation for the root to 2 significant figures.

Solve sin2

cos

for 0

2

lies

1

2

2

End of Question 11

5

Question 12


sin 2

Marks

a)

Evaluate

b)

(i)

From a group of 6 boys and 6 girls, 8 are chosen at random to form a group. How many different groups of 8 people can be formed?

1

(ii)

How many of these groups consist of 4 boys and 4 girls?

1

(iii)

4 boys and 4 girls are chosen and placed around a circle. What is the probability that the boys and girls alternate?

2

c)

3

The rate of change of the temperature (T) of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding medium(C), ie

An object is heated and placed in a room of temperature 20° to cool. After 10 minutes its temperature is 36° . After 20 minutes the temperature is 30° .

d)

is a solution to the differential equation above.

(i)

Show

(ii)

Find the value of

(iii)

What was the temperature of the object when it was first placed in the room?

and the value of

to 3 decimal places.

1 3 1

Prove !

1 3 5 … … … … … … … …(2n+1) =

!

induction.

End of Question 12

6

for

0 by mathematical

3

Question 13

a)

The polynomial

6

8 has roots ,

Marks

and .

Find : (i)

1

(ii)

2

(iii)

b)


if

has a triple root.

2

PN is the normal to the parabola 4 at the point P (2 , ). The normal intersects the line SN which is parallel to the tangent at P. S is the focus of the parabola. y

N 2

P(ap, P 2 ap , ) S

x

2

(i)

Show the equation of the normal PN is

.

(ii)

Find the equation of the line SN.

(iii)

Show that N has coordinates (

(iv)

Find the equation of the locus of N as P moves on the parabola.

2 1

,

.

Question 13 continues on the following page 7

2 2

c)

Sand is falling on the ground forming a conical pile whose semi apex angle is 30°. / .(

The volume of the pile is increasing at a rate of

30°

h

r

(i)

Show that the volume of the pile is given by:

1

(ii)

Find the rate at which the height of the pile is increasing when the height of the

2

pile is 2 metres.

End of Question 13

8

Question 14


a)

V 10m  100m

tan

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

b)

away. Let acceleration due to gravity be =10ms .

5

and

.

2

(i)

Show that the equations of motion are

(ii)

Find the initial velocity, V of the projectile.

2

(iii)

At what speed is the projectile travelling the instant it clears the wall?

2

Copy or trace the diagram below in your exam booklet. C

E

Q R

x° D

A

is the midpoint of the arc diameter

and meets

. The radius

B

P

O

meets

at .

is perpendicular to the

at .

(i)

If ∠

°, show ∠

2 °.

(ii)

Prove

is a cyclic quadrilateral.

Question 14 continues on the following page

9

2 2

c)

Below is the graph of

√

ln y

e+1 e

2

x

(i)

Show that the equation of the inverse function is given by

(ii)

Hence find the area of the shaded region above.

End of Paper. 10

.

2

3

Name: ________________________________ Teacher: _______________________________ Class: _______________________________

FORT STREET HIGH SCHOOL

2015

HIGHER SCHOOL CERTIFICATE COURSE

ASSESSMENT TASK 3: TRIAL HSC

Mathematics Extension 1 Time allowed: 2 hours (plus 5 minutes reading time) Syllabus Assessment Area Description and Marking Guidelines Outcomes Chooses and applies appropriate mathematical techniques in order to solve problems effectively HE2, HE4 Manipulates algebraic expressions to solve problems from topic areas such as inverse functions, trigonometry, polynomials and circle geometry. HE3, HE5 Uses a variety of methods from calculus to investigate HE6 mathematical models of real life situations, such as projectiles, kinematics 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 section Section II 60 Marks Attempt Questions 11‐14, Allow about 1 hour 45 minutes for this section

General Instructions:

Section I

Total 10

Q1‐Q10

Questions 1‐10 11, 12 13 14 Marks

Section II

Total 60

Marks

Q11

/15

Q12

/15

Q13

/15

/15

Percent

 Questions 11‐14 are to be started in a new booklet.  The marks allocated for each question are indicated. Q14  In Questions 11 – 14, show relevant mathematical reasoning and/or calculations.  Marks may be deducted for careless or badly arranged work.  Board – approved calculators may be used.

1 | P a g e

SECTION I (One mark each) Answer each question by circling the letter for the correct alternative on this sheet. Allow about 15 minutes for this section. 1

Which expression is a correct factorisation of 4 16 (A) ( 4 8 16 (B) ( 4 (C) ( 4 4 16 4 16 (D) ( 4

2

Which expression is equal to

3

(A)

sin 3

(B)

sin 3

(C)

sin 6

(D)

sin 6

64

?

Which inequality has the same solutions as | 2 | | 3 | 5 ? (A) 6 0

3

4

5

(B)

(C)

| 2

1|

(D)

0 5

1

A Mathematics department consists of 5 female and 5 male teachers. How many committees of 3 teachers can be chosen which contain at least one female and one male? (A) 100 (B) 120 (C) 200 (D) 2500 Consider the function

(A) (B)

‐3

(C)

(D)

3

and its inverse function

. Evaluate

3 .

3 | P a g e

6 Which group of three numbers could be the roots of the polynomial equation 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. What is the probability that the two oldest members of the family sit together? (A)

! !

!

(B)

! !

!

(C)

! !

!

(D)

! ! !

8) Let x  1 be a first approximation to the root of the equation cos x  log e 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 3 6

sec 2 x dx ? Use the substitution u  tan x . tan x

0.6009

(A) (B)

0.6913

(C)

log e 3

(D)

log e 3

4 | P a g e

10

Let | |

1. What is the general solution of

(A)

(B)

(C)

(D)

2

1

2

?

, is an integer

, is an integer

, is an integer

, is an integer

Question 11 ( 15 marks) a) Evaluate lim

→

Use a NEW writing booklet.

1

2

2

2

b) Find

c) Find

3

d) Find the acute angle between the lines 3 e) The points 2 , and 2 , (i) The equation of the chord

is

2

5

9.

lie on the parabola

4

8, and

.

. (Do NOT prove this).

If the chord passes through 0, , show that 1. 1 (ii) Given the chord passes through 0, and the normals at and intersect at the point whose coordinates are , 2 . Find the equation of the locus of . 2 5 | P a g e

f)

The sketch shows the graph of the curve The area under the curve for 0 (i) Find the intercept. (ii) Find the domain and range of

2 cos

where

.

3 is shaded. 2 cos

(iii) Calculate the area of the shaded region.

1

.

2

2

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 . 1 (ii) Find + 2 b) A particle moves in a straight line and its position in metres at anytime seconds is given by 3 cos 2 4 sin 2 (i) Express the motion in terms of cos . 2 (ii) Find the particle’s greatest speed. (Answer to the nearest whole number). 2 6 | P a g e

c)

A coffee maker has the shape of a double cone 60cm high. The radii at both ends are 4cm. / . Coffee is flowing from the top cone at the rate of 5

(i) Show that radius ( in the bottom cone is

1

(ii) How fast is the level of coffee in the bottom cone rising at the instant when the coffee in this cone is 6 cm deep? 3 d) In the diagram, is tangent to both the circles at . The points and are on the larger cicles, and the line is a tangent to the smaller circle at . The line intersects the smaller circle at .

Copy or trace the diagram into your answer booklet. i) Explain why AXD  ABD  XDB ii) Explain why AXD  TAC  CAD iii) Hence show that bisects BAC

1 1 2

7 | P a g e



a) In a bag there are 6 red, 4 white and 3 black balls. Three balls are drawn simultaneously. What is the probability that these are: (i) all red. 1 (ii) exactly 2 white balls. 1 b) In the diagram, , is a point on the unit circle 1 at an angle from the positive

axis, where ‐

1 at . The points 0 ,

. The line through 0,1 and intersects the line

and 0 , 1 are on the

axis.

and ∆

(i) Using the fact that ∆

show that

are similar,

sec + tan .

2

1

1

1

(ii) Show that

= sec + tan .

(iii) Show that SNP 

 2



(iv) Hence, show that tan

 4

8 | P a g e

c) A freshly caught fish, initially at 18°C, is placed in a freezer that has a constant unknown temperature of ° . The cooling rate of the fish is proportional to the difference between the temperature of the freezer and the temperature ° , of the fish. It is known that satisfies the equation

,

where is the number of minutes after the fish is placed in the freezer. (i) Show that =

satisfies this equation.

1

3

(ii) If the temperature of the fish is 10° after 7 minutes, Show that the fish’s temperature after minutes is given by 2

T  x  (18  x)e

log 15

e

 10  x  t  18  x 

(iii) Find the temperature of the fish after 15 minutes when the initial freezer temperature is 5° . Answer to the nearest degree.

1

d) Use the principle of mathematical induction to show that 4 1 7 0 for all integers 2 .

3

9 | P a g e

Question 14 (15 marks)


a) From a point is due south of a tower, the angle of elevation of the top of the tower , is 23°. From another point , on a bearing of 120°, from the tower, the angle of elevation of is 32°. The distance is 200 metres.

i)

ii)

Copy or trace the diagram into your writing booklet, adding the given information to your diagram. 1 Hence find the height of the tower to the nearest metre. 3

b) A particle is projected horizontally from a point P, metres above , with a velocity of metres per second. The equation of the motion of the particle are = 0 and = .

(i)

If the horizontal position of the particle at time is is given by

.

, show that the vertical position 1

10 | P a g e

A canister containing a life raft is dropped from a helicopter to a stranded sailor. The helicopter is travelling at a constant velocity of 216 km/h, at a height of 120 metres above sea level, along a path that passes above the sailor. How long will the canister take to hit the water? (Answer to one decimal place). 2 (Take g 10 / ). A current is causing the sailor to drift at a speed of 3.6 km/h in the same direction as the plane is travelling. The canister is dropped from the plane when the horizontal distance from the plane to the sailor is metres. What values can take if the canister lands at most 50 metres from the stranded sailor? 4

(ii)

(iii)

c) The depth of water metres on a tidal creek is given the time being measured in hours. (i) Draw a neat sketch of

5

5

4 cos , for 0

4 .

2

4 cos , showing all important features.

If the low tide one day is at 1.00 p.m., when is the earliest time that a ship requiring 3 m of water can enter the creek? Give your answer in hours and minutes. 2

(ii)

END

11 | P a g e

Girraween High School 2015 Year 12 Trial Higher School Certificate

Mathematics Extension 1 General Instructions • Reading tjmc - 5 mjnutcs

• Working time- 2 hours

l

Total marks - 70

a11

( Section I

l 1,

)

• Write using black or blue pen Black pen is preferred

i:

10 marks

!J

• Boa.rd-approved calculators may be used

• Attempt Questions 1-10 • Allow about 15 minutes for this section

n

• A table of standard integrals is provided at the back of this paper • In Questions 11-14, show relevant mathematical reasoning and/or calculations

>I

i

IIq 11

I

( Section II ) 60 marks • Attempt Questions 11-14 • Allow about 1 hour and 45 minutes for this section

• For Section II: Questions 11- 14 MUST be returned in clearly marked separate sections. • On each page of your answers, clearly write: ~

the QUESTION being answered

~

YOURNAME

~

your Mathematics TEACHER'S NAME.

• Start each new question on a NEW PAGE. • You may ask for extra pieces of paper if you need them.

For questions 1-10, fill in the response oval corresponding to the correct answer on your Multiple choice answer sheet. 1. What is the acute angle between the lines y

= 2x -

3 and 3x + 5y -1 = 0 , to the

nearest degree? A)32°

B) 50°

D) 86°

C)82°

2. The number of different arrangements of the letters of the word REGISTER which begin and end with letter R is:

A)

6! (2!)'

B) 8! 2!

D) 8! 2!2!

C) 6! 2!

3. The middle tenn in the expansion (2x - 4)4 is A) 81

B) 216x 2

C) 384x

2

D) -96x 3

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

C)P(x) =x 3 (x-2)

D) P(x)

= -x 3 (x + 2)

5. The coordinates of the points that divides the interval joining (-7,5) and (-1,-7) externally in the ratio 1:3 are A) (-10,8)

B)(-10,11)

C) (2,8)

D)(2,11)

!'.. 8

6. Which of the following represents the exact value off cos' xdx? 0

A) re-2.J2

B) re-2.fi.

16

C) re+ 2.J2

8

16

7. Which of the following represents the derivative of y = cos-' (

I A)----

x~x'-I

B)

-1 ~x 2 -1

D) re+ 2.J2

C)

1 ~x 2 -1

8

!)

?

D)

I

x~x 2 -I

. he value o fI- + I + -I ? 8. Let a,,B,ybe the roots of 2xi+, x -4x+ 9 = 0 .What 1st

a,B

D)

,By

ay

__!__

2

9. If cose = _I and O< e
1 B) -or 3 3

C) -2

D)2

I 0. A particle is moving in Simple Harmonic Motion and its displacement, xunits, at time t seconds is given by the equation x = A cos(nt) + 2 .The period of the motion is 4re seconds and the particle is initially at rest,12 units to the right of the origin. Find the values of A and n . I A)A=IOn=, 2

B)A=I0,n=2

I C)A=l2,n=2

D) A=12,n=2

Marks

Questionll.(15 marks)-show all necessary working) 5 x-1

2

(a)Solvefor x: - > 2

(b) Find the value of e, such that

-/3 cos e - sine = 1, where

f sin2x

O:;:; e:;:; 2:r. 3

'3

. . u = sm . ,- x to evaluate (c) U se t h e sub st1tut10n

.

l+sm 2 x

dx

'4 3

Give your answer in simplest form.

(d) Use the mathematical induction to show that for all positive integers n 2:: 2, 2x1+3x2+4x3+ ............... +n(n-1)=

(e) The coefficients of x' and

x-'

n(n 2 -1) 3

in the expansion of

4

(ax-:,)'

are the same, where a and b are non-zero. Show that a+ 2b = O.

3

Question 12.(15 marks)

a) i) Find !!_cos-' dx

(x -IOI 0 J.

2

10

ii) Hence, evaluate

f-J 5

1 20x-x 2

2

x

b) Two points P(2ap,ap 2 ) and Q(2aq,aq 2 ) lie on the parabola x 2

= 4ay.

The general tangent at any point on the parabola with parameter t is given by

= tx- at 2

y

i)

(DO NOT prove this).

Find the coordinates of the point of intersection T of the tangents to the parabola at P and Q.

2

You are given that the tangents at P and Q intersect at an angle of 45°.

ii)

Show that

p - q = 1 + pq

2

By evaluating the expression x' - 4ay, or otherwise, find the locus of the

iii)

point T when the tangents at P and Q meet as described in part (ii) above.

c) The velocity v

111

Is of a particle moving in simple hannonic motion along

the x-axis is given by v2

i)

ii)

3

= 8 + 2x- x'.

Between what two points is the particle oscillating?

. What is the amplitude of the motion?

1

1

iii)

Find the acceleration of the particle in tenns ofx.

1

iv)

Find the period of oscillation.

1

Question 13.(15 marks)

a) Let ABPQC be a circle such that AB= AC, AP meets BC at X and AQ meets BC at Y as shown below. Let LEAP= a and LABC

= f3.

B

7

I

I

x

A~

\_JP I

•c

y

Q

i)

Copy the diagram in your writing booklet, marking the information given above.

1

a + fJ .

ii)

State why LAXC

iii)

Prove that L.BQP = a .

1

iv)

Prove that L.BQA

= f3 .

1

v)

Prove that the quadrilateral PQYX is cyclic.

=

1

2

b) When the polynomial P(x) is divided by x 2 -1, the remainder is 3x + 1. What is the remainder when P(x) is divided by x + 1?

2

c)

A 205 m

NOTTO SCALE A is 205 metres above the horizontal plane BPQ . AB is vertical. The angle

of elevation of A from P is 37° and the angle of elevation of A from Q is 22

°.P

°

is due East of B and Q is south 47 east from B . Calculate the

distance from P to Q, to the nearest metre.

3

d) Four people visit a town with four restaurants A, B, C and D. Each person chooses a restaurant at random.

i)

Find the probability that they all choose different restaurants.

2

ii)

Find the probability that exactly two of them choose restaurant A.

2

Question 14.(15 marks). a) The graph of y =I+ 2 sin -i (2x -1) is shown in the diagram.

y ,":\ c·

I___..........

,, \/ Determine the values of a, b and c.

Question 14 continues on the next page

2

b) 1.) Stale th e range o f y = tan -1

ii) Find dy for the function

dx

.Jx2 - 4 .

1

2

.Jx2

-4 y=tan-1 - - 2

2

c) Find the volume of the solid when the region enclosed entirely by the curves y = sin x and y = sin 2x over the domain Os x

s re 2

is rotated about the

3

x axis.

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

Vms- 1 at an angle a to the horizontal.

On its ascent, the projectile just clears one edge of the wall and on its decent it clears the other edge of the wall, as shown in the diagram. The equations of motion of the projectile are

x = Vt cos a and y = Vt sin a - g t 2 (Do not prove this.) 2

i)

V 2 sin2a Show that the horizontal range R of the projectile is - - - g

Question 14 continues on the next page

1

'

ii)

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

y=x(1-;}ana. iii)

2

The projectile is fired at 45° and the wall of the fort is !Ometres high. Show that the x coordinates of the edges of the wall are the roots of the equation x2

iv)

-

Rx+ I OR= 0.

1

If the wall of the fort is 4.5 metres thick, find the value of R.

End of examination!!!

