2015 Bored of Studies Trial Examinations

Mathematics Written by Carrotsticks and Trebla.

General Instructions

Reading time – 5 minutes.

Working time – 3 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.

Show all necessary working in

Questions 11 – 16.

Total Marks – 100

Section I Pages 1 – 5

10 marks

Attempt Questions 1 – 10

Allow about 15 minutes for this section.

Section II Pages 6 – 18

90 marks

Attempt Questions 11 – 16

Allow about 2 hours 45 minutes for this section.

– 1 –

Shade your answers in the appropriate box in the Multiple Choice answer sheet provided.

1 Let 1100

nr

A P

. If 16000A , 1000P and 4n , then which of the following

values of r satisfies the equation?

(A) 300 .

(B) 1 .

(C) 1.

(D) 300 .

2 Which of the following statements is always true?

(A) An isosceles triangle, which has an angle of 45 , is a right angled triangle.

(B) Two triangles, which have two corresponding sides equal and one corresponding

angle equal, are congruent.

(C) A quadrilateral which has its diagonals bisecting each other is a parallelogram.

(D) The interior angle of a hexagon is 120 .

3 Which of the following two functions do NOT satisfy f x g x ?

(A) 2

sin cosf x x x and 2sin cosg x x x .

(B) 2tanf x x and 2secg x x .

(C) logef x x and log 5eg x x .

(D) 2sinf x x and 2cosg x x .

Total marks – 10

Attempt Questions 1 – 10

All questions are of equal value

– 2 –

4 Let f x be a continuous function with a derivative at 0x x .

Which of the following is the correct expression for 0f x ?

(A) 0 0

0limh

f x h f x

h

.

(B)

0

0 0

00

limx

f x h f x

x

.

(C) 0

0limh

f x h f h

h

.

(D)

0

0

00

limx

f x h f h

x

.

5 Let f x be a quadratic polynomial with roots and .

Which of the following statements is always true?

(A) 0f .

(B) 02

f

.

(C) 0f .

(D) 02

f

.

– 3 –

6 Consider two lines with equations 1 1 1 0a x b y c and

2 2 2 0a x b y c , where all

coefficients are non-zero.

Which of the following statements about the lines is FALSE?

(A) If 1 2 2 1 0a b a b , then the lines are parallel.

(B) If 1 2a a ,

1 2b b and 1 2c c , then the lines cannot coincide.

(C) If 1 2 1 2 0a a bb , then the two lines intersect at right angles.

(D) If 1 2a a ,

1 2b b and 1 2c c , then the lines must coincide.

7 Consider the region bounded by the curves 2y x , 2

2y x and the y axis.

Which of the following regions defined below have the same area as ?

(A) 2y x , 2

2y x and 0y .

(B) 2y x , 2

2y x and 0y .

(C) 4 4y x , 0 1x and 0y .

(D) 4 4y x , 0 1x and 0y .

– 4 –

8 Let f t be a continuous function in the interval 0 t T .

A bucket of water has a hole and continues to leak water until it is empty. Initially, it has

V litres of water and it is empty after T seconds. The rate at which water leaks out at any

given time t is f t litres per second.

Which of the following statements is correct?

(A) The volume of water remaining in the bucket after 0t seconds, where

00 t T ,

is given by 0

0

.

t

f t dt

(B) The initial volume of water can be expressed as 0f T f .

(C) The volume of water at any time t is given by f t .

(D) The rate that the bucket loses water is zero, just before it is empty.

9 Let ,P x y be a point on the parabola 2 4y ax with focus S. Let M be the foot of the

perpendicular from P to the directrix.

Which of the following coordinates of P satisfy PS PM ?

(A) 4 ,a a .

(B) , 4a a .

(C) 2 ,a a .

(D) , 2a a .

– 5 –

10 In an experiment, the probability of outcome A occuring is P A and the probability of

outcome B occuring is P B . The two outcomes are mutually exclusive events.

Which of the following is always true?

(A) The probability of A and B occuring together is 0.

(B) The probability of A and B occuring together is P A P B .

(C) The probability of A and B occuring together is P A P B .

(D) The probability of A and B occuring together is 1.

– 6 –

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

In Questions 11 – 16, your responses should include relevant mathematical reasoning and/or

calculations.

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

(a) Solve the following equation for x. 2

2 3 4log log logx x x

(b) Show that 2

sin 0 sin1 sin 2 sin 3 ... sin 901

cos 0 cos1 cos 2 cos3 ... cos90

.

(c) The rate that a population P grows at any time t is given by

dPkP

dt ,

where k is a positive constant.

At time 0t t , the population is

0aP , where a is a positive constant and 0P is

the initial population.

(i) Verify that 0

ktP P e satisfies the differential equation. 1

(ii) Hence, show that 0

0

t

tP P a . 1

Question 11 continues on page 7

Section II

90 marks

Attempt Questions 11 – 16

Allow about 2 hours and 45 minutes for this section

– 7 –

Question 11 (continued)

(d) Solve the following equation for 0 2x . 3

3 3sin cos sin cosx x x x .

(e) On the number plane, sketch the set of all points ,P x y that are equidistant 3

from both 0ax by and 0bx ay .

(f) Let 2 2 0x xy y , where 0x y . 3

Simplify

2015 2015

x y

x y x y

.

End of Question 11

– 8 –

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

(a) Show that log logb bx aa x . 2

(b) The diagram below shows the point 6, 2A on the number plane. 4

The point B is chosen in the first quadrant so that ABO is an isosceles

right angled triangle, as shown in the diagram below.

Find the coordinates of B.

(c) Let tan2

xf x

.

(i) Sketch the graph of y f x for x . 2

(ii) The region bounded by the curve y f x , the y axis and the line 3

1y is rotated about the x axis to form a solid.