3

STUDENT NUMBER

GOSFORD HIGH SCHOOL 2015 TRIAL HSC EXAMINATION

EXTENSION 1 MATHEMATICS General Instructions:

Total marks: - 70

• Reading time: 5minutes • Working time: 2 hours • Write using black or blue pen • Board-approved calculators may be used • A table of standard integrals is provided • In Questions 11-14, show relevant mathematical reasoning and/or calculations

Section I (10 marks) Attempt Questions 1- 10. Answer on the Multiple Choice answer sheet provided Allow about 15 minutes for this section Section II (60 marks) Attempt Questions 11-14 Start each question in a separate answer booklet Allow about 1 hour and 45 minutes for this section

MULTIPLE CHOICE

/10

QUESTION 11

/15

QUESTION 12

/15

QUESTION 13

/15

QUESTION14

/15

TOTAL

/70

Section I. Total marks (10). Attempt Questions 1-10. Allow about 15 minutes for this section. Answer on the multiple choice answer sheet provided. Select the altemative A, B, C, D that best answers the question. Fill in the response oval completely.

1. The point A has coordinates ( -1,4) and the point B has coordinates (5, -2). Find the coordinates of the point which divides AB externally in the ratio 1:3.

A. (-4, 7)

C. (7,-4)

B. (4, -7)

D. (-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.

-1:5x
D.

x :5 -1, 0 :5 x :5 2

3. A committee of three is to be chosen from a group of five men and seven women. How many different committees can be fo11ned if the committee is to contain at least one man and at least one woman?

A.

220

B.

c.

175

70

105

D.

4. If the acute angle between the lines 2x - y == 2 and kx - y == 5 is 45°, then the value of k is -1

-1

1

B. -3 or -

A. 3 or -

3

C. 3 or -

3

3

1

D. -3 or -

3

5. The acceleration of a paiiicle moving along a straight line is given by a == -ze-x where x metres is the displacement from the origin. If the velocity of the paiiicle is given by v, then

A. v 2 == ze-x

+c

B. v 2 == 2ex

+c

C. v 2 == 4e-x

+c

D.

v 2 == 4ex + c

6

·

1 f 1+9x

2

1

-

A.

27 1

c.

dx= 1

B. - tan- 1 3x

tan- 1 3x + c

9

D. tan- 1 3x + c

- tan- 1 3x + c 3

dN dt

= -0.4(N -

7. If -

100) and N

+c

= 0 when t = 0, the value of N correct to 2

decimal places when t = 20 is B. 99.97

A. 7.69

C. 100.03

D. 192.31

8. Victoria made an error proving that 32" -1 is divisible by S (where n is an integer greater than 0) using mathematical induction. Part of the proof is shown below.

Step 2: Assume the result true for n = k 32 k -1 = SP where P is an integer. 2

Hence 3 k

Line 1

= SP+ 1

To prove the result is true/or n =k+ 1 32
Line 2

LHS = 32
= 32k x3 2 -1 = (SP+l)x3 2 -1

Line 3

=72P+l-1

Line4

=72P

= S(9P) =SQ

= RHS Which line did Victoria make an enor?

(A) Line 1

(B) Line 2

(C) Line 3

(D) Line 4

9.

Which of the following is an expression for f 2cos 2 x dx. A.

1 . 2 x--sm x+c 2

C.

x - sinZx + c

D.

x + sinZx + c

10. One approximation to the solution of the equation

"+ tan·' x-x 2 =O is x =1. What

4 is another approximation to this solution using one application ofNewton's method?

A. x == 1.3805

B. x == 1.3914

c.

x == 1.4125

D. x == 1.4156

Section II. Total marks (60). 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 11 (15 marks) Use a SEPARATE writing booklet. d Find- (x sin 2x). dx

(a) (i)

(ii)

(1)

Hence or otherwise find

f x cos 2x dx .

(b) Consider the function f(x) = 2 cos- 1

(3)

x

-.

3

(i)

Evaluate f(O).

(ii)

Draw the graph of y =

(iii)

State the domain and range of y = f(x).

(1)

f (x).

(c) If ex,~ and y are the roots of 2x 3

-

6x 2

(1)

+x +2 =

(2)

0, find the value of

(i)

a+ p + y.

(1)

(ii)

(a - 1)(/3 - 1)(y - 1).

(2)

(d) Evaluate

f16 x-Jx + 3 dx

by means of the substitution u 2 = x

+ 3.

(4)

Question 12 (15 marks) Use a SEPARATE writing booklet. (a) Two points P(2ap, ap 2 ) and Q(2aq, aq 2 ) lie on the parabola x 2 = 4ay. (i)

Show that the equation of the tangent to the parabola at P is y

= px-ap

2

(2)

•

(ii)

The tangent at P and the line through Q parallel to the y axis intersect (2) at T. Find the coordinates ofT.

(iii)

Write down the coordinates of M, the midpoint of PT.

(1)

(iv)

Determine the locus of M when pq = -1.

(1)

(b) The diagram below shows a cyclic quadrilateral MNKL with MN II LK.

p NOTTO SCALE

PN is a tangent to the circle and LMNK = 2LKNP.

Copy the diagram into your writing booklet and prove that t,.LMK is isosceles. Hence, show that MK bisects LLMN.

(4)

(c) The point P(Z;-1) divides the interval joining A(-2,3) and 8(8, -7) internally the ratio m: n. Find the values of m and n. (3)

(d) Differentiate

(2)

Question 13 (15 marks) Use a SEPARATE writing booklet. dv Showthat-

d 1 = ( v dt dx 2

(a) (i)

2 ).

(2)

(ii)

A particle is moving along a straight line. At time, t seconds, its displacement, x metres, from a fixed point O on the line is such that t = x 2 - 3x + 2. Find an expression for its velocity v in terms of x. (I)

(iii)

Hence, find an expression for the particle's acceleration a in terms of x.

Express ...f3cosx - sinx in the form Rcos(x + a) where O :5 a :5

(b) (i)

Tr

2.

Hence, or otherwise, solve ...f3cosx - sinx = 1.

(ii)

(c) How many 4-letter "words" consisting of at least one vowel and at least one consonant can be made from the letters of the word EQUATION?

(2)

(2) (2)

(2)

(d) The region bounded by the curve y = cos 2x and the x-axis between x = 0 and Tr

x = - is rotated about the x-axis. Find the exact value of the volume of the solid of 4

revolution generated.

(4)

Question 14 (15 marks) Use a SEPARATE writing booklet.

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

L r(r!)

= (n + 1)! - 1.

r=l

(4)

(b) A particle moves in a straight line so that its displacement, x metres, at time t seconds, is given by x = 4 - 2sin 2 t.

(c)

(i)

Show that the motion is simple harmonic.

(2)

(ii)

Find the period and the centre of the motion.

(2)

(iii)

Show that the velocity v of the particle in te1ms of its displacement can be expressed as v 2 = 4(-8 + 6x -x 2 ). (2)

(i) Show that the range offligbt of a projectile fired at an angle of a to the horizontal with velocity v is

v 2 sin2a

B

where g is the acceleration due to

gravity.

(2)

The equations describing the trajectory of the projectile are:

x

= vt cos a, y = vt sm. a -

-12 gt 2 .

(You ar·e NOT required to prove these equations)

(ii) A cannon fires a shell at an angle of 45° to the horizontal and strikes a point SOm beyond its target. When fired with the same velocity at an angle of 30° it strikes a point 20m in front of the target. Calculate the horizontal distance between the cannon and the target co11·ect to 2 decimal places. (3)

END OF PAPER

2015 Assessment Task 4 Trial HSC Examination

Mathematics Extension 1 Examiners - Mrs D. Crancher, Mrs S. Gutesa, Mr S. Faulds, Ms P. Biczo

General Instructions

Total marks (70)

II

!section

o Reading Time - 5 minutes o

Total marks (10)

Working Time - 2 hours

o

o Write using a blue or black pen.

Attempt Questions 1 - IO

o

Board approved calculators and mathematical templates and instruments maybe used.

o Answer on the Multiple Choice answer sheet provided on the last page of this question booklet.

o

Show all necessary working in Questions 11,12,13 and 14

o Allow about 15 minutes for this section

o This examination booklet consists of 13 pages including a standard integral page and a multiple choice answer sheet.

lsection

nl

Total marks (60) o

Attempt questions 11 to 14

o

Answer each question in the writing booklets provided.

o

Start a new booklet for each question with your student name and question number at the top of the page.

o

All necessary working should be shown for every question

o Allow about I hour 45 minutes for this section

Student N a m e : - - - - - - - - - - - - - - - - - - Teacher: - - - - - - - - - - - - - - - - - - -

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

2

The solution to the inequality x( 2- x )( x + 1) ::0: 0 is (A)

x S: -2 or OS: x S: 1

(C)

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

(B) (D)

-2 S: x S: 0 or x ::0: I -1 S: x S: 0 or x ;:,: 2

A committee of3 men and 3 women is to be fonned from a group of8 men and 6 women. How many ways can this be done? (A)

48

(B)

1120

(C)

40320

(D)

3003

3

c NOTTO SCALE

D

In the diagram, AB is a tangent to the circle, BC= 6cm and CD What is the length of AB?

4

(A)

6Ji cm

(B)

6./3 cm

(C)

72cm

(D)

108cm

= 12cm.

What is the equation of the tangent at the point ( 4 p, 2 p 2 ) on the parabola x (A)

y=px-p'

(B)

x+py=2p+p 3

(C)

x+ PY =4p+ p'

(D)

y

2

= 8 y?

= px-2p 2

Hur/stone Agricultural High Schoo/ 2015 Trial HSC Mathe111atics Extension 1 Exa11li11atio11

Page 3

5

6

7

What is the acute angle to the nearest degree that the line 2x-3 y + 5 = 0 makes with the y-axis?

(A)

27°

(B)

34°

(C)

56°

(D)

63°

Which of the following statements is FALSE.

(A)

cos- 1(-8) = -cos- 1

(C)

tan- 1 (-8) =

(C)

9

tan-I fJ

(B)

sin-I (-8) = -sin-I fJ

(D)

cos- 1 (-8) = n - cos- 1 8

(B)

-3 ( 3x-' -I )' +c

(D)

-2x ( 3x 2 -I )5 +c

The ptimitive of 2x( 3x 2 - I)' ts:

(A)

8

e

1 ( 3x·' -1 )' +c

15

2x

5

2

s

(3x -l) +c

5

15

The equation(s) of the horizontal asymptote(s) to the curve y = (A)

y=O

(B)

x=±I

(C)

y=l

(D)

x=l only

x:x- +-11 are

What are the coordinates of the point that divides the interval joining the points A(2, 2) and B(4,5) externally in the ratio 2:3? (A)

(-2,-4)

(B)

(-2,11)

(C)

(8,-4)

(D)

(8,11)

Hurlstoue Agricultural High School 2015 Trial HSC Mathe111atics Extension I Exa111inatio11

Page 4

10

Which of the following equations is shown in the sketch below

y

2TI 3

-, 3n --

-n:

3n 2

IT

--2

l{

--

'

.}

2TI .-

3

-TI

I

T

(A)

y

. ) = cos -I ( smx

(B)

y

= sin- 1 (cosx)

(C)

y

= sin -i (x) + sin(x)

(D)

y

= cos· 1 (x) + cos(x)

- End of Section I -

Hur/stone Agricultural High School 2015 Trial HSC Mathentlllics Extension I Exa111inatio11

Page 5

Section II 60 marks Attempt Questions 11 to 14 Allow about 1 hour 45 minutes for this section Answer each question in the appropriate writing booklet. All necessary working should be shown in every question.

Marks

Question 11 (15 marks) >l?

3

3

(a)

Solve the inequality

(b)

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

x(2x-1)

2

X (-1,3)to Y (4,8)?

(c)

(i)

Show that the curves y at the point (-2,

(ii)

(d)

(i)

(ii)

(e)

=x 3 -

x and y

=x -

x' intersect

-6)

Dete1111ine the acute angle between the curves y

1

=x3 -

x

and y = x - x' at the point of intersection, to the nearest minute.

3

A class of25 students is to be divided into four groups consisting of 3, 4, 5 and 6 students. How many ways can this are done? Leave your answer in unsimplified fonn.

2

Assume that the four groups have been chosen. How many ways can the 25 students be arranged around a circular table if the students in each group are to be seated together? Leave your answer in unsimplified fonn.

2

Five different fair dice are thrown together. What is the probability the five scores are all different?

Hur/stone Agricultural High Schoo/ 2015 Trial HSC Mathe1natics Extension 1 Exa111i11utio11

2

Page 6

Marks


(a)

Consider the function

l(x) = (x - I)'

= l(x).

(i)

Sketch y

(ii)

Explain why

l(x)

1

does not have an inverse function for

all x in its domain. (iii)

1

State a domain and range for which function

l(x)

has an inverse

1-' (x).

x;:,: I

l

1-' (x).

(iv)

For

(v)

Hence , on a new set of axes, sketch the graph of y

f

find the equation of the function

(b)

. d F111

(c)

Find the exact value of tan( 2 tan_,

(d)

Find the general solution to 2 cos x

dx

= 1-' (x).

1

2

.J9-4x 2

ij

2

-Ji.

Leave your answer in terms of re .

(e)

2

2

Differentiate (with respect to x)

and hence find the exact value of JJ tan-'.:'.'.

I

3 Ix x +9 2

3

0

Hur/stone Agricultural High School 2015 Trial HSC Mathe111atics Extension I Exa111inatio11

Page 7

Marks


(a)

The points P ( 2ap, ap') and Q (2aq, aq 1 ) lie on the parabola x 2

= 4ay

such that OP is perpendicular to OQ. y

P(2ap, a/)

x

0

(i)

Prove that pq = -4.

(ii)

R is the point such that OPRQ is a rectangle.

2

Explain why the co-ordinates of R are ( 2a (p + q), a ( p' + q (iii)

(b)

(c)

1

)).

Show that the locus of Risa parabola.

2

2

Find by division of polynomials, the remainder when x' + 4 is divided by x-3.

1

a, f3 and y aretherootsoftheequation x3 -3x'-6x-1=0. ' p-+r. a' ' . d a-+ F111

2

Question 13 continued next page ....

Hurlsto11e Agricultural High School 2015 Trial HSC Mathe111atics Extension I Exa11d11atio11

Page 8

Question 13 continued ....

(d)

(i)

(ii)

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 nature.

.f(x)=sin 2 x-x+l has a zero near x, =

3

ir.

2

Use one application of Newton's method to obtain another approximation x 2 , to this zero.

2

(iii)

rr/2

The graph off ( x) = sin 2 x-x + 1 is shown in the vicinity of x =

ir.

2

By using this diagram, detennine if x 2 is a better approximation than x1 to the real root of the equation. You must justify your answer.

Hur/stone Agricultural High School 2015 Trial HSC Mathe111atics Extension I Exa111i11ation

1

Page 9

Marks

Question 14 (15 marks) (a)

F

E

In the diagram above, FG is a common tangent and FBIIGD.

(b)

(i)

Prove that FAIIGC.

2

(ii)

Prove that BCGF is a cyclic quadrilateral.

2

(i)

Find:

_:'.._ ( x sin 3x) dx

2

,T

f 6

(ii)

Hence, evaluate:

xcos3xdx

3

0

(c)

Use the substitution y =

f (d)

Fx to find

dx ~x(l-x)

Use mathematical induction to prove the inequality:

3

3

n ! > 2", for all positive integral values of n;:: 4

- End of Section II -

Hur/stone Agricultural High School 2015 Trial HSC Mathe111atics Extension I Exa111i11atio11

Page 10

Year 12 Mathematics Extension I T1ial 2015 Question No. 11

Solutions and Marking Guidelines

Outcon1es Addressed in this Question PE3 Solves problen1s involving pem1utations and co1nbinations, inequalities and polynon1ials. HS Ann lies annropriate techniaues from the studv of geo1netry.

Outcome

Markin<' Guidelines

Solutions (a)

3 marks Co1Tect solution 2 marks

3

PE3

--->! x(2x I)

Substantial progress to\vards

Multiply by the square of the denominator

con-ect solution 1 n1a1·k

3x(2x - 1) > x 2 (2x -1 )2

Saine progress to,vards

coJTect solution

2

2

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

2

!I

5

.,

;\ I\

25

(b) A(-1,3) B(4,8) P(l4,18)

H5

Ratio AB: BP, m: 11 assume 1: k .Xo

=

n1x,

-

+ nx1

111 +n

](4)+/c(-1) l+k 14+ 14/c = 4-k

2 111arks Conect solution. 1 mark Substantial progress to,vnrds correct solution.

14

-2

k=3 -2

l:k=l:3

:. Ratio is -3:2 (c)(i) Substitute x = -2 into y = x 3 - x

HS

RHS = (-2) 3 -(-2) =-6

: . Satisfies the curve. Substitutex= -2 into y = x-x' RHS = -2-(-2) 2 =-6

:. Satisfies the curve. :. (-2, -6) is the point of intersection.

1 mark

CotTect solution

(c)(ii)

For y

= x 3 -x, y'=3x2 -1

when x = -2, 111, = 3(-2)" -1 = 11 Fory=x-x

2

,

y'=l-2x

whenx=-2,1112 =1-2(-2)=5 tane

1111 -1112

3 marks C01Tcct solution \Vith co1Tect rounding

. 2 marks Substantial progress to,vards co1Tect solution

1 n1ark Some progress to,vards correct solution

l+m1m2

11-5 , tan e = ,____ I+ (11)(5) 6 ) 56 :. e = 6°7' (to the nearest minute)

:.B=tan- 1 (

PE3

25! or----3!4!5!6!7!

Substantial progress towards correct solution

(d) (ii)

2 n1arks

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

Co1Tect solution.

PE3

(e)

HS

2 marks CotTect solution l mark

P(E)

6x5x4x3x2 6' 5 54

1 n1ark Substantial progress towards cmTcct solution.

2 n1arks Correct solution. 1 n1ark Substantial progress to\l1anls correct solution.

Task 4 Trial HSC Mathematics Extension I Solutions and Marking Guidelines Outcomes Addressed in this Ouestion uses the relationship between functions, inverse functions and their derivatives

Year 12 2015 Question No. 12 HE4

Outcome

HE4

Markin!! Guidelines

Solutions Question 12 a) (i)

\

41 I Mark for correct sketch

;

HE4

I

'

(ii) 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.

HE4

(iii) Domain: x;:: I Range: y ;:: 0

HE4

(iv) x=(y-1)

I Mark for correct explanation

l Mark for c01Tect answer

2 Marks for complete correct solution

1

.J; = v-1

l Mark for paitial correct solution

y=l+.J; :.r'(x)=I+.J;

HE4

(v) I Mark for c01Tect sketch

6

5 4 3 2 1 • 0 -4 .3 -2 ·1 -1

I' ,

2

3

4

5

6

7

8

-2 I

HE4

(b)

I

dx


../9-4x

-)~ +;"'[f}c

I Mark for paitial correct solution

or

I sm .

2

-•(2x) +C 3

HE4

(c)

3 :. tanB=4


now,

tan( 2 tan_,

i)

= tan(2B)

1 Mark for partial correct solution

2tanB l-tan 2 B

2(i) 1-(i )'

=--'--'--

24

=-

7

HE4

(d)


-./3 -./3 cosx=2cosx =

1 Mark for pmiial correct solution

2

:.x=2111r±cos-I --./3 2

Jr

.

.

= 2111r±-, wheren1sanymteger.

6

HE4

(e) d

-1

' x -

-:(tan -) d.,

3

=

-l

-

x

3

l


I 2(1an -) ----:,

3 [ I+~ 9

3 ,) -( _, x)( 3 9+xtan-' 5J

2 Mark for substantial working that could lead to a correct solution with only one error.

=2 tan -

=6 9+x2

(

l Mark for correctly

no,v,

differentiating !!:_(tan-'

Jj[ tan_,xJ l fJj (tan_,xJ --? dt=6 ------}- d,· 9+.x f 9+x6 0

0

=i [(tanI [( tan

6

1

;r]~

_,Jj)' ( _,o)' ) 3 3

=i ((i)'-oJ 216

-tan

d,

!:)3

2

Year 12 Trial Higher School Certificate Extension 1 Mathematics Solutions and Marking Guidelines Question No. 13

Examination 2015

Outcomes Addressed in this Question PE3 solves problems involving polynomials and parametric representations PES determines derivatives which require the application of more than one rule of differentiation H6 uses the derivative to determine the features of the graph of a function HE7 evaluates mathematical solutions to problems and communicates them in an appropriate form. Markin!! Guidelines Outcome Solutions PE3

(a) (i) OP

= -1.

2 marks : correct solution I mark : significant progress towards COITect solution

ap + aq, ap' ; aq' )

2 marks : correct solution 1 mark : significant progress towards co1Tect solution

OQ, :. mOP xm OQ

_l_ 2

ap aq' :.-x-=-l 2ap 2aq :. p xi=-1

2 2 :.pq=-4.

(ii) Midpoint PQ = PE3

(

As the diagonals bisect one another in a rectangle, OR will also have the same midpoint as PQ.

If 0(0, 0), R has midpoint ( ap +aq, ap' ;aq'), then 2

Ris (2a(p+q), a(p +q PE3 (iii)

At

2

)).

R, 1::::;P + q) 2

y=a(p +q

2

2 marks : c01Tect

~;]

[3]

)

From [2], x 2 =4a'(p+q)' :. x

2

solution I mark : significant progress towards c01Tect solution

=4a 2 (p 2 +q' +2pq)

Substituting [1] and [3],

2

x =4a'(; + 2x-4),

:. x 2 = 4a (y-8a), which is a concave up parabola with

ve1iex ( 0, 8a). (b)

PE3

x+3 x-3)x' + Ox+ 4 x

2

-3x

Remainder is 13.

3x+4 3x-9 13 (c) From x

3

2 marks : c01Tect solution I mark : significant progress towards c01Tect solution

2

3x - 6x- l = 0, -b c a+/J+r=-=3, afJ+/Jr+aA.=-=-6. a a -

2

2

2

(a+ /3 + r ) = a + /3 + r' + 2 ( afJ +/Jr+ ar)

2 marks : c01Tect solution I mark : significant progress towards correct solution

:. a'+ /3 2 + r' =(a+ /J + r)' -2(a/J + /Jr+ar) 2

:. a'+ /3 2 + r' =(3) -2(-6) = 21. H6, PE5

(d) (i) f(x) = sin 2 x-x+ I f'(x) = 2sinxcosx-l :. f'(x)

3 marks : coITect solution 2 marks : substantial progress towards COITect solution I mark: significant progress towards correct solution

= sin2x-l

f'(x) = 0 for stationary points. Solving sin 2x- l = 0, sin2x = I For O:,; x:,; n-, 0 s 2x s 2n-.

. 2 7r o vmg, x = - , SI

:.

Testing x =

f' (x) =sin 2x-l,

2

for

7r,

4

f'(x)

7r

.

7r

-

-

3

~

J5-2

0

J5-2

.

4

"4

"6

-

x

.

one stationary pomt, at x = -

-2

2

J5-2 is negative, there is a horizontal point of

As

2

. fl . 111 ex10n at x

7r = -.

4

2 marks : c01Tect solution l mark : significant progress towards c01Tect solution

. ' n- n- I n- =sm----+ =2-(11.. ) f (n-)

PE5, HE7

.

2

f'(; )=

2

2

2

sinn--1 =-1. 7r

Newton's method: x, -

7r

=-

:. x, =

2-- __ 2

2 2

-1

(iii)

I mark : c01Tect explanation

HE7 I I I

I n/2

real root ~x,

\~

x, is where the tangent at " meets the x axis. 2 This is closer to the real root, : . a better approximation.

Year 12 Mathematics Extension 1 Trial Examination 2015 Question No. 14 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 derivatives HE6 determines integrals by reduction to a standard form throu!!h a !!iven substitution Outcome Solutions Markin!! Guidelines PE3 (a)(i) Let LDGH= a 2 marks .. LBFG = a (corresponding angles, FBIIGD) Correct solution \Vith full reasoning. 1 mark No,v, LGCD = a (angle bet,veen a chord and Substantial progress to\vards a correct tangent is equal to the angle in the alternate segment)

solution.

Similarly, LFAB = a Since LGCD = LFAB = a, FAIIGC (corresponding angles are equal)

(ii) Since LGCD PE3 also,

=

a

(shown above) (angles on a straight line (sho,vn above)

LGCB = 180°- a LBFG = a

L.GCB + L.BFG = 180° - a+ a = 180°

2 marks Correct solution ,vith full reasoning. 1 mark Substantial progress to,vards a correct solution.

:. BCGF is a cyelie quadrilateral (opposite angles supplementary)

HE4

(b) (i)

!!... xsin 3x = x.3cos 3x+ sin 3x.l dx

= 3xcos3x+sin3x

2 marks Correct application of product ntlc to find correct answer. 1 mark Demonstrates kno,vlcdge of product rulc in making substantial progress to a full solution.

(ii)

HE4

If ~xsin3x = 3xcos3x+ sin3x

3 marks Correct solution. 2 marks Correctly finds the required primitive function. t n1ark Substantial progress to\vards finding the required pri1nitive function.

dx

then 3xcos3x=~xsin3x-sin3x

dx I d . I . xcos.x=--xs1n 3 3x--s1n 3x 3dx 3 Integrating both sides, -n6

-rr6

f

x cos 3:rdx = .!. 3

O

-6n

f

f

O

O

!.!...x sin 3xdx- .!. dx 3

sin 3xdx

= .!.[xsin 3x+ .!.cos3x]~ 3 3 0 =.!.[(!!_sin!!.-+ .!.cos!!.-)-( OsinO + .!.cosO )] 3 6 2 3 2 3

=1[(~-o )-( o+i)] =

-1.(!:_-1.) 3 6 3 n:-2 18

=--

(c)

HE6

3 rnarks

y=J;

Let

Correct solution.

:.x=y"

2 marks

dx = 2v dy .

Uses the given substitution correctly and makes substantial progress to,vards a correct solution.

dx=2ydy

I I I.v~6-y') I

1 mark

dx

~x(l-x) -

2ydy

Uses the given substitution correctly.

~/(!-/)

2vdv

=

dv

=2

J(1~y')

= 2sin- 1 y+ c but

I

y=J; 2sin- 1 J;+c

d,

. . ~x(l-x)

(d)

HE2

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

Prove true for 11 = 4 LHS=4!

RHS= 2'

=24

=16

24 > 16

:. True for n = 4

3 marks Correct solution. 2 marks Prove the relationship is true for n=4 and 1nakes substantial progress to,vards a correct solution.

1 mark Correctly proves the relationship true for n=4.

Assmne true for n = k

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') Now, since k > 4, (k-1) > 0

k!>O :.(k-1).k!>O

Also, k!-2k > O. from the assmnption

Hence. (k-l}k!+2(k!-2')>0 (k+l)!-2''' >0 :.(k+1}!>2'''

..

By the process ofn1athematical induction,

11! > 2" is t1ue for all positive integral values of 11;?. 4

Section 1 (10 marks) Attempt questions 1 ‐10. Use the multiple‐choice answer sheet provided. 1.

Evaluate lim x 0

(A)

3

3 sin 7 x 5x

(B)

2.

21 5

(C)

(A)

(B)

(C)

(D)

For what values of x is

(D)

15 7

0

x4  6? x 1

3.

The interval joining the points A 3,2 and B 9, y  is divided externally in the ratio 5:3 by the point P x,13. What are the values of x and y ?

(A)

x  27, y  22

(B)

x  18, y  4

(C)

x  6, y  12

(D)

x  27, y  4

4.

A circle with centre O has a tangent TU, diameter QT, STU = 25o and RPS= 22o .

What is the size of RTQ ? (A) (B) (C) (D)

22° 25° 43° 47°

5.

For the polynomial equation 6  4 x  10 x 2  8 x 3  0 , the sum of its roots, when divided by the product of its roots would be:

(A)

5 3

(B)

4 3

(C)

1 2

(D)

5 4

JRAHS 2015 TRIAL EXT 1 MATHEMATICS

Page 1

6. 7.

A particle moves such that when it is x metres from the origin its acceleration is given by 1 a   e  x . What is its velocity when x  3 , given that v  1 when x  0 ? 2 (A) 0.050 ms 1 (B) 0.070 ms 1 (C) 0.158 ms 1 (D) 0.223 ms 1

Which of the following is the correct expression for



dx 36  x 2

?

x c 6

(A) cos -1

(B) cos -1 6x  c

(C) sin -1

(D) sin -1 6x  c

8.

Eden, Toby and four friends arrange themselves at random in a circle. What is the probability that Eden and Toby are not together?

(A)

x c 6

1 120

(B)

2 5

(C)

3 5

(D)

119 120

9.

If t  tan

(A)

 2

which of the following expressions is equivalent to 4 sin   3 cos   5 ?

2t  2 1 t2 2

(B)

t  42 1 t2

2t  2 1 t2 2

(C)

(D)

t  42 1 t2

10.

An expression for the general solution to the trigonometric equation tan 3 x   3 where n is any integer is:

(A)

x

n   3 9

(B)

x

n   3 3

(C)

x

n   3 3

(D)

x

n 2  3 9


Page 2

Section II (60 marks) Attempt all questions from 11‐14. Answer each question on a separate page.

Question 11 (15 marks) (a)

The number of animals in a local farm who will be infested with a virus adheres to the equation p where n = the number of animals infested by the virus n 1  Ce  kt 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 of 200 is infested with the virus. After one month the number of animals infested with the virus increases to 5.

(i)

(ii) Show that k = 1.63 (to 3 significant figures) (iii) How many animals can the farmer expect to be infested after 3 months.

(b)

(i)

(ii)

Show that after t months, n 

Find

(d)

1

1 2 2

d  2x  x   tan 1    2 dx  4  x  2  2

Hence evaluate  0

(c)

200 1  199e  kt

dx

4  x 

2 2

3

A spherical metal ball is being heated such that the volume increases at a rate of 5 mm3 / min . At what rate is the surface area increasing when the radius is 3 mm . Find an expression for 

3

e3 x dx using the substitution u  1  e x . x 1 e

3

Question 12 (15 marks) Start a new page

(a)

A group of 15 students from a local school is selected for training in soccer to represent the school at grade sport. However only a team of 11 players is to be chosen for the Wednesday game. The probability that a player will not be available to play on Wednesday due to injury or other commitments is 0.14.

(i)

Find the probability that 3 students will not be available for the Wednesday grade sport in soccer. Answer to 3 decimal places.

2

(ii)

Write the numerical expression for the probability that the team will be unable to make up a team of all fit 11 players. You do not have to simplify the answer.

2


Page 3









(b)

Let P 2ap, ap 2 and Q 2aq, aq 2 be two points on the parabola x 2  4ay. The secant PQ passes through the point Aa,0 , and the tangents at P and Q meet at R.

(i)

Show that p  q  2 pq.

2

(ii)

Find the coordinates of R in terms of p and q.

3

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

1

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

1

(c)

Use mathematical induction to prove that for all integers n  3,

4

2  2  2  2  2  . 1  1  1  ...........1     n  nn  1  3  4  5 

Question 13 (15 marks) Start a new page (a) (i) Using the auxiliary angle method express 3 sin 2t  2 cos 2t in the form r sin2t   . A particle moves horizontally in a straight line so that its position x from a fixed point at time t is given by: x  3 sin 2t  2 cos 2t  2 Displacement is measured in metres and time in hours. (ii) Find an equation to represent the acceleration of this particle and prove that it is moving in simple harmonic motion. (iii) Given that the particle is at the origin at noon, between what times will the particle be more than one metre to the right of the origin for the first time (Let the time at t = 0 be noon). Give your times correct to the nearest minute. (b) (c)

Consider the function y 

2

2

2

(i) (ii) (iii)

Find the domain and range of the function. Sketch the graph of the function showing clearly the coordinates of the end points. 1 The region in the first quadrant bounded by the curve y  cos 1  x  1 and the coordinate 2 axes is rotated about the y axis. Find the volume of the solid of revolution, giving your answer in simplest exact form.  x What is the exact value of the definite integral  3 sin 2 dx ? 4 0


1 cos 1  x  1. 2

Page 4

2 1 3

3

Question 14 (15 marks) Start a new page (a)

In a BMX dirt bike competition the take‐off point O for each competitor was located at the top of  the downslope. The angle between the downslope and the horizontal is . The biker takes off 3  from O with velocity V m / s at an angle  to the horizontal, where 0    . The biker lands 2 on the downslope at some point Q, a distance D metres from O.

The flight path of the biker is given by 1 x  Vt cos  and y   gt 2  Vt sin  2 where t is the time in seconds after take‐off. (DO NOT PROVE THIS)

(i)

(ii)

Show that the Cartesian equation of the flight path of the biker is given by gx 2 y  x tan   2 sec2  . 2V

3

Show that V2 D  4 cos  3 cos   sin  . g (iii) Show that dD V2 4 cos 2  3 sin 2 . d g (iv) Show that D has a maximum value and find the value of  for which this occurs. (i) Considering the identity 1  x n 1  x n  1  x 2 n , where n is a positive integer,







(b)

2

2



3 2

show that for integer values of r , 2r

  1 k 0

(ii)

k n

Ck nC2 r  k   1 nCr provided 0 r

Hence show that   1

Ck nC2 r  k 

k n

k 0

(iii)

.

r

6

Hence evaluate   1 k 0

k



12





1  1r nCr 1 nCr for 0 2

2

.

1



2

Ck as a basic numeral.

END OF THE EXAMINATION JRAHS 2015 TRIAL EXT 1 MATHEMATICS

Page 5

NORTHERN BEACHES SECONDARY COLLEGE

MANLY SELECTIVE CAMPUS HIGHER SCHOOL CERTIFICATE Trial Examination 2015

Mathematics Extension 1 General Instructions       

Reading time – 5 minutes Working time – 2 hours Write using black or blue pen Write your Student Number at the top of each page Answer Section I- Multiple Choice on Answer Sheet provided Answer Section II – Free Response in a separate booklet for each question. Board approved calculators and templates may be used.

Section I Multiple Choice  

10 marks Attempt all questions

Section II – Free Response   

60 marks Each question is of equal value All necessary working should be shown in every question.

Weighting: 40%

Manly Selective Campus 2015 HSC Mathematics Extension 1 Trial

Multiple Choice: Answer questions on provided answer sheet. Q1.

The diagram shows a circle with centre O. The line PT is tangent to the circle at the point T. ∠TOP = 4x° and ∠TPO = x°.

O

x°

4x°

P

T 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

Q3.

The point P divides the interval AB in the ratio 3:7. In what external ratio does the A divide the interval PB (A)

3:10

(B)

3:4

(C)

7:3

(D)

10:3

Page 2 of 10

Manly Selective Campus 2015 HSC Mathematics Extension 1 Trial Q4.

What is the obtuse angle between lines (A)

15o

(B)

75o

(C)

105o

(D)

165o

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 and then arranged in a line so that the two oldest people in the selected group are at 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 (A)

1

(B)

2

(C)

x+2

(D)

x+1

is equal to

Page 3 of 10


The power of x in the 7th term of the expansion of

Q8. (A)

3

(B)

-3

(C)

5

(D)

-5

Q9.

is

The velocity of a particle is given by the equation is initially located at the origin, what displacement at t = 3? (A)

3

(B)

8

(C)

15

(D)

16

Q10.

The diagram show the graph of a cubic function

Which is a possible equation of this function? (A) (B) (C) (D) End of Multiple Choice Page 4 of 10

.