Find the volume of the solid.

Question 12 continues on page 9

O

B

A

y

x

– 9 –

Question 12 (continued)

(d) The line y mx b is a tangent to the parabola 2 4x ay , where 0a .

(i) Show that 2 0am b . 2

(ii) Hence, or otherwise, find the equations of possible tangents to the 2

parabola which have their y intercepts on the directrix.

End of Question 12

– 10 –

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

(a) Let p and q be any real numbers.

(i) Show that if p q p q , then 0pq . 2

(ii) Hence, or otherwise, find the set of values of x such that 2

x a x b a b ,

where a b .

Question 13 continues on page 11

– 11 –

Question 13 (continued)

(b) The diagram below shows the point ,T p q in the first quadrant.

A monic quadratic polynomial P x with two real roots has coefficients in terms of p

and q. Let the real roots be and , where 0 .

When the two real roots are plotted on the x axis, they form an equilateral triangle with

the point T.

(i) Show that 3

qp and state a similar expression for . 3

(ii) Hence, show that the quadratic polynomial is 2

2

2 223

qpx xx pP

.

Question 13 continues on page 12

x

y

– 12 –

Question 13 (continued)

(c) A parallelogram ABCD has three of its sides AB, BC and AD also being sides of

equilateral triangles ABZ, BCQ and ADP, as shown in the diagram below.

Let X and Y be the centres of ADP and BCQ respectively.

(i) Prove that XAZ YBZ . 4

(ii) Deduce that XYZ is equilateral. 2

End of Question 13

A

X

Y

Z

B

C D

P

Q

– 13 –

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

(a) A parallelogram ABCD has diagonals AC and BD intersecting at the point O.

Let AOD .

(i) Show that 1

2 2 2 2 cosAB OB OA OA OB .

(ii) Hence, show that 3

2 2 2 2 2 2A BC CD AD DB AC B .

(b) Find the domain and range of 21f x x x . 3

Question 14 continues on page 14

B A

C D

O

– 14 –

Question 14 (continued)

(c) The diagram below shows the point 1 3

,2 2

on the semi-circle 21y x .

Let be the angle that the point 1 3

,2 2

makes with the positive x axis.

(i) Find the value of . 1

(ii) By considering the areas of segments, show that 3

1

2

1 2

4 3 31 .

24x dx

(iii) Use the trapezoidal rule with three function values to estimate 3

1

2

1 2

1 ,x dx

and hence obtain an approximation for , correct to four significant

figures.

Question 14 continues on page 15

x

y

O

– 15 –

Question 14 (continued)

(iv) Jennifer uses the trapezoidal rule with three function values to estimate 1

the value of

1 2

2

0

1 x dx

and similarly to part (iii), she obtains another approximation for .

Yvonne thinks that her estimation of , using the result of part (iii), is

more accurate.

Without explicitly calculating her approximation, explain why Jennifer

has the more accurate estimation of .

End of Question 14

– 16 –

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

(a) Let 2n 1lf x x x .

(i) Show that f x is an odd function. 3

(ii) Show that f x has no stationary points. 2

(iii) Find the coordinates of the point of inflexion. 2

(iv) Hence, sketch the graph of y f x . 2

(b) Show that for integer values of 2n , 3

1 1 1 1 1...

1 2 1 2 3 1 2 3 4 1 2 3 ... 1

n

n n

.

(c) Player A and B compete against each other in a series of games. 3

Player A has probability p of winning each game and player B has

probability q of winning each game

To win a series, a player must win two games in a row.

Show that the probability of Player A winning the series is

2 1

1

p q

pq

.

End of Question 15

– 17 –

Question 16 (15 marks) Use a SEPARATE writing booklet.

(a) Eric takes out a loan of P dollars and makes monthly repayments of $M at the end of

each month at 100 %r per month, compounded monthly.

Let n be the number of months such that the loan is completely repaid, where n is an

even number of months.

(i) Show that 2

1 1

1n

n

r PM

r

r

.

(ii) After m months into the course of the repayment, Eric receives 4

a one-off lump sum of $B, which is put entirely into the loan.

He then continues with the monthly repayments of $M until the

loan is completely repaid.

The total time taken to repay the loan, with the inheritance, was halved.

Show that

21log 1 1

n

r

Brm

P

.

Question 16 continues on page 18

– 18 –

Question 16 (continued)

(b) Prove that if lny f x attains a local maximum at x , then y f x 3

also attains a local maximum at x .

(c) A square with side length d is drawn on a number plane such that three of its points

00

2,x x , 11

2,x x and 22

2,x x , where 0 1 2x x x , lie on the parabola 2y x , as

shown in the diagram below.

Let be the acute angle of inclination of the square with the horizontal, as shown

above, where 02

.

(i) Show that

1

sin cos sin cosd

. 3

(ii) Use part (b), or otherwise, to show that the minimum area of the 3

square ABCD is 2 square units.

You do not need to verify that it is a minimum.

End of Exam

y

x

O

– 19 –

STANDARD INTEGRALS

1

2

1

2 2

2 2

1,

1

1ln , 0

1,

1cos sin ,

1sin cos ,

1sec tan ,

1sec tan sec ,

1 1tan

1; 0, if 0

,

1

0

0

n

0

0

0

0

si

n n

ax ax

x dx x nn

dx x xx

e dx e aa

ax dx ax aa

ax dx ax aa

ax dx ax aa

ax ax dx ax aa

xdx a

a x a a

dxa

n

x

x

1

2 2

2 2

2 2

2 2

,

1

0

ln ,

,

0

1ln

a x a

x

xa

a

dx x x ax a

dx xx a

a

x a

NOTE: ln log , 0ex x x

© Bored of Studies NSW 2015