. If the particle


Question 11: Start A New Booklet

(a)

Evaluate

(b)

(i)

Verify that (αβ+αγ+ βγ)² = α²β²+α²γ²+ β²γ² + 2αβγ(α+β+γ).

(ii)

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

(c)

(d)

15 Marks

.

(3)

(1) ,

evaluate

(3)

(i)

Determine the vertical asymptotes for

(2)

(ii)

Hence sketch the curve

(2)

Find the general solution of the equation (4)

Page 5 of 10


Question 12 Start A New Booklet

(a)

Use the substitution

(b)

A particle moves with acceleration

15 Marks

to show

(3)

. Initially, the particle is one metre

to the right of the origin and its velocity is 4m/s. Find the displacement of the particle when it is at rest.

(c)

(3)

ABCD is a cyclic quadrilateral. The tangents from Q touch the circle at A and B. The diagonal DB is parallel to the tangent AQ, and QA produced intersects CD produced at P. Let < QAB = α.

C NOT TO SCALE D

B α

P

(d)

A

Q

(i)

Prove that ΔBAD is isosceles, giving reasons.

(2)

(ii)

Find < DCB in terms of α, stating reasons.

(1)

(iii)

Show that P,C, B and Q are concyclic points

(2)

Show that x = 1.8 is a reasonable approximation for the x value of the point of intersection of y = 2sin x and y = x in the domain 0.78 < x < 2.35.

(1)

Use Newton’s method with one application to find a better approximation to the x value of this point on intersection.

(3)

(i)

(ii)

Page 6 of 10


Question 13 Start A New Booklet (a)

15 marks

The diagram shows the parabola

The tangent to the parabola at P cuts the y-axis at N.

cuts the x-axis at T and the normal at

The equation to the tangent is given by (i)

Show the coordinates of N are

(2)

(ii)

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

(3)

Use mathematical induction to prove that ( n + 1) + n − 1 is divisible by 2 2

(b)

for all integers n ≥ 1 .

(c)

(3)

A school band is to be formed with a brass section containing 8 students and a percussion section containing 4 students. (i)

(ii)

In how many ways can the band be formed if 12 students audition for the brass section and 10 students audition for the percussion section?

(1)

In how many ways can the band be formed if it is certain that Maria will be successful for the brass audition and Marcus will be successful for the percussion audition?

(2)

Question 13 continues on next page

Page 7 of 10


Question 13 continued (d)

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

for some

constants a > 0 , b > 0. The initial population size is 20 and the limiting population size is 100.

(i)

Show that

(2)

(ii)

Find the values of a and b.

(2)

Page 8 of 10


Question 14

(a)

15 marks

Warehouse A has 100 computers and the probability that of selecting a computer which is defective is 0.02. Warehouse B has 100 computers, two of which only, are defective. Joe buys three computers from Warehouse A and three computers from Warehouse B. What is the probability that exactly one of the computers he has bought is defective?

(b)

(3)

Two towers T1 and T2 have heights h metres and 2h metres respectively. The second tower is due south of the first tower. The bearing of tower T1 from a surveyor is 292°. The bearing of the tower T2 from the surveyor is 232°. The angle of elevation from the surveyor to the top of tower T1 is 30° while the angle of elevation from the surveyor to the top of tower T2 is 60°.

Show that the distance d between the two towers is given by d =

Question 14 continues on next page

Page 9 of 10

21h 3 metres.

(3)


Question 14 continued. (c)

A ball is projected vertically from the ground with a speed of 49 m/s. The height y of −4.9t 2 + 49t. (Do NOT show this.) the ball at time t is given by y = At the same time, a second ball is projected from the ground into the air with an angle of projection θ. Its horizontal displacement is given by x = 98t cos θ and its −4.9t 2 + 98t sin θ . (Do NOT show this.) height is given by y = (i)

Find the maximum height of the ball that was projected vertically.

(2)

(ii)

Find the value of θ at which the second ball should be projected if it is to hit the first ball when the first ball reaches its maximum height.

(2)

Find the horizontal distance between the two balls when they are first projected into the air. Give your answer in exact form.

(1)

(iii)

(e)

Consider the binomial expansion

Show that, if n is even:

(4)

End of Examination

Page 10 of 10

2015 MSC HSC X1 Trial Solutions

Q1

B

Q2

D

Q3

A

Q4

C

Page 1 of 14


Q5

C

Q6

C

Q7

D

Q8

B

Page 2 of 14


Q9

Q10

C

Cubic of form

C

Page 3 of 14

2015 MSC HSC X1 Trial Solutions Q11

3 marks – correct solution

2 marks - correct integrand

1 mark – simplification to

1 mark – expansion demonstrated

Page 4 of 14


3 marks – correct solution 2 marks – correct rearrangement of initial term 1 mark – correct values for

2 marks – correct answer 1 mark – equating denominator to zero

2 marks - shape - x-intercept -

Page 5 of 14


4 marks - correct solution 3 marks -correct solution for without stating reason for excluding cot ϴ 2 marks – fully factorised and use of trig identity 1 mark –first factorisation

Page 6 of 14

2015 MSC HSC X1 Trial Solutions Q12

3 marks – correct solution 2 marks – final integral simplified to cos2ϴ 1 mark – correct initial substitution including limits and dϴ

Page 7 of 14


3 marks – correct solution 2 marks – attaining constant correctly 1 mark – use of

2 correct solution 1 mark – one correct use of geometrical principle.

c-i

c-ii

1 – correct solution

Page 8 of 14


2 marks – correct solution c-iii

1 mark – determination of size of

1mark – correct solution

d-i The solution is close to zero therefore reasonable approximation.

3 marks – correct solution 2 marks – correct expression for xa

d-ii

1 mark – incorrect substitution into initial formula

Page 9 of 14


Q13 2 marks-correct solution showing all steps 1 mark- correct equation of normal

a)i)

3 marks- correct equation showing all steps 2 marks-correct values for midpoint 1 mark- correct x value for T

ii)

Page 10 of 14

2015 MSC HSC X1 Trial Solutions 3 marks- correct solution showing all steps 2 marks-partial correct with insertion of assumption into S(k+1)

b)

1 mark- correct for n=1

1 mark- correct solution

c)i)

2 marks- correct solution

ii)

1 mark- correct selections for brass/percussion only 2 marks –correct solution 1 mark- correct initial expression for dN/dt

d)i)

Page 11 of 14

2015 MSC HSC X1 Trial Solutions 2 marks- correct solution 1 mark- only one correct value

ii)

a

Question 14 P(exactly one computer defective) =P(1 defective from A, 0 from B) +P(0 from A, 1 from B)

3 marks: correct solution 2 marks: obtaining a correct binomial prob. 1 mark: 1st line – showing understanding of options required.

b

3 marks: correct solution 2 marks:correct expressions for triangles, attempting to use them in cos rule 1 mark: correct expressions for triangles

Page 12 of 14


i

2 marks: correct solution 1 mark: finding solving

and

ii 2 marks: correct solution 1 mark: substituting t=5 and y=122.5 into2nd equation

iii 1 mark: correct solution

Page 13 of 14

2015 MSC HSC X1 Trial Solutions d

4 marks: correct solution incl. explanation of for n even 3 marks: correct solution without justifying sign of last term 2 marks: differentiating twice and substituting either x=1 or x=-1 1 mark: differentiating twice

Page 14 of 14

BOSTES Number: ____________________________ CLASS (Please circle): 12M1 12M2 12M3 12M4 12M5

NORMANHURST BOYS HIGH SCHOOL N E W

S O U T H W A L E S

2015 H IGH ER SCHOOL C ERTIFICATE T R IA L E X A M IN A T I O N

Mathematics Extension 1 General 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 Begin each question on a separate writing booklet

Total marks - 70 Section I

Pages 2-4

10 marks  Attempt Questions 1-10  Answer on the Multiple Choice answer sheet provided  Allow about 15 minutes for this section Section II

Pages 5-9

60 marks  Attempt Questions 11-14  Allow about 1 hour 45 minutes for this section

Students are advised that this is a school-based examination only and cannot in any way guarantee the content or format of future Higher School Certificate Examinations.

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 y NOT TO SCALE

y = f '( x)

-4

-2

2

4

x

The diagram above represents a sketch of the gradient function of the curve y  f ( x ) . Which of the following is a true statement? The curve y  f ( x ) has (A)

a minimum turning point occurs at x  4

(B) a horizontal point of inflexion occurs at x  2 (C) a horizontal point of inflexion occurs at x  4 (D) a maximum turning point occurs at

2

Solve (A)

3 1 x  2 x  1 x  1, 0  x 

(B) 1  x  0 and

1 3 and x  2 2 1 3 x 2 2

(C)

3 1   x   and 0  x  1 2 2

(D)

1 3 x   ,   x  0 and x  1 2 2 2

0.

3

,

If

, evaluate

2 .

(A) (B) (C)

(D)

4 Find the acute angle between the tangents to the graphs y  x and y  x3 at the point (1,1).

(A) 27° (B) 30° (C) 45° (D) 63° 5

The polynomial Find the value of k.

2

5

2 has

(A) –7 (B) 7 (C) –12 (D) 12 6

Find (A)

(B)

√1

sin

2 1  x2 1

1  x2 (C) cos 1 x (D) sin 1 x 3

2 as a factor.

7

Find the domain and range of 3 cos 3 Range: 0 (A) Domain: 0 (B) Domain: 1 (C) Domain: (D) Domain:

8

1 Range: 0

Range: 0

3

Range: 0

3

A stone is thrown at an angle of α to the horizontal. The position of the stone at time t 1 seconds is given by x  Vt cos  and y  Vt sin   gt 2 where / is the 2 acceleration due to gravity and v m/s is the initial velocity of projection. What is the maximum height reached by the stone? V sin  (A) g (B)

g sin  V

V 2 sin 2  (C) 2g g sin 2  2V 2 The volume of a sphere of radius 8 mm is increasing at a constant rate of 50 mm 3 . (D)

9

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 (D) 12.50 mm2 /s 10 The acceleration of a particle is defined in terms of its position by 2 particle is initially 2 to the right of the origin, travelling with velocity 6 minimum speed of the particle. (A) 6

(B) 16

1

(C) 20 (D) 36

1

4

4 . The . Find the

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 calculations.


Use a SEPARATE writing booklet.

(a)

.

(b)

Marks

(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 approximation for e . Give your answer correct to 3 decimal places.

1

.

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

1 2

2

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

(c)

3 2 The polynomial P( x)  x  bx  cx  d has roots 0, 3 and –3.

(i) (ii)

What are the values of b, c and d? Without using calculus, sketch the graph of y  P ( x ) .

(iii)

Hence or otherwise, solve the inequality

x2  9 0 x

(d)

tan . Find the slope of the tangent to the Consider the function curve where the function y  f ( x ) cuts the y-axis.

(e)

Find the exact value of



1

1

4  x2 dx , using the substitution x  2sin  .

5

2 1 1

2

3



Marks

(a)

28,19 divides the interval AB externally in the ratio : 5 . The point Find the value of k if A is the point 4,3 and B is the point 2, 1

(b)

(i)

 sin x  cos x Show that sin( x  )  4 2

(ii)

Hence or otherwise, solve

2

(c) (i)

,

and

2

,

2

1

sin x  cos x 3  for 0  x  2 . 2 2

2

2 are two points on the parabola x  4ay .

Show that the equation of the normal to the parabola at P is given by

1

x  py  2ap  ap3 . (ii)

Find the co-ordinates of R, the point of intersection of the normals at P and Q, in terms of p and q.

2

(iii)

If

2

(d) Evaluate

(e)

2, find the cartesian equation of the locus of R.

√1

3

using the substitution

1

3

.

2

After t years the number of animals, N, in a national park decreases according to the equation: dN  0.09( N  100) dt The initial number of animals in the national park is 400. (i)

Verify that N  100  Ae 0.09 t is a solution of the above equation, where A is a constant.

1

(ii)

How many years does it take for the number of animals to reach 150?

2

6



Marks

(a)

The diagram shows a cylindrical barrel of length l and radius r. The point A is at one end of the barrel, at the very bottom of the rim. The point B is at the very top of the barrel, half-way along its length. The length of AB is d.

(b)

(i)

Show that the volume of the barrel is

(ii)

Find l in terms of d if the barrel has maximum volume for the given d.

.

1 2

Two circles are intersecting at P and Q. The diameter of one of the circles is PR.

Copy this diagram into your writing booklet. (i) (ii)

Draw a straight line through P, parallel to QR to meet the circle PRQ at S and the other circle at T. Prove that QS is also diameter of the circle. Prove that the circles have equal radii if TQ is parallel to PR.

Question 13 continues on page 9 7

2 2

Question 13 (continued) (c)

A rocket is fired from a pontoon on the sea. The rocket is aimed at a 60m high cliff, 240m from the pontoon. The angle of projection of the rocket is 45º and its initial velocity is 40√2 . (i)

Taking the point of projection as the origin O, derive expressions for the horizontal component x and vertical component y of the position of the rocket at time t seconds. (Assume the acceleration due to gravity is 10

2

(ii)

Show that the path of the rocket is given by the equation

1

(iii)

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

2

(iv)

Find the exact velocity of the rocket when it reaches this point. (Hint: velocity includes magnitude and direction)

3

8



(a)

Use mathematical induction to prove that 3 1. for all integers

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

(c)

7

is divisible by 4

(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?

Marks 3

1 3 1

A particle moves in a straight line and its displacement x metres from a fixed point O at any time t seconds is given by the equation x  4 cos 2 t  1 . (i)

Prove that the particle is undergoing simple harmonic motion.

2

(ii)

State the period of the motion.

1

(iii)

Sketch the graph x  4 cos 2 t  1 for 0  t   . Clearly show the times when the particle passes through O.

2

(iv)

Find the time when the velocity of the particle is increasing most rapidly for 0  t   .

2

End of paper

9

STANDARD INTEGRALS n

 x dx



1

1 n+1 x , n  -1; x  0, if n < 0 n 1

 x dx

 ln x, x > 0

e

dx



1 ax e , a0 a

 cos ax dx



1 sin ax, a  0 a

 sin ax dx

1  - cos ax, a  0 a

 sec

ax dx



1 tan ax, a  0 a

 sec ax tan ax dx



1 sec ax, a  0 a

1 dx  x2



1 -1 x tan , a  0 a a

ax

a   

2

2

1 a2  x2 1 x a 2

2

1 x a 2

2

dx

x  sin -1 , a  0, -a < x < a a

dx

 ln x  x 2  a 2 , x  a  0

dx

 ln x  x 2  a 2









NOTE: ln x  loge x, x  0

10

2015 HSC ASSESSMENT TASK 3

Mathematics Extension 1 General Instructions Class Teacher:

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   

(Please tick or highlight)

     

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

Student Number:_________________________ (To be used by the exam markers only.) Question No

Mark

1-10

11

12

13

14

Total

Total

10

15

15

15

15

70

100

Section I 10 marks Attempt Questions Allow about 15 minutes for this section Use the multiple choice answer sheet for Questions

_____________________________________________________ 1.

What is the value of

(A) 0 (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? (A) (B) (C) (D)

4.

What are the coordinates of the point that divides the interval joining the points A( B( externally in the ratio 1:3? (A) (B) (C) (D)

5.

Which of the following is the solution to (A) (B) (C) (D)

?

and

6.

The polynomial

has roots ,

and . What is the value of ?

(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 circle at B and C.

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

9.

Find the gradient of the normal to the parabola (A) (B) (C) (D)

,

at the point where

.

10.

An approximate solution to the equation is Newton’s method, a more accurate approximation is given by: (A)

(B)

(C)

(D)

. Using one application of

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 reasoning and/or calculations.

______________________________________________________________________ Question 11 (15 Marks) Start a NEW page. (a)

When the polynomial is the value of a?

is divided by

the remainder is 7. What 2

(b) (i) 1

(ii) 1

(iii) 2 (iv) 2 (c)

Find the acute angle between the lines

(d)

Evaluate

and

2

3

(e)

Find the general solution to

2

Question 12 (15 Marks) Start a NEW page.

(a) (i)

(ii)

(b)

Without using calculus, sketch the graph of

2

Hence solve

1

Using the substitution

find the exact value of:

3

(c) (i)

A chef takes an onion tart out of the fridge at into a room where the air temperature is . The rate at which the onion tart warms follows Newton’s law, that is:

where k is a positive value, time t is measured in minutes and temperature T is measured in degrees Celsius. Show that

is a solution to

and find the value of A.

2

(ii)

The temperature of the onion tart reaches

in 45 minutes. Find the exact value of k.

2

(iii)

Find the temperature of the onion tart 90 minutes after being removed from the fridge.

1

Question 12 continues on page 8

(d) (i)

ABC is a triangle inscribed in a circle. MAN is the tangent at A to the circle ABC. 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

1

(iii)

Hence show that DE is parallel to MAN.

3

End of Question 12

Question 13 (15 Marks) Start a NEW page (a)

Is the graph of

identical to

? Give a reason for your answer.

(b) (i)

A particle is moving in a straight line. At time t seconds it has displacement x metres from a fixed point O on the line, velocity and acceleration given by . Initially the particle is 5m to the right of O and moving towards O with a speed of 6 . Explain whether the particle is initially speeding up or slowing down.

1

(ii)

Find an expression for

(iii)

Find where the particle changes direction.

1

Express

2

(c) (i)

(ii)

in terms of .

1

2

in the form

Hence, or otherwise, solve

2

for

(d) (i)

A square ABCD of side 1 unit is gradually ‘pushed over’ to become a rhombus. The angle at A decreases at a constant rate of 0.1 radian per second. At what rate is the area of rhombus ABCD decreasing when

? 3

(ii)

At what rate is the shorter diagonal of the rhombus ABCD decreasing when 3

Question 14 (15 Marks) Start a NEW page. (a)

Prove that

is a multiple of 10 for all positive integers

3

(b) (i)

Show that

2

(ii)

Hence, using a similar expression, find a primitive for

1

(iii)

The curves

and

The curve

also intersects with the x axis at Q.

intersect at

3

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

(iii)

A line with gradient m passes through

and state its geometrical significance

1

2

and cuts the parabola at distinct points Q and R.

Find the range of possible values for m.

End of Examination.

3

NORTH SYDNEY GIRLS HIGH SCHOOL

2015 TRIAL HSC EXAMINATION

Mathematics Extension 1 General Instructions

Total marks – 70



Reading Time – 5 minutes

Section I



Working Time – 2 hours

10 marks



Write using black or blue pen Black pen is preferred Board approved calculators may be used A table of standard integrals is provided at the back of this paper

 Attempt Questions 1 – 10  Allow about 15 minutes for this section

In Questions 11 – 14, show relevant mathematical reasoning and/or calculations

 Attempt Questions 11 – 14  Allow about 1 hour and 45 minutes for this section

  

NAME:______________________________

Section II

1–10

Pages 7 – 14

60 Marks

TEACHER:___________________

STUDENT NUMBER:__________________________

QUESTION

Pages 2 - 6

MARK

11

/10 total/15

12

/15

13

/15

14

/15

TOTAL

/70

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

2

-

2 x3  3x 2  5 x  7 x  4 x  x3

What is the value of lim

(A)

1 2

(B)

2

(C)



(D)

2

1 2

Which of the following is equivalent to

(A)

  2sin     6 

(B)

  2sin     6 

(C)

  2sin     3 

(D)

  2sin     3 

3 sin   cos  ?

2

3

A curve is defined by x  2t and y  log e t . Which of the following is the value of

4

-

(A)

1 4

(B)

1 2

(C)

1

(D)

2

What is the value of lim x 0

(A)

2 3

(B)

3 2

(C)

4 9

(D)

1

( )

dy at the point 2, 0 ? dx

2sin 2 x ? 3 tan 3x

3

5

What is the value of sin  , given that ACD   in the diagram below?

(A)

-

3 5



2 3

(B)

2 5  5 3

(C)

2 2  5 3

(D)

6

2

4 3 5



1 3

 dx ? What is the correct expression for   4  x2

(A)

x sin 1    c 4

(B)

x sin 1    c 2

(C)

1 1  x  sin    c 4 4

(D)

1 1  x  sin    c 2 2

4

7

The graph below represents the depth of water in a channel (in metres) as it changes over time (in hours).

Which of the following is NOT true?

8

-

(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 when the depth is 8 m

Which of the following are the roots of the equation x3  4 x 2  x  6  0 ? (A)

1, 3, 2

(B)

1,  3,  2

(C)

1, 1,  6

(D)

1,  1, 6

5

9

10

What is the value of cos 1  sin   where 2     ? (A)

 

(B)

  2

(C)

  2

(D)

  2

x 1 2 within the natural domain, three students obtain the  x 1 x following inequalities. In solving

Student I:

 x  1

Student II:

 x  1

Student III:

 x  1

2

2 x

3

 2  x  1 x

3

x  2 x  x  1

Which students will obtain the correct solution to the original inequality?

-

(A)

Student I only

(B)

Student II only

(C)

Student III only

(D)

Student II and Student III

6

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 booklets are available. In Questions 11 to 14, your responses should include relevant mathematical reasoning and/or calculations.


(a)

Differentiate x cos1  ex  with respect to x.

(b)

Find ô sin 2 3x dx

(c)

The point P divides the interval joining A -1, 5 to B 2, 3 externally in the ratio 4 : 3 . Find the coordinates of P.

2

(d)

Find the size of the acute angle between the line y  2 x and the curve

3

2

ó

2

õ

(

)

( )

y  x 2 at the point of intersection  2, 4  . Give your answer to the nearest degree.

1

(e)

dx  Use the substitution u  x to determine  . 1 1  x  x

3

3

Give your answer in exact form.

(f)

(i) (ii)

-

x  Sketch the graph of y  sin   for the domain 3  x  3 .  2  Hence, or otherwise, find for what positive values of m, the equation x  sin    mx , has exactly three solutions.  2  7

1 2


(a)

Angela is preparing food for her baby and needs to use cooled boiled water. The equation y = Aekt describes how the water cools, where t is the time in minutes, A and k are constants and y is the difference between the water temperature and the room temperature at time t, both measured in degrees Celsius. The temperature of the water when it boils is 100C and the room temperature is a constant 23C. (i)

Find the value of A.

1

(ii)

The water cools to 88C after 5 minutes. Find the value of k correct to three significant figures.

2

(iii) Angela can prepare the food when the water has cooled to 50C. How much longer must she wait?

(b)

A particle’s displacement satisfies the equation t  x 2  5 x  4 , where x is measured in cm and t is in seconds. Initially, the particle is 4 cm to the right of the origin.

1 . 2x  5

(i)

Show that the velocity is given by v 

(ii)

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

1

2

(iii) Find the position of the particle 10 seconds after the start of the motion.

2

(iv)

1

Briefly describe the motion of the particle.


-

2

8

Question 12 (continued)

(c)

BAC and BAD are two circles such that the tangents at C and D meet at T on AB produced.

A

C

B D T

T

Copy or trace the diagram into your writing booklet.

If CBD is a straight line prove that: (i)

TCAD is a cyclic quadrilateral

3

(ii)

TC = TD

1

End of Question 12

-

9


(a)

()

(i)

Fully factorise P x = x 3 - x 2 - 8x +12 .

(ii)

Hence, find any values of k, such that Q x ³ 0 for all real x,

()

( )(

3

()

)

1

where Q x = P x 2x - k .

(b)

In the diagram below, OL is a road that runs due east. OM is another road and intersects OL at 30. Both roads are on flat ground. On OM there are three equally spaced vertical telegraph poles AB, CD and EF of equal height h m. The distance between adjacent poles is twice the height of the poles. From an observer at P, the bearing of the first pole AB is 300T. The angles of elevation of A and C from P are 45 and 30 respectively.

(i)

Explain why triangle BDP is right angled.

2

(ii)

Deduce that PF  h 13 .

3


-

 10 


(c)

Consider the parabola x 2  4 y . P  2 p, p 2  and Q  2q, q 2  lie on the parabola.

Q  2q, q 2  P  2 p, p 2 

(i)

Find the equation of the chord PQ.

2

(ii)

Show that if PQ is a focal chord then pq  1 .

1

(

)

(

)

(iii) T 2t,t 2 , t  0 and R 2r,r 2 are two other points on the parabola distinct from P and Q. If TR is also a focal chord and P, T, Q and R are concyclic, show that p 2 + q 2 = t 2 + r 2 .

End of Question 13

-

 11 

3


(a)

A particle is undergoing simple harmonic motion such that its displacement x centimetres from the origin after t seconds is given by :

  x  2  4sin  2t   . 3 

(b)

(i)

Between which two positions is the particle oscillating?

1

(ii)

At what time does the particle first move through the origin in the positive direction?

3

Use the principle of mathematical induction to prove 3n  7  4n for all

3

integers n  3 .


-

 12 


(c)

Consider the region enclosed by the circle  x  a   y 2  a 2 and the line x  b 2

shown in the diagram below, where 0  b  2a . y

x b

(i)

Show that the volume of the spherical cap formed by rotating this region around the x-axis is given by

V

(ii)

a

 b2 3

 3a  b 

cubic units

A spherical goldfish bowl of radius 10 cm is being filled with water at a constant rate of 75 cm3 per minute. Using part (i) or otherwise, find the rate at which the water level in the bowl is rising when the bowl is half full of water.


-

2

 13 

2


(d)

Consider the function f  x   x 

(i)

By restricting the domain of the original function to x  0 , find the equation of f

(ii)

1 whose graph is shown below. x

1

 x .

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

x 1  16 . Leave your answer in exact form. x

No marks will be awarded for solving the equation directly for x.

End of paper

-

2

 14 

2

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 ie

2  2 . 1

2. B Using auxiliary angle method, this is o the form

R sin     where R 

2

3  (1) 2  2 .

B using standard integrals table

7. C The amplitude is the distance from the centre of motion to the extreme of motion which is 12 – 8 = 4. 8. B Sum of roots = 4 and product of roots is 6. 9.

1  tan     . 6 3

3.

6.

cos

C

1

 sin    cos 1  sin      ;     cos 1  cos  2        ;

dy dy dx   . dx dt dt dy 1 dx dy 1 dy 1  ;  2;  . At  2, 0  , t  1;  . dt t dt dx 2t dx 2

C

2sin 2 x 2 sin 2 x lim   lim x  0 3 tan 3 x 3 x 0 tan 3 x 2 2 sin 2 x 3x    lim  lim x  0 tan 3 x 3 3 x 0 2 x 4  9 5.

A



ˆ  ACB ˆ sin   sin BCD 

 2

     acute

 cos 1  cos   2   ;   2 acute

B

   2

Use parametric differentiation.

4.

acute



2 5 1 2 2 2 .  .   5 3 5 3 3 3 5

using compound angle result

Alternately sub a second quadrant angle into your calculator and verify which option works. 10.

D

As

x  0 it is not necessary to multiply by the square

of

x only by the square of x  1 as Student II has

done. However by multiplying by the square of x Student III does not generate extra solutions because x  0 is not an admissible solution.

Mathematics Extension 1 Trial HSC 2015 – Suggested Solutions Section II Question 11 (a) Using chain rule,

d 1 x cos 1  ex    1.cos 1  ex   x e  2 dx 1   ex   cos 1  ex   (b)

1  e2 x 2

Using double angle results,

 sin

(c)

ex

2

  1  cos 6 x  3 xdx    dx 2   1 sin 6 x  = x c 2 6 

Using a ratio of 4 : 3

 4  2  3  1 4  3  3  2  P ,   11, 3 4  3 4  3   (d)

y  2 x; m1  2 and y  x 2 ; y '  2 x; m2  4 at x  2 tan  

m1  m2 24 2   1  m1m2 1 8 9

  13 (nearest degree) (e)

du 1 dx   2du  dx 2 x x 1 1 x u  ; x 1 u 1 3 3

u x

1

dx  1  1  x  x 3

1

1  2.du   2  tan 1 u  1 2  1 1 u 3 3

    =2     4 6 6

(f)

(i) (ii)

The upper bound of m to ensure exactly three solutions is found by finding the gradient of the tangent at x  0 and is

 2

. As we need positive values of m, then the

required range of values for m is 0  m 

 2

.

Question 12 (a)

(i)

At t  0; y  100  23  77

77  Ae0  A  77

(ii)

t  5; T  88  23  65 65  77e5 k 65 77  65  5k  ln    77  1  65  k  ln    0.0339 (4dp) 5  77  e5 k 

(iii)

50  23  77ekt 27 e kt  77  27  kt  ln    77  1  27  t  ln   k  77  t  30.928  30m 55s

Therefore, she must wait another 25 min 55 sec.

(b)

(i)

(ii)

t  x2  5x  4 dt  2x  5 dx dx 1 v  dt 2 x  5





d 1 2 v dx 2  d 1 1 a    dx  2  2 x  5 2  1 2   2 2  2 x  5 3 a



2

 2 x  5

3

(iii)

When t  10

10  x 2  5 x  4 x2  5x  6  0

 x  6  x  1  0 x  1, 6 Initially, x  4 so v 

1 1  0 2(4)  5 3

So the particle is moving to the right. v can never be zero so the particle never turns around. So it can never be at x  1 .  x  6 . (iv)

(c)

Initially the particle is 4 units to the right and moving to the right. the acceleration at this time is negative, so the particle is slowing down.

(i) Let TCB   and TDB  

TCB  CAB   (Angle between tangent and chord equal to angle in alternate segment) TDB  DAB   (Angle between tangent and chord equal to angle in alternate segment) CAD  CAB  DAB     (1) Now, CTD  180      (Angle sum of CTD ) (2)

CAD  CTD  180 (adding (1) and (2)) TCAD is a cyclic quadrilateral. (opposite angles are supplementary)

(ii)

CAT  CDT (angles in the same segment in circle TCAD)    This means that TCB  TDB TC  TD (equal sides opposite equal angles in TCD )

Question 13 (a)

(i)

P  2   0   x  2  is a factor

P  x    x  2   x 2  x  6  by inspection. [Alternately use long division]. P  x    x  2  x  3 x  2  P  x    x  2   x  3 2

(ii)

Q  x   P  x  2 x  k 

Q  x    x  2   x  3 2 x  k  2

k 2   2  x  2   x  3  x   2  k  If Q  x   0 for all x, then  x     x  3 2  Or k  6

(b)

(i)

BP  h cot 45  h and

DP  h cot 30  h 3 BP 2  DP 2  h 2  3h 2  4h 2   2h   BD 2 2

 BDP is right angled at P (converse of Pythagoras Theorem)

 3h   h   60   In BPF , BF  2h  2h  4h (ii) DBP  tan 1 

Using the cosine rule,

ˆ PF 2  BP 2  BD 2  2.BP.BD.cos PBF  h 2   4h   2 .h.4h. 2

1 2

 17 h 2  4h 2  13h 2  PF  13h as required

(c)

q  p q  p p  q q2  p2   2q  2 p 2 2 q  p

(i) mPQ 

Equation of PQ is:

y  p2 

pq  x  2 p 2

2 y 2 p 2   p  q  x 2 p 2  2 pq y

pq x  pq 2

(ii) Focus is  0,1 . Sub into eqn of chord PQ

1  0  pq

or

pq  1

(iii) PQ and TR are chords of circle PTQR and intersect at the focus S.

PS  SQ  TS  SR (product of intercepts of intersecting chords) But PS  p 2  1 using the locus definition; distance from focus = distance from directrix Similarly, QS  q 2  1 etc

  p 2  1 q 2  1   t 2  1 r 2  1 p2q2  p2  q2  1  t 2r 2  t 2  r 2  1 2 2 But pq  1  p 2 q 2  1 and tr  1  t r  1

 p2  q2  t 2  r 2

Question 14 (a)

(i)

(ii)

Centre of motion is – 2 . Amplitude is 4. Hence, oscillates between – 6 and + 2. Solving for x  0

  4sin  2t    2 3   1  sin  2t    3 2    5 13 2t   , , ,... 3 6 6 6   11 2t   , , ,... as t  0 6 2 6  11 t , ,... 4 12 Graph of displacement is as below:

Hence, crosses the origin in a positive direction the second time ie at t  crosses in a positive direction.

(b)

To prove 3n  7  4n for n  3

Test if true for n = 3: LHS = 33  7  27  7  34 and RHS = 43  64 LHS < RHS, hence true for n  3 . Assume the result is true for some n = k where k  1; k 



ie assume that 3k  7  4k Prove true for n = k + 1 ie prove that 3k 1  7  4k 1

3k  7  4k by assumption. Multiply both sides by 3. 3k 1  21  3.4k 3k 1  7  14  3.4k 3k 1  7  3.4k 3k 1  7  4.4k 3k 1  7  4k 1 Hence, the proposition is true for all n  3 by Mathematical Induction.

11 . Alternately, use v  0 to find when it 12

(c)

(i) y 2  a 2   x  a 

2

b

V    y 2 dx 0





   a 2   x  a  dx b

0



b

0

a

2

2



 x 2  2ax  a 2 dx b

  x3 ax 2    2  2 0  3   b 3       ab 2    0     3   b3 3ab 2      3   3  b2 V  3a  b  3 (ii) a  10 and

V

 b2

dV  75 dt

 30  b   10 b2 



b3

3 3 dV   20 b   3 b 2 db 3 dV dV db (Chain Rule)   dt db dt db dV dV   dt dt db db  75   20 b   b 2  dt When the bowl is half full, b  10 db 75 3  75   200  100    dt 100 4 (d)

(i) y  x 

1 x

For the inverse: x  y 

1 y

Multiply by y

xy  y 2  1

y 2  xy  1  0 x  x2  4 2 As y  0 for the inverse, then y

y

x  x2  4 2

(ii)

f

x 1 1  16  x   16 x x

 x   16

x  f 1 16  



x  8  65



2

16  162  4  8  65 2

 129  16 65 End of solutions

PENRITH HIGH SCHOOL 2015 HSC TRIAL EXAMINATION

Mathematics Extension 1

General Instructions:

• Reading time – 5 minutes • Working time – 2 hours • Write using black or blue pen Black 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 • Answer each question on a new sheet of paper

Student Number:

Total marks–70 SECTION I

Pages 3–5

10 marks • Attempt Questions 1–10 • Allow about 15 minutes for this section SECTION II

Pages 6–9

60 marks • Attempt Questions 11–14 • Allow about 1 hours 45 minutes for this section

Teacher Name:

This paper MUST NOT be removed from the examination room

Assessor: T Bales 1

Section I 10 marks Attempt Questions 1–10 Allow about 15 minutes for this section

Use the provided multiple–choice answer sheet for Questions 1–10 1

For 𝑥𝑥 > 1, 𝑒𝑒 𝑥𝑥 − ln 𝑥𝑥 is: (A) (B) (C) (D)

2

=0 >0 <0

= 𝑒𝑒

Consider the function 𝑓𝑓(𝑥𝑥) given in the graph below: f(x)

−𝑎𝑎

0

𝑎𝑎

Which domain of the function f(x) above is valid for the inverse function to exist?: (A)

𝑥𝑥 > 0

(C)

0 < 𝑥𝑥 < 𝑎𝑎

(B)

(D) 3

𝑥𝑥 < 0

What is the acute angle between the lines 2𝑥𝑥 − 𝑦𝑦 − 7 = 0 and 3𝑥𝑥 − 5𝑦𝑦 − 2 = 0 ? (A)

(B) (C) (D) 3

−𝑎𝑎 < 𝑥𝑥 < 𝑎𝑎

4∘ 24′

32∘ 28′

57∘ 32′

85∘ 36′

𝑥𝑥

4

A particle is moving in a straight line with velocity 𝑣𝑣 𝑚𝑚/𝑠𝑠 and acceleration 𝑎𝑎 𝑚𝑚/𝑠𝑠 2 . Initially the particle started moving to the left of a fixed point 𝑂𝑂. The particle is noticed to be slowing down during the course of the motion from 0 to 𝐴𝐴. It turns around at 𝐴𝐴, keeps speeding up for the rest of the course of motion, passing 𝑂𝑂and 𝐵𝐵 and continues. The particle never comes back. Take left to be the negative direction. A

𝑂𝑂

B

𝑥𝑥

During the course of the particle’s motion from 𝑂𝑂 to 𝐴𝐴, which statement of the following is correct?

(A)

𝑣𝑣 > 0 and 𝑎𝑎 > 0

(C)

𝑣𝑣 < 0 and 𝑎𝑎 > 0

(B)

(D)

5

𝑣𝑣 > 0 and 𝑎𝑎 < 0

𝑣𝑣 < 0 and 𝑎𝑎 < 0

A particle is moving in a straight line with velocity,

𝑣𝑣 =

1

1+𝑥𝑥

𝑚𝑚/𝑠𝑠

where 𝑥𝑥 is the displacement of the

particle from a fixed point 𝑂𝑂. If the particle was observed to have reached the position 𝑥𝑥 = −2 𝑚𝑚 at a certain moment of time, then this particle: (A) (B) (C) (D)

6

4

will definitely reach the position 𝑥𝑥 = 1𝑚𝑚 may reach the position 𝑥𝑥 = 1𝑚𝑚

will never reach the position 𝑥𝑥 = 1𝑚𝑚

will come to rest before reaching 𝑥𝑥 = 1𝑚𝑚

If the rate of change of a function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) at any point is proportional to the value of the function at that point then the function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) is a:

(A)

Polynomial function

(B)

Trigonometric function

(C)

Exponential function

(D)

Quadratic function

7

Let 𝛼𝛼 and 𝛽𝛽 be any two acute angles such that 𝛼𝛼 < 𝛽𝛽. Which of the following statements is correct ? (A) (B) (C) (D)

8

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 < 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 < 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 < 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 < 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Which of the following is a primitive function of 𝑠𝑠𝑠𝑠𝑠𝑠2 𝑥𝑥 + 𝑥𝑥 2 ?

(B)

(D)

1 𝑥𝑥 2

− 𝑠𝑠𝑠𝑠𝑠𝑠2𝑥𝑥 + 2𝑥𝑥 + 𝑐𝑐

+ 𝑐𝑐

1 4

𝑛𝑛

𝑐𝑐1 + 2 𝑛𝑛𝑐𝑐2 + ⋯ + 𝑛𝑛 𝑛𝑛𝑐𝑐𝑛𝑛 = 𝑛𝑛2𝑛𝑛−1

𝑛𝑛

𝑐𝑐1 + 𝑛𝑛𝑐𝑐2 + ⋯ + 𝑛𝑛𝑐𝑐𝑛𝑛 = 2𝑛𝑛−1

𝑛𝑛

(B) (C)

𝑐𝑐1 + 2 𝑛𝑛𝑐𝑐2 + ⋯ + 𝑛𝑛 𝑛𝑛𝑐𝑐𝑛𝑛 = 𝑛𝑛2𝑛𝑛+1

𝑛𝑛

(D)

𝑐𝑐1 + 𝑛𝑛𝑐𝑐2 + ⋯ + 𝑛𝑛𝑐𝑐𝑛𝑛 = 2𝑛𝑛+1

Using the substitution, 𝑢𝑢 = 1 + √𝑥𝑥, find the value of

(A) (B) (C)

(D)

5

𝑥𝑥 3 3

− 𝑠𝑠𝑠𝑠𝑠𝑠2𝑥𝑥 +

Consider the binomial expansion (1 + 𝑥𝑥)𝑛𝑛 = 1 + 𝑛𝑛𝐶𝐶1 𝑥𝑥+𝑛𝑛𝐶𝐶2 𝑥𝑥 2 + ⋯ + 𝑛𝑛𝐶𝐶𝑛𝑛 𝑥𝑥 𝑛𝑛 . Which of the following expressions is correct? (A)

10

+ 𝑐𝑐

1 𝑥𝑥 2

1 2

1 4

𝑥𝑥 3 3

𝑥𝑥 − 𝑠𝑠𝑠𝑠𝑠𝑠2𝑥𝑥 + 2𝑥𝑥 + 𝑐𝑐

(C)

9

1 2

𝑥𝑥 − 𝑠𝑠𝑠𝑠𝑠𝑠2𝑥𝑥 +

(A)

6 5

1 3 2 3

3 2

4

∫1

1

2

1

�1+√𝑥𝑥� √𝑥𝑥

𝑑𝑑𝑑𝑑 is:

Section II 60 marks Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section Begin each question on a new sheet of paper. Extra sheets of paper are available. In questions 11–14, your responses should include relevant mathematical reasoning and/or calculations

Question 11 (15 marks) Start a new sheet of paper. (a)

(𝑥𝑥 + 1) and (𝑥𝑥 − 2) are factors of 𝐴𝐴(𝑥𝑥) = 𝑥𝑥 3 − 4𝑥𝑥 2 + 𝑥𝑥 + 6. Find the third factor.

2

(b)

Find the coordinates of the point 𝑃𝑃 which divides the interval 𝐴𝐴𝐴𝐴 internally in the ratio 2: 3 with 𝐴𝐴(−3, 7) and 𝐵𝐵(15, −6)

3

(c)

Solve the inequality

(d)

Use the method of mathematical induction to prove that, for all positive integers 𝑛𝑛: 𝑛𝑛 12 + 32 + 52 + 72 + ⋯ + (2𝑛𝑛 − 1)2 = (2𝑛𝑛 − 1)(2𝑛𝑛 + 1)

1

4𝑥𝑥−1

< 2,

graphing your solution on a number line.

3

3

3

(e)

T

P

•B •Q A

PT is the common tangent to the two circles which touch at T. PA is the tangent to the smaller circle at Q, intersecting the larger circle at points 𝐵𝐵 and 𝐴𝐴 as shown. i)

ii)

State the property which would be used to explain why 𝑃𝑃𝑃𝑃 2 = 𝑃𝑃𝑃𝑃 × 𝑃𝑃𝑃𝑃 If 𝑃𝑃𝑃𝑃 = 𝑚𝑚, 𝑄𝑄𝑄𝑄 = 𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 = 𝑟𝑟, prove that m =

Proceed to next page for question (12) 6

nr n−r

1

3


The equation 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1 − 2𝑥𝑥 has a root near 𝑥𝑥 = 0.3. Use one application of Newton’s methods to find a better approximation, giving your answer correct to 2 decimal places.

(b)

Five couples sit at a round table. How many different seating arrangements are possible if:

3

i)

there are no restrictions?

1

ii)

each person sits next to their partner?

2

𝑥𝑥 6 5

(c)

In the expansion of (4 + 2𝑥𝑥 − 3𝑥𝑥 2 ) �2 − � , find the coefficient of 𝑥𝑥 5

(d)

i)

Write the binomial expansion for (1 + 𝑥𝑥)𝑛𝑛

2

ii)

Using part (i), show that

2

iii)

Hence show that

n n 1 n 1 x dx Ck 3k +1 + = ∑ ∫0 ( ) k =0 k + 1

3

3

n

1

n = Ck 3k +1 ∑ 1 k + k =0


1  n +1  4 − 1 n +1 

2


Use the substitution 𝑢𝑢 =

𝑥𝑥

�1−𝑥𝑥2

(You can use the result that ∫

3

(b)

Give the exact value for −

(c)

∫

to show that 1

𝑎𝑎2 +𝑥𝑥 2

𝑥𝑥

�𝑡𝑡𝑡𝑡𝑡𝑡−1 �√1−𝑥𝑥2 �� = 𝑑𝑑𝑑𝑑 𝑥𝑥

1

√1−𝑥𝑥 2

3

𝑑𝑑𝑑𝑑 = 𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡−1 𝑎𝑎 + 𝑐𝑐 . You do not need to prove this).

dx 3 + x2 3

3

The distinct points P, Q have parameters 𝑡𝑡 = 𝑡𝑡1 and 𝑡𝑡 = 𝑡𝑡2 respectively on the parabola 𝑥𝑥 = 2𝑡𝑡, 𝑦𝑦 = 𝑡𝑡 2 . The equations of the tangents to the parabola at P and Q respectively are given by: 𝑦𝑦 − 𝑡𝑡1 𝑥𝑥 + 𝑡𝑡1 2 = 0 and 𝑦𝑦 − 𝑡𝑡2 𝑥𝑥 + 𝑡𝑡2 2 = 0 (You do not need to prove these)

2

ii)

Show that the equation of the chord 𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 2𝑦𝑦 − (𝑡𝑡1 + 𝑡𝑡2 )𝑥𝑥 + 2𝑡𝑡1 𝑡𝑡2 = 0

2

∝) Prove that for any value of 𝑡𝑡1 , except 𝑡𝑡1 = 0, there are exactly two values of 𝑡𝑡2 for which M lies on the parabola 𝑥𝑥 2 = −4𝑦𝑦.

3

i)

iii)

Show that M, the point of intersection of the tangents to the parabola at P and Q, has coordinates �𝑡𝑡1 + 𝑡𝑡2 , 𝑡𝑡1 𝑡𝑡2 �.

𝛽𝛽)

Find these two values of 𝑡𝑡2 in terms of 𝑡𝑡1 .


1

𝑑𝑑

2


A particle 𝑃𝑃 is moving in simple harmonic motion on the 𝑥𝑥 axis, according to the law 𝑥𝑥 = 4𝑠𝑠𝑠𝑠𝑠𝑠3𝑡𝑡 where 𝑥𝑥 is the displacement of 𝑃𝑃 in centimetres from 𝑂𝑂 at time 𝑡𝑡 seconds.

i)

State the period and amplitude of the motion.

2

ii)

Find the first time when the particle is 2cm to the positive side of the origin and it’s velocity at this time.

2

iii)

Find the greatest speed and greatest acceleration of 𝑃𝑃

3

(b)

𝑦𝑦 𝜃𝜃

𝑂𝑂

𝑉𝑉 𝑃𝑃

𝑥𝑥

Q A projectile is fired from 𝑂𝑂, with speed 𝑉𝑉𝑉𝑉𝑠𝑠 −1, at an angle of elevation of 𝜃𝜃 to the horizontal. After 𝑡𝑡 seconds, its horizontal and vertical displacements from 𝑂𝑂 (as shown) are 𝑥𝑥 metres and 𝑦𝑦 metres, repectively. i) ii) iii)

1 2

Prove that 𝑥𝑥 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 and 𝑦𝑦 = − 𝑔𝑔𝑡𝑡 2 + 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 Show that the time taken to reach 𝑃𝑃 is given by

𝑡𝑡 =

2𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑔𝑔

The projectile falls to 𝑄𝑄, where its angle of depression from 𝑂𝑂 is 𝜃𝜃. Prove that, in its flight from 𝑂𝑂 to 𝑄𝑄, 𝑃𝑃 is the half-way point in terms of time.

End of paper

9

3 2 3

STANDARD INTEGRALS

∫x

n

1

∫ x dx ∫e

1 x n +1 , n ≠ −1; x ≠0 , if n < 0 n+1

dx =

ax

= ln x , x > 0

dx =

1 ax e , a≠0 a 1 sin ax , a ≠ 0 a

∫ cos ax dx

=

∫ sin ax dx

= −

∫ sec

ax dx =

2

∫ sec ax tan ax dx 1

∫ a2 + x2

∫ ∫ ∫

dx =

1 a2 − x2 1 x −a 2

2

1 x +a 2

2

1 cos ax , a ≠ 0 a 1 tan ax , a ≠ 0 a =

1 sec ax , a ≠ 0 a

1 x tan −1 , a ≠ 0 a a

dx = sin −1

x , a ≠ 0, − a < x < a a

dx = ln( x + x 2 − a 2 ), x > a > 0

dx = ln( x + x 2 + a 2 )

NOTE: ln x = loge x, x > 0

10

Student Number:_________________

11

Teacher Name:_____________________

St George Girls High School Trial Higher School Certificate Examination

2015

Mathematics Extension 1 General Instructions

Total Marks - 70

• • • • • •

Section I - Pages 2 - 5 10marks • Attempt Questions 1-10. • Allow about 15 minutes for this section. • Answer on the sheet provided.

Reading time - 5 minutes Working time - 2 hours Write using blue or black pen. Write your student number on each booklet. Board-approved calculators may be used. A table of standard integrals is provided at the back of this paper. • The mark allocated for each question is listed at the side of the question. • Marks may be deducted for careless or poorly presented work.

Section II - Pages 6 - 11 60marks • Attempt Questions 11- 14. • Allow about 1 hour 45 minutes for this section. • Begin each question in a new booklet. • Show all necessary working in Questions 11- 14.

Students are advised that this is a Trial Examination only and does not necessarily reflect the content or format of the Higher School Certificate Examination.

St George Girls High School Trial HSC Examination - Mathematics Extension 1 - 2015

Page2

Section I 10marks Attempt Questions 1 - 10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1-10. 1.

In the diagram, AB is a diameter of the circle and MN is tangent to the circle at C. LCAB = 35°. What is the size of LMCA? M

(A) 35°

A

(B) 45°

(C) 55°

N

(D) 65°

2.

Which function is graphed below? I

I

I

I

I

I

--+------~-----+---

I

2

I

------""i------~----I

'

(A) 2ir sin 3x 1

''

' '

' ' --+------~-----+--' ' ''

TI

' ' ------""i------~ '

(B) Zn sin- 1 - x '

3

' '

(C) 4 sin- 1 3x

-----...f

(D) 4 sin- 1

_,

.,

'

'

--T-----

2

-1

''

I ----- 1 ' ---

,

'

' ' ------~-----+---

'

------ ,------1 ' '

' '

- - - - - 1'

'

.

''

~ ------ ,------,' -----''

1

3x

St George Girls High School Trial HSC Examination - Mathematics Extension 1- 2015

Page3

Section I (cont'd)

3.

4.

Find 1- 1 (x), given f(x)

3x-3 =x-2

3y-3

(A)

f- 1 (x) = -

(B)

f- 1 (x) = -x-3

(C)

f- 1 (x)

= 3x-3

(D)

f- 1 (x)

3-3x = -

x-2

Zx-3

x-2

2-x

Which diagram best represents y = -x(Z - x) 3 (x

(A)

(B)

·2

,1

(C)

2

(D)

+ 1) 2 ?

St George Girls High School

Page4

Trial HSC Examination - Mathematics Extension 1 - 2015

Section I (cont'd) 5.

Find k given x - 2 is a factor of P(x)

=x3 -

3x 2

+ kx + 12

= -4

(A) k

(B) k = 0 (C) k = 2

=4

(D) k

6.

The acute angle between 11 : 2x - y - 3 and 12 : y = 3x + 7 is closest to: (A) 15° (B) go (C) 82°

(D) 45°

7.

J2cos x dx 2

+x +C sin 2x + x + C

(A) sin x cos x

(B)

-i 2

(C) -cos 3 x 3

(D)

8.

-2

Y1-x 2

+C +c

The velocity of a particle at a position x is

x=

x

2e-, metres per second.

Calculate the particle's acceleration when its displacement is -2 metres. (A) -e m/s 2 4 I z (B) --ms eZ

(C) -2e 2 m/s 2 (D)

e 2 m/s 2

St George Girls High School Trial HSC Examination - Mathematics Extension 1- Z015

Page 5

Section I (cont'd) 9.

Find the exact value of sin 15° (A)

(B) (C)

(D)

1

3..fj, 2-.,fi. 2.,fi.

2(-../6 - ~) ../6 -.,fi. 4

10. Given the curve below, Eden intends to use Newtons Method to find an approximation to the root shown. Which initial estimate will not produce a good approximation with this method?

y

(A)

Xo

(B)

Xo

(C)

Xo

(D)

Xo

b

End of Section I

=a =b =C =d


Page 6

Section II 60marks Attempt Questions 11 - 14 Allow about 1 hours 45 minutes for this section Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11 - 14, your responses should include relevant mathematical reasoning and/or calculations. Question 11 (15 marks) Use a SEPARATE writing booklet 1

> -21

2

a)

Solve the inequality - 1x-1 1

b)

Sketch the intersection of y >

c)

Given A(-2, 3) and B(lO, 11), find the coordinates of the point P which divides the interval AB in the ratio 3: 1.

lxl -

1 and y

<1

d)

You are given 3.6 as an approximate root of the equation x 3 - 50. Use one application of Newton's method to find a better approximation. (to 2 decimal places)

e)

If y = sin(ln x), find

(i)

dy dx

Marks

(ii)

d2y dx 2

3

2

2

1,2


Page7

Marks

Question 11 (continued) BC is tangent to the circle at B. Find the value of

f)

A

x, giving reasons.

3

St George Girls High School

Trial HSC Examination - Mathematics Extension 1- 2015

Page 8

Marks

Question 12 (15 marks) Use a SEPARATE writing booklet a)

Solve 3 sinx

+ 4cosx =

3

2.5, 0:::::; x:::::; 2rr

b)

p

..e=...~~~~~~~~~~~..,._~~~___;;:::....~...;;;;,,=-~~----lQ A

The tangents from Q touch the circle at A and B. PC and PQ are straight lines LBAQ = a. (i)

Copy or trace the diagram into your writing booklet.

1

(ii) Given PD = 5 cm and DC = 7 cm, calculate the exact length of AP.

1

(iii) Show that LBCD = 2a.

3

(iv) Show that PQBC is a cyclic quadrilateral.

2


Page 9

Marks


c)

The rate at which a body warms in air is proportional to the difference between its temperature T and the constant temperature A of the surrounding air. This rate can be expressed by the differential equation dT

- = k(T-A)

dt

where t is time in minutes and k is constant. Show that T =A+ Cekt where C is a constant, is a solution of the differential equation.

1

(ii) A glass of milk warms from 4°C to 8°C in 15 minutes. The air temperature is 25°C. Find the temperature of the glass of milk after a further 45 minutes, correct to the nearest degree.

3

(iii) With reference to the equation for T, explain the behaviour of T as t becomes very large.

1

(i)


Page 10

Marks


Evaluate

b)

(i)

fo

1

x 3 (-Jx4 +

1) dx

using the substitution u = x 4

+ 1.

By considering its second derivative, show that y = ex - 4x is always concave up.

3

2

(ii) Use the trapezoidal rule with 3 function values to find an approximation

h 5

to

(e:L- 4x) dx,

3

correct to 4 significant figures.

(iii) Is this approximation too large or too small? Justify your answer? c)

1

y

r

SOm

l

0

x 200m---------'>

A projectile is launched from the top of a 50 m high building with an initial speed of 40 m/s. It is launched at an angle of a 0 above the horizontal, as in the diagram. Acceleration due to gravity is 10 m/s2. (i)

d 2x d 2y Given that - = 0 and - = -10, show that x = 40t cos a and dt 2 dt 2

y =-5t 2 + 40t sin a

+ 50 where x and y

are the horizontal and vertical

displacements of the projectile in metres from O at time t seconds after launching.

3

(ii) The projectile lands on the ground 200 metres from the base of the building. Find the two possible values of a. Give your answers to the nearest degree.

3


Page 11

Marks


b)

c)

P(2ap, ap 2 ) and Q(2aq, aq 2 ) are points on the parabola x 2 If p + q = 4, find the locus of M, the mid point of PQ.

= 4ay.

Given that x is a positive integer, prove by the method of mathematical induction that (1 + x)n - 1 is divisible by x for all positive integers n 2:: 1.

3

3

The velocity v ms- 1 of a particle moving on a horizontal line is given by v 2 = 252 (i)

+ 216x -

36x 2

Show that the particle is performing simple harmonic motion.

1

(ii) Find the centre of the motion.

1

(ii) Find the amplitude of the motion.

1

(iv) Find the period of the motion.

1

(v) Find the maximum speed of the particle.

1

(vi) Initially the particle is at one of the extreme points of the motion. Where will it be when t = Brr seconds.

2

12

(vii) Find its average speed during the first

1 :;

End of Examination

seconds.

2

~OLUTIONS

;a ,s-

. £~-,-

1

HSC

,. =. s .',... - I

-\

:>(..

~ ~ {· J

-Jr L. v -

~ I

-~~x.~1

<.!!° -

-i

v

Jj

-J

S~

-

11

zr

~ '1 ~ ... );r

-'

.! 3

.,,('

'1

- 4 _,

'1 -.... ~

~,,..

I

- II -<

2.

')(

3

-D.

• •

~1-

.,!

3. ')(. -3. ~ i.

')(-

~,.... t,i10,-

.

,J

'>l

i

")(.

":...

3'-7 - ~ I

·; ... ')_,

1 - ) x ~ 1 ',_ - 3 ("K -3.) :. 2-~-3 ,

-:.

..

:x.-3

/

So Le

TrctJ

~l

,<. =- 0

.._J-.

")<. :. -

.,._J..

')A-

~

ALL

I

A

2

A,

,

--

A

••

SJ

~(1)

,,.Q

d- 3

-

3.(~)'l- + k(2-)

~

-

I;..

-tJ.k.

1-

l'.l

::<'.)

=O

+12.

.)}.__

-f ;. -1-

~

k

A

")

"" :. .2 I

(''-,_ : 3 j

\

1

')

1 ) eas i: =::.

l

"-

el1t

jL 1 ( ,

-f <-DS .,'}~)

)

( I X-

+ +

c..,_s

2')\. )

l. .s ,---...

I

J d~ J '}(.

.l n. -+- c.

...

•

J3, - I :)_ J2---

Ji_ ')(

-

J'i-

~_Ji

'+

•••

D

'""'--o """ -- a.

A

SYDNEY BOYS HIGH SCHOOL MOORE PARK, SURRY HILLS

2015 HIGHER SCHOOL CERTIFICATE TRIAL PAPER

Mathematics General Instructions

Extension 1 Total Marks – 70

•

Reading time – 5 minutes.

•

Working time – 2 hours.

10 Marks

•

Write using black or blue pen.

•

Attempt Questions 1–10

•

Board approved calculators may be used.

•

•

All necessary working should be shown in

Allow about 15 minutes for this section.

• •

Pages 2–4

every question if full marks are to be awarded.

Section II

Marks may NOT be awarded for messy or

60 marks

badly arranged work.

•

Attempt Questions 11–14

Leave your answers in the simplest exact form,

•

Allow about 1 hour and 45 minutes for this section

unless otherwise stated. •

Section I

Start each NEW question in a separate answer booklet.

Examiners:

Pages 6–11

R. Elliot & J. Chen

This is an assessment task only and does not necessarily reflect the content or format of the Higher School Certificate.

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

The roots of 3x3 – 2x2 + x – 1 = 0 are α, β and γ. What is the value of α 2 βγ + αβ 2γ + αβγ 2 ? (A) (B) (C) (D)

2

1 9 2 − 9 1 2 9 −

What is the minimum value of (A)

–2

(B)

–4

(B)

–16 7 −3

(D) 3

4

7 sin x − 3cos x ?

⎛ 2x ⎞ What is the domain and range of y = sin −1 ⎜ ⎟ ? ⎝ 5⎠ (A) Domain: −1 ≤ x ≤ 1 ; Range: −π ≤ y ≤ π (B)

Domain: −1 ≤ x ≤ 1 ;

Range: −

π π ≤ y≤ 2 2

(C)

5 5 Domain: − ≤ x ≤ ; 2 2

Range: −

π π ≤ y≤ 2 2

(D)

5 5 Domain: − ≤ x ≤ ; 2 2

Range: −π ≤ y ≤ π

Evaluate lim x→0

(A) (B) (C) (D)

sin3x . 2x

0 2 3 1 3 2 –2–

5

In the diagram below, AB is a tangent to the circle BCD. Also, CD is a tangent to the circle ABD. ∠ BAD = θ and ∠ BCD = φ. NOT TO SCALE B

C A D Which of the following is a true statement?

6

(A)

ΔABD ≡ ΔBDC

(B)

ABCD is a cyclic quadrilateral

(C)

ΔABD ||| ΔBDC

(D)

AB || CD

A particle moves in simple harmonic motion so that its velocity, v, is given by v2 = 6 − x − x2 .

Between which two points does it oscillate?

7

(A)

x = 6 and x = 3

(B)

x = –2 and x = 3

(C)

x = 1 and x = 2

(D)

x = 2 and x = –3

⌠ Which of the following is an expression for ⎮ cos3 x sin x dx ? ⌡ 4 (A) − cos x + c

(B)

1 − cos 4 x + c 4

(C)

cos 4 x + c

(D)

1 cos 4 x + c 4

–3–

8

Which of the following is the correct expression for the inverse of f (x) = e1−2 x ? (A) (B) (C) (D)

9

10

f −1 (x) = −2e1−2 x 1 f −1 (x) = − e1−2 x 2 1 f −1 (x) = − log e (1− 2x) 2 1 f −1 (x) = (1− log e x) 2

Three Mathematics study guides, four Mathematics textbooks and five exercise books are randomly placed along a bookshelf. What is the probability that the Mathematics textbooks are all next to each other?

(A)

4! 12!

(B)

9! 12!

(C)

4!3!5! 12!

(D)

4!9! 12!

A particle moves on the x-axis with velocity v m/s, such that v2 = 16x – x2. Which of the following is the particle’s maximum speed and the position of where this maximum speed occurs? (A)

Maximum speed = 16 m/s at x = 0

(B)

Maximum speed = 8 m/s at x = –8

(C)

Maximum speed = –8 m/s at x = 8

(D)


–4–

Section II 60 marks Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section Answer each question in a NEW writing booklet. Extra pages are available In Questions 11–14, your responses should include relevant mathematical reasoning and/or calculations. Question 11 (15 Marks)

Start a NEW Writing Booklet

(a)

Differentiate sin −1 (log e x) .

1

(b)

⌠ Find ⎮ ⌡

1

(c)

(i)

Simplify sin ( A + B ) + sin ( A − B ) .

(ii)

⌠6 Hence, evaluate ⎮ sin3x cos x dx. ⌡0

1 4 − 9x 2

dx .

1

π

(d)

(e)

(

2

)

The point P 6 p, 3p 2 is a point on the parabola x 2 = 12 y . (i)

Find the equation of the tangent at P.

2

(ii)

The tangent at P cuts the y-axis at B. The point A divides PB internally in the ratio 1 : 2. Find the locus of the point A as P varies.

3

A piece of meat at temperature T° C is placed in an oven, which has a constant temperature of H° C. The rate at which the temperature of the meat warms is given by dT = − K (T − H ) , dt where t is in minutes and for some positive constant K.

(i)

Show that T = H + Be− Kt , where B is a constant, is a solution of the differential equation above.

1

(ii)

If the meat warms from 10° C to 50° C in the oven, which has a constant temperature of 180° C, in 30 minutes, find the value of K.

2

(iii)

How long will it take the meat to get to a temperature of 150° C? Express your answer correct to the nearest minute.

2

–6–

Question 12 (15 Marks) (a)

(b)


(i)

Solve cos x − 3sin x = 1 for 0 ≤ x ≤ 2π .

2

(ii)

Hence, or otherwise, find a general solution to cos x − 3sin x = 1 .

1

(i)

On the same set of axes sketch the graphs of y = cos 2x and y =

(ii)

Use the graph to determine the number of solutions to the equation

x +1 3

2

1

3cos 2x = x + 1

(iii)

One solution of the equation 3cos 2x = x + 1 is close to 0.5. Use one application of Newton’s Method to find another approximation, correct to 3 decimal places.

3

π

(c)

(d)

⌠4 Evaluate ⎮ sin 2 2x dx ⌡0

3

When x cm from the origin, the acceleration of a particle moving in a straight line is given by:

d 2x 5 =− 2 3 dt ( x + 2) It has an initial velocity of 2 cm/s at x = 0. If the velocity is V cm/s, find V in terms of x.

–7–

3

Question 13 (15 Marks) (a)


In the diagram below, DC is a diameter of the larger circle centred at A. AC is a diameter of the smaller circle centred at B. DE is tangent to the smaller circle at F and DC = 12.

4

Copy the diagram to your answer booklet. Determine the length of DE.

E

F

D

(b)

A

B

C

(i)

Simplify k! + k × k!

1

(ii)

Prove, by mathematical induction, that

3

1× 1! + 2 × 2! + 3× 3! + ...+ n × n! = (n + 1)! − 1

for all positive integers n.

(c)

(i)

⌠ Using the substitution x = 3+ 3sin θ find ⎮ ⎮ ⌡

(ii)

Let R be the region bounded by the curve y = 4 x ( 6 − x ) and the x-axis.

x ( 6 − x ) dx

Find the volume of the solid of revolution generated by revolving R about the x-axis.

End of Question 13

–8–

4

3

15

Question 14 (15 Marks)

SPECMATH EXAM 1 PT1


n 18 (a) 8 cm

1.2 m

10 cm 4m m hhcm

The figure above shows an inverted conical cup with base radius 8 cm and height 10 cm. rted cone, as shown in the diagram, is initially full of water. The water 2flows out through a hole at the π Some cm3 per minute. 3 water is poured into the cup at a constant rate of at the rate of 0.1 h m per hour, where h m is the depth of water remaining after t hours. The volume 5 3 water is given by VLet = 0.03 πh . of the water be h cm at time t minutes. the depth

t hours,

dh Find is given bythe rate of change in the area of the water surface when h = 4 dt

3

3

.9πh 2

(b)

9πh

3 2

5

.009πh 2

A particle is projected horizontally at 30 ms −1 from the top of a 100 m high wall. Assume that acceleration due to gravity is 10 ms −2 and that there is no air resistance. The flight path of the particle is given by: x = 30t, y = 100 − 5t 2 (Do NOT prove this)

1 3

0.9πh 2 1

9πh

3 2

where t is the time in seconds after take-off. (i)

Find the time taken for the particle to reach the ground.

(ii) Find the angle and speed at which the particle strikes the ground. n 19 le starts from rest and moves in a straight line with acceleration 6 sin(2t) m/s2 at time t s. acement from its starting point, in metres, Question at time t is by on page 11 14given continues 4 sin(2t) sin(2t) – 3t 3 cos(2t) 5 sin(2t) – 1.5 sin(2t) – 10 –

1 2

Question 14 (continued) (c)

The diagram below shows a tetrahedron such that VA = VB = AB = 2a, CA = CB = 3a and VC = 5a . O is the foot of the perpendicular from V to the base ABC. M is the midpoint of AB. P is a point on BC such that BP = ra where 0 ≤ r ≤ 3. ∠ VMC = θ and ∠ VPO = ϕ. V

5a 2a

C φ

O

A

P

3a

θ

M

B

6 . 4

(i)

By considering Δ VMC, show that cosθ =

(ii)

Hence find the exact value of VO.

(iii)

Show that VP 2 =

(iv)

Hence show that sin ϕ =

(v)

Hence, or otherwise, find the maximum value of ϕ as r varies.

(

1

)

1 2 3r − 8r + 12 a 2 3

(

2

45

8 3r − 8r + 12 2

3

)

End of paper

– 11 –

1

2

SYDNEY BOYS HIGH SCHOOL M O O R E PA R K , S U R RY H I L L S

2015 HIGHER SCHOOL CERTIFICATE TRIAL PAPER

Mathematics

Extension 1

Sample Solutions Question Teacher Q11 RB Q12 BK Q13 BD Q14 PB

MC Answers Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10

D B C D C D B D D D

Section I

1

10 marks

The roots of 3x3 – 2x2 + x – 1 = 0 are ,  and . What is the value of  2    2   2 ? (A) (B) (C) (D)

1 9 2  9 1 2 9 

ANSWER: D 3x 3 – 2x 2  x – 1  0 d b          a a 1 2   3 3

 2    2   2         1 2   3 3 2  9

2

What is the minimum value of (A)

–2

(B)

–4

7 sin x  3cos x ?

ANSWER: B (B)

–16

7 sin x  3cos x (D)

7 3

r

 7

2

 32

 79 4 Let

7 sin x  3cos x  r sin      4sin    

No matter what the value of  1  sin  x     1

4  4sin  x     4 Therefore the minimum value is x  4.

–2–

3

(A)

Domain: 1  x  1 ;

 2x  ?  5  Range:   y  

(B)

Domain: 1  x  1 ;

Range: 

(C)

Domain: 

5 5 x ; 2 2

Range: 

(D)

Domain: 

5 5 x ; 2 2

Range:   y  

What is the domain and range of y  sin 1 

 2

 2

 y  y

 2

 2

ANSWER: C

 2x   2x  y  sin 1    sin y     5   5 

Domain:  1  sin y  1 2x 1 5 5 5 x 2 2   Range:   y  as this is the range of 2 2 1 y  sin x 1 

4

sin3x . x0 2x

Evaluate lim (A) (B) (C) (D)

0 2 3 1 3 2

ANSWER: D

lim x 0

–3–

sin 3x 3 sin 3x 2  lim  2x 2 x 0 2 x 3 3 2sin 3x  lim 2 x 0 2  3 x 3  1 2 3  2

5

In the diagram below, AB is a tangent to the circle BCD. Also, CD is a tangent to the circle ABD.  BAD =  and  BCD =  NOT TO SCALE B

C A D Which of the following is a true statement? (A)

ABD  BDC

(B)

ABCD is a cyclic quadrilateral

ANSWER: C

(C)

ABD ||| BDC

(D)

AB || CD

In ABD and DCB : DCB  DBA (angle in the alternate segment) ie DCB  ABD BAD  BDC (angle in the alternate segment) ie BAD  CDB BD is but not respective to angles. Therefore ABD  DCB Hence, triangles are equiangular they are similar and ABD ||| DCB

6

A particle moves in simple harmonic motion so that its velocity, v, is given by

v2  6  x  x2 . Between which two points does it oscillate? (A)

x = 6 and x = 3

(B)

x = –2 and x = 3

(C)

x = 1 and x = 2

(D)

x = 2 and x = –3

ANSWER: D

v2  6  x  x2 For the particle to reach its oscillation points v  0. v2  6  x  x2 0  6  x  x2 0  (3  x)(2  x)  x  3 and 2 –4–

7

 Which of the following is an expression for  cos 3 x sin x dx ?  (A)

 cos 4 x  c

(B)

1  cos 4 x  c 4

(C)

cos 4 x  c

(D)

1 cos 4 x  c 4

ANSWER: B

 3  cos x sin x dx , testing solutions:  d cos 4 x   4 cos 3 x   sin x  dx 1 d   cos 4 x   cos 3 x sin x dx 4 1   cos 4 x   cos 3 x sin x dx 4 

8

Which of the following is the correct expression for the inverse of f (x)  e12 x ? (A) (B) (C) (D)

f 1 (x)  2e12x 1 f 1 (x)   e12 x 2 1 f 1 (x)   log e (1 2x) 2 1 f 1 (x)  (1 log e x) 2

ANSWER: D

Let y  e1 2 x

y  e12 x ln y  1  2 x 2 x  1  ln y 1 1  ln y  2 1  f 1 ( x)  1  ln y  2 x

–5–

9

Three Mathematics study guides, four Mathematics textbooks and five exercise books are randomly placed along a bookshelf. What is the probability that the Mathematics textbooks are all next to each other?

(A)

4! 12!

ANSWER: D

(B)

9! 12!

(C)

4!3!5! 12!

(D)

Since there are 9 elements counting the textbooks as 1 element, hence these can be arranged in 9! ways. Also the textbooks can be arranged in 4! ways. As there are 12 separate elements, the divisor for population can be counted in 12! ways.

4!9! 12!

Therefore, the probability is

10

9!4! 12!

A particle moves on the x-axis with velocity v m/s, such that v2 = 16x – x2. Which of the following is the particle’s maximum speed and the position of where this maximum speed occurs? (A)


(B)

Maximum speed = 8 m/s at x = –8

ANSWER: D

(C)

Maximum speed = –8 m/s at x = 8

(D)


v 2  16 x  x 2 x2 1 2 v  8x  2 2 2x d 1 2  v   8 2 dx  2   8 x  x  8 x x  0 the speed is the greatest so, When   x 8 x 0 8 x x 8 At x  8, v 2  16 x  x 2  16(8)   8 

2

 64 v  8 As v, velocity can take positive and negative values, but the speed can only be positive, the maximum speed is 8 m/s. –6–

(i) Most students realised they needed the auxiliary angle method. Common errors included: -evaluating tan incorrectly and having 1/sqrt(3) -not finding all the solutions in the given domain. (ii) Some had the incorrect formula.

An inaccurate graph resulted in the wrong number of solutions. Using Newton's Method done well on the whole. Some students did not use the given starting value and so were incorrect.

The most common mistake was not using the double angle formula correctly.

The most common error was to differentiate the given function in terms of t. Half a mark was deducted if no statement about sign of v was included.

Solutions Q13 X1 THSC 2015 Average mark: 11.31/15

0 6

0.5 18

1 138

Mean 0.91

0 7

0.5 1 1.5 2 2.5 3 3.5 4 Mean 6 11 14 25 8 20 22 49 2.70

0 3

0.5 5

1 1.5 11 4

2 2.5 3 3.5 4 Mean 20 15 25 24 55 2.93

0 2

0.5 8

1 2

1.5 3

2 0

2.5 4

3 Mean 143 2.77

0 15

0.5 10

1 21

1.5 4

2 24

2.5 23

3 65

Mean 2.05

Sydney Girls High School 2015 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Extension 1 General Instructions • • • • • •

Reading Time – 5 minutes Working time – 2 hours Write using black or blue pen Black 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

Total marks – 70 Section I

Pages 3 – 5

10 Marks • Attempt Questions 1 – 10 • Answer on the Multiple Choice answer sheet provided • Allow about 15 minutes for this section Section II

Pages 6 – 9

60 Marks • Attempt Questions 11 – 14 • Answer on the blank paper provided • Begin a new page for each question • Allow about 1 hour and 45 minutes for this section

THIS IS A TRIAL PAPER ONLY Name: ……………………………………………………………………..……..

Teacher: ………………………………………………………………………...

It does not necessarily reflect the format or the content of the 2015 HSC Examination Paper in this subject.

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. A committee of six is to be formed from seven women and nine men. Find the number of committees possible if exactly two members of the committee are to be men. (A)

1260

(B)

2646

(C)

36 036

(D)

60 480

85° . What is the size of angle ∠ACB ? 34° and ∠ADE = 2. In the diagram below ∠BAC =

(A)

51°

(B)

56°

(C)

61°

(D)

60°

ex 3. Use the substitution u = e x to determine which of the following is an expression for ⌠ dx .  ⌡ 1 + e2 x

(A)

tan −1 ( e x ) + C

(B)

tan −1 ( e 2 x ) + C

(C)

(D)

−1

2 (1 + e x )

−e x

(1 + e )

x 2

2

+C

+C

~3~

4. The radius of a spherical balloon is increasing at the rate of 2 cm/s. The rate at which the volume of the balloon is increasing when the radius is 10 cm is : (A)

200π cm3 / s

(B)

400π cm3 / s

(C)

800π cm3 / s

(D)

100π cm3 / s

5. A stone is thrown vertically upwards with a speed of 21 m/s. How long is the stone in the air before it reaches its maximum height? (Assume acceleration due to gravity is 10 m/s2.) (A)

4.2 s

(B)

0.48 s

(C)

0.95 s

(D)

2.1 s

6. The polynomial equation f ( x ) = x3 + x − 1 has a root near x = 0.5 . Using this as the initial approximation, determine another approximation (correct to four decimal places) to the root using one application of Newton’s method. (A)

x = 0.7141

(B)

x = 0.7142

(C)

x = 0.7143

(D)

x = 0.7144

dy a 7. If y = sin −1   , then = dx x

(A)

(B)

(C)

(D)

−a x2 − a2 a x2 − a2 −a x x2 − a2 a x x2 − a2

~4~

20

8. What is the value of

∑ k =1

(A)

1 048 574

(B)

1 048 575

(C)

1 048 576

(D)

1 048 577

20

Ck ?

9. Given that a , b and c are the roots of the equation x 3 − 3 x 2 + mx + 24 = 0 , and that − a and −b 2 are the roots of the equation x + nx − 6 = 0 , then the value of n is : (A)

1

(B)

−1

(C)

7

(D)

−7

10. The sum 14 + 24 + 34 + 44 + ... + n 4 is given by the expression The value of a − b is :

(A)

−25

(B)

25

(C)

−5

(D)

5

~5~

6n5 + an 4 + bn3 − n . 30

Section II Total 60 marks Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section Answer all questions, starting each question on a new page. In Questions 11–14, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (a) Solve

Marks

(15 marks)

2 ≤3 . x −5

3

x (b) Sketch the function f ( x ) = π cos −1   , clearly indicating the domain and range of 2

3

the function. (c) The velocity of a particle when x m from the origin is given by= v 2 x 2 e3 x + 4 .

2

Find the acceleration of the particle when x = 1 .

(d) Find the general solution to the equation

(e) Find the value of

3 tan θ + 1 = 0.

lim 5 x + sin 3 x . x→0 2x

2

2

(f) In the diagram, AOB is the diameter of

3

the circle with centre at O . TC is a tangent to the circle at the point C such that AC bisects ∠TAB . Copy the diagram onto your writing paper. Prove that AT is perpendicular to TC .

End of Question 11 ~6~

Question 12

Marks

(15 marks)

(a) By considering the derivative of ln ( tan x ) , find ∫ cosec 2x dx .

3

10

 2  (b) In the expansion of  x + 2  , find the coefficient of x . x  

3

(c) P ( 4 p, 2 p 2 ) and Q ( 4q, 2q 2 ) are two variable points on the parabola x 2 = 8 y . The tangents at P and Q intersect at the point T . (i)

Derive the equation of the tangent at P .

1

(ii)

Hence, show that the point T has the coordinates ( 2 ( p + q ) , 2 pq ) .

2

(iii)

Given that p 2 + q 2 =, 10 determine the cartesian equation of the locus of T .

2

(d) Find

∫ ( sin

2

x + 2 cos 2 x + 3 tan 2 x ) dx .

2

1 (e) Prove that tan −1 ( x + 1) + cot −= x tan −1 ( − x 2 − x − 1) for x > 0 .

End of Question 12

~7~

2

Question 13

Marks

(15 marks)

(a) The growth rate per month of the number N of birds on a property during a drought is −20% of the excess of the bird population over 1000. (i)

Express the information in the form of a differential equation and show that = N 1000 + Ae

−0.2 t

2

(where t is the time in months) is a solution to this

differential equation. (ii)

Given that initially there are 8000 birds on the property, find the amount of time that will elapse before the population is reduced to half.

(b) Ansett Airlines offer two options on all flights for their meal service – chicken or

2

2

beef (vegetarians choose not to fly with Ansett). If 60% of the time Ansett passengers select the chicken dish, what is the probability that out of 7 randomly selected passengers at least 2 will select chicken for their meal?

(c) An iPhone is thrown from the top of a building, 6 metres high, with an initial velocity of 8 m/s at an angle of 30° to the horizon. (i)

Using 10 m/s2 for acceleration due to gravity, derive the horizontal and

2

vertical equations of motion for the iPhone. (ii)

Determine the greatest height of the iPhone above ground level.

(iii)

Find the velocity and direction of the iPhone’s path after 1 second.

(d) Prove the following statement is true by mathematical induction for all integers n ≥ 1 . n

∑ r ( r !=) ( n + 1)! − 1

r =1

End of Question 13

~8~

2 2

3

Question 14

Marks

(15 marks)

(a) A particle moves in a straight line with simple harmonic motion. At time t seconds, π  its displacement x metres from a fixed point O , is given by x = 2 + 5sin  3t +  . 4  −9 ( x − 2 ) . (i) Show that x = (ii) Determine the maximum speed of the particle and its displacement at this time.

(b) How many different arrangements of the word MAMMOTH can be made if only five

1 2

2

letters are used? (c) Use the substitution u 2= x + 1 to find the volume of the solid formed by rotating the x −1 area bounded by the curve y = , the x axis and the lines x = 3 and x = 8 x +1 about the x axis. Express your answer in exact form. (d) Use the expansion of (1 + x ) to prove that

4

n

n + ( −1) ( −1) 1n 1 = C1 − nC2 +  + n +1 2 3 n n

3 n n

Cn −1 .

(e) Given that f ( x ) = Ax3 + Bx 2 + Cx + D is a function with a double zero at x = 1 , and with a minimum value of − 4 when x = −1 , find the values of A, B, C and D .

End of paper

~9~

3

CANDIDATE NUMBER

SYDNEY GRAMMAR SCHOOL

2015 Trial Examination

FORM VI MATHEMATICS EXTENSION 1 Wednesday 5th August 2015

General Instructions

Collection

• Writing time — 2 hours

• Write your candidate number on each answer booklet and on your multiple choice answer sheet. • Hand in the booklets in a single wellordered pile.

• Write using black or blue pen.

• Board-approved calculators and templates may be used. • A list of standard integrals is provided at the end of the examination paper. Total — 70 Marks • All questions may be attempted. Section I – 10 Marks • Questions 1–10 are of equal value. • Record your answers to the multiple choice on the sheet provided. Section II – 60 Marks

• Hand in a booklet for each question in Section II, even if it has not been attempted. • If you use a second booklet for a question, place it inside the first. • Write your candidate number on this question paper and hand it in with your answers. • Place everything inside the answer booklet for Question Eleven.

• Questions 11–14 are of equal value. • All necessary working should be shown. • Start each question in a new booklet.

Checklist • SGS booklets — 4 per boy

• Multiple choice answer sheet • Candidature — 112 boys

Examiner PKH

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 2 SECTION I - Multiple Choice Answers for this section should be recorded on the separate answer sheet handed out with this examination paper.

QUESTION ONE Which of the following is an odd function? (A) f(x) = tan−1 x (B) f(x) = cos x (C) f(x) = sin(x −

π ) 4

(D) f(x) = cos−1 x

QUESTION TWO Suppose θ is the acute angle between the lines y − 2x = 3 and 3y = −x + 2. Which of the following is the value of tan θ? (A) 7 (B) −7 (C) 1 (D) −1 QUESTION THREE

B

A

C

70º O

What is the size of 6 ABC? (A) 110◦ (B) 145◦ (C) 140◦ (D) 130◦ Exam continues next page . . .

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 3 QUESTION FOUR What is the inverse function of f(x) = x2 + 1 for x ≤ 0? √ (A) f −1 (x) = − x − 1, for x ≤ 0 √ (B) f −1 (x) = x − 1, for x ≤ 0 √ (C) f −1 (x) = − x − 1, for x ≥ 1 √ (D) f −1 (x) = x − 1, for x ≥ 1 QUESTION FIVE x2 + x − 6 . Find lim x→2 x−2 (A) ∞ (B) −∞ (C) −5 (D) 5

QUESTION SIX

x 5 7

Find the length of x. √ (A) 35 √ (B) 12 √ (C) 60 √ (D) 84

Exam continues overleaf . . .

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 4 QUESTION SEVEN 1 If f(x) = tan−1 , find f 0 (x). x x2 (A) 1 + x2 (B) − (C)

1 1 + x2

1 1 − x2

(D) −

x2 1 − x2

QUESTION EIGHT 1 How many solutions does the equation x 3 = |x − 2| − 3 have? (A) 0 (B) 1 (C) 2 (D) 3 QUESTION NINE The parametric form of a parabola is (6t, −3t2 ). Its focal length is: (A)

1 4

(B) −

1 4

(C) −3 (D) 3 QUESTION TEN The polynomial P (x) has degree 4 and the polynomial Q(x) has degree 2. If you divide P (x) by Q(x), the remainder has degree: (A) 1 (B) 2 (C) 0 or 1 (D) 0, 1 or 2 End of Section I Exam continues next page . . .

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 5 SECTION II - Written Response Answers for this section should be recorded in the booklets provided. Show all necessary working. Start a new booklet for each question.

(15 marks) Use a separate writing booklet.

QUESTION ELEVEN

(a) Let A = (−1, 4) and B = (5, −5). Find the co-ordinates of the point P which divides interval AB in the ratio 1 : 2. (b) Solve the inequation

x < 2. 2x + 1

Marks

2

2

(c) Sketch the graph of y = 2 cos−1 (x − 1), clearly marking the domain and range.

2

(d) Differentiate etan x ln x.

2

(e) Find the coefficient of a3 in the expansion of (2a − 1)20 .

2

(f) Taking x = 1·4 as a first approximation, use one application of Newton’s method to find a better approximation to 3 sin 2x − x = 0. Give your answer correct to 3 significant figures.

2

(g) (i) Prove that

sin 2A = cot A. 1 − cos 2A

(ii) Hence find the values of a and b if cot

1 √ 3π = a + b for integers a and b. 8


2

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 6 QUESTION TWELVE


(a) Use the substitution u = tan x to evaluate

Z

sec2 x dx . tan2 x + 3

(b) Prove by Mathematical Induction that, for n ≥ 1,

Marks

2 3

2 × 21 3 × 22 n 2n−1 2n 1 1 × 20 + + + ... + = − . 2×3 3×4 4×5 (n + 1)(n + 2) n+2 2

√ 3 1 (c) Find the area bounded by y = √ , the line x = 0, the line x = and the 6 1 − 9x2 x-axis. (d) Consider the function y = x2 + (i) Find

3

16 . x

dy . dx

1

(ii) Find the co-ordinates of any stationary points and determine their nature.

2

(iii) Show that there is a point of inflexion at the x-intercept.

2

(iv) Sketch the graph y = x2 +

16 , showing the above information. x

Exam continues next page . . .

2

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 7 QUESTION THIRTEEN


sin ax . x→0 x

(a) Find lim

Marks

1

(b) (i) Show that x ¨=

d 1 2 ( v ). dx 2

1

(ii) If v 2 = 24 − 6x− 3x2, find the acceleration of the particle at the particle’s greatest displacement from the origin.

3

(c) Let α, β and γ be the roots of the equation x3 − px + q = 0. In terms of p and q find 1 1 1 an expression for + + . α β γ

2

(d) Show that tan−1 1 + tan−1 2 + tan−1 3 = π.

2

(e)

3

N Q

xm

q R

120 m

P

E

An observer stands at P , 120 metres East of R. A second person is at Q, x metres due North of R and continues to move North. Let angle RP Q = θ. Suppose θ is changing at 0·2 radians/minute. Find the rate at which x is changing when x = 90 metres.


SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 8

D

(f)

A

B

O

S C

P Two diameters AB and CD of a circle, with centre O, are at right angles. Diameter DC is produced to P and P B cuts the circle again at S. (i) Prove that AOSP is a cyclic quadrilateral.

1

(ii) Prove that 6 BCS = 6 SP O.

2

Exam continues next page . . .

SGS Trial 2015 . . . . . . . . . . . . . . Form VI Mathematics Extension 1 . . . . . . . . . . . . . . Page 9 (15 marks) Use a separate writing booklet.

QUESTION FOURTEEN

Marks

1

(a) Consider the function f(x) = ln(x x ), for x > 0. (i) Show that f 0 (x) =

1 (1 − ln x). x2

2

(ii) Find the range of f(x), giving full reasons. (b)

3

y q

h

x

O

R

A projectile is fired from the top of a cliff of height h above a horizontal plane with initial speed V at an angle of elevation θ. The horizontal range of the projectile is R. The magnitude of the gravitational acceleration of the projectile is g. Take the origin at the base of the cliff directly below the launch point of the projectile. It is known that the vertical and horizontal displacements satisfy x = V cos θ t

and

y = h + V sin θ t − 12 gt2 .

(i) Show that the Cartesian equation of motion is y = h + x tan θ −

2

gx2 sec2 θ. 2V 2

V2 V2 tan θ − 2h = 0. g g 4 2 V V2 V2 2 (iii) Show that R = + 2h − R tan θ − . g2 g g (ii) Show that R2 sec2 θ − 2R

(iv) Deduce that the maximum range is

1p 4 V + 2hV 2 g . g

(v) Show that the angle of elevation satisfies tan θ = range. (vi) Show that tan 2θ =

V2 where R1 is the maximum gR1

R1 . h

2 2 1 1

2 End of Section II

END OF EXAMINATION

2015_3u_papers

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