Extension 1 Mathematics

General Instructions

• Reading time – 5 minutes

• Working time – 2 hours

• Write using black pen

• Board-approved calculators maybe used

• A reference sheet is provided atthe back of this paper

• In Questions 11–14, show relevantmathematical reasoning and/orcalculations

Total marks – 70

Section I Pages 1–3

10 Marks

• Attempt questions 1–10• Allow about 15 minutes for thissection

Section II Pages 4–10

60 Marks

• Attempt questions 11–14• Allow about 1 hour and 45 minutesfor this section

Section I

10 marks

Attempt Questions 1–10

Allow about 15 minutes for this section

1. Which expression is the correct factorisation of x3 � 8(A) (x� 2)(x2 � 2x+ 4)(B) (x� 2)(x2 � 4x+ 4)(C) (x� 2)(x2 + 2x+ 4)(D) (x� 2)(x2 + 4x+ 4)

2. A particle is moving in simple harmonic motion with displacement x. Its velocityv is given by

v2 = 9(4� x2)

What is the amplitude, A, and the period, T , of the motion?

(A) A = 2 and T =⇡

3

(B) A = 2 and T =2⇡

3

(C) A = 3 and T =⇡

3

(D) A = 3 and T =2⇡

3

3. What is the derivative of sin�1(3x2)

(A)6xp

1� 9x4

(B)�6xp1� 9x4

(C)x

6p1� 9x4

(D)�x

6p1� 9x4

1

4. The polynomial P (x) = x3 � 5x2 + 7x+ k has a factor x� 1. What is the valueof k?

(A) �3(B) �1(C) 4

(D) 7

5. What is the value of the following definite integral

Z 10

�10xe�x

4dx

(A) �e20

(B) 0

(C) 1

(D) e20

6. Which expression is equal top3 cosx� sin x

(A)p2 cos

⇣x+

⇡

6

⌘

(B)p2 cos

⇣x� ⇡

6

⌘

(C) 2 cos⇣x+

⇡

6

⌘

(D) 2 cos⇣x� ⇡

6

⌘

7. The letters of the word SMARTER are arranged in a row. Find the probabilitythat there are three letters between S and T

(A)1

8

(B)1

7

(C)1

4

(D)1

3

2

8. Which expression is the correct factorisation of x3 � 8(A) (x� 2)(x2 � 2x+ 4)(B) (x� 2)(x2 � 4x+ 4)(C) (x� 2)(x2 + 2x+ 4)(D) (x� 2)(x2 + 4x+ 4)

9. A particle is moving in simple harmonic motion with displacement x. Its velocityv is given by

v2 = 9(4� x2)

What is the amplitude, A, and the period, T , of the motion?

(A) A = 2 and T =⇡

3

(B) A = 2 and T =2⇡

3

(C) A = 3 and T =⇡

3

(D) A = 3 and T =2⇡

3

10. What is the derivative of sin�1(3x2)

(A)6xp

1� 9x4

(B)�6xp1� 9x4

(C)x

6p1� 9x4

(D)�x

6p1� 9x4

3

Section II

60 marks

Attempt Questions 11–14

Allow about 1 hour and 45 minutes for this section

Answer each question in the appropriate writing booklet. Extra writing bookletsare available.

In Questions 11–14, your responses should include relevant mathematical reasoningand/or calculations.

Question 11 (15 marks)

(a) 1Sketch the graph of y = |2x� 1|.

(b) 2Hence, or otherwise, solve |2x� 1| |x� 3|.

(c) 2The interval AB, where A is (4, 5) and B is (19,�5), is divided internally in theratio 2 : 3 by the point P (x, y). Find the values of x and y

(d) 2Calculate the size of the acute angle between the lines y = 2x+5 and y = 4�3x.

(e) 3Evaluate Zsec7 x tan x dx

using the substitution u = sec(x)

(f) A particle moving in a straight line has the equation of motion:

v2 = �8x2 � 32x� 7

(i) 2Show that the particle is moving in Simple Harmonic Motion

Hence, find the following:

(ii) 1The centre of motion

(iii) 1The amplitude of motion

(iv) 1The period of motion

4

Question 12 (15 marks)

(a) Suppose f(x) = ex � e�x

(i) 2Show that f(x) is increasing for all x

(ii) 1Using part (i) or otherwise explain why f(x) has an inverse

(iii) 3Find an expression for f�1(x)

(iv) 1Findd

dx

�f�1(x)

�

(v) 1Hence find Zdxpx2 + 1

(b) 2Use two applications of Newton’s method to solve cosx = x to 3 decimal places,with an initial estimate of 0.5.

x

y

O

Question 12 Continues on Next Page

5

(c) The diagram shows the parabola x2 = 4ay. The point P (2ap, ap2), where p 6= 0,is on the parabola.

The tangent to the parabola at P , meets the y-axis at I. S is the focus of theparabola. The point closest to P , that lies on the directrix is M .

x

y

O

I

P (2ap, ap2)

M

(i) 2Find the equation of the tangent at P and the coordinates of I

(ii) 3Show that PI bisects the angle SPM

End of Question 12.

6

Question 13 (15 marks)

(a) Water is projected horizontally, at speed vms�1 from the top of a building. Thewater strikes the ground at an angle of 30 degrees, L metres from the base ofthe tower as shown in the diagram.

The equations that describe the trajectory of the ball are x = vt and y = h�12gt2

(DO NOT PROVE THESE)

L

30�

(i) 1Prove that the ball strikes the ground at time t =

r2h

g.

(ii) 2Hence or otherwise show that d = 2p3h.

(iii) 1How high up the building must the fireman start the hose, if he wishes toput out a fire 13 metres away from the base of the building. Give youranswer to the nearest metre.

(b) It is assumed that the temperature of tea will decay in proportion to the dif-ference between the current temperature T and the ambient temperature T

a

,i.e,

dT

dt= �k(T � T

a

)

(i) 1Show that T = Ta

+ Ae�kt satisfies the di↵erential equation, where t is thetime in minutes, and T

a

and A are constants.

(ii) 3Andre, Tatjana, and Misaki have tea bags immersed in hot water at 100�C.The room temperature is 25�C. The tea is brought o↵ the boil at t = 0.Misaki’s tea has a temperature of 50�C after 7 minutes.

Find the constants Ta

, A and k.

(iii) 2Andre and Tatjana both prefer to have milk in their tea. Andre decidesto add milk to his tea immediately after it is o↵ the boil. Tatjana insteaddecides to wait for two minutes before adding her milk.

At T = 10 minutes whose drink will be the warmest, and whose will be thecoldest? Justify your answer mathematically. You may assume that addingmilk results in a constant temperature drop �T

m

.

Question 13 Continues on Next Page

7

(c) Suppose N and k are integers.

• An integer N is even if it can be written in the form 2k, e.g. 6 = 2 ⇥ 3,hence 6 is even

• An integer N is odd if it can be written in the form 2k+1, e.g. 5 = 2⇥2+1,hence 5 is odd

(i) 2Show that if a number N is an integer, then N(N + 3) is always even. Itwill be helpful to consider the following two cases:• Case 1: N is even, i.e. N = 2k• Case 2: N is odd, i.e. N = 2K + 1

(ii) 3Show that n3+3n2+2n is divisible by 6 using the principle of mathematicalinduction, and the results of part (i).

End of Question 13.

8

Question 14 (15 marks)

(a) Three cards are dealt from a pack of 52.

(i) 2Find the probability that one club and two hearts are dealt, in any order.

(ii) 2Find the probability that one club and two hearts are dealt in that order.

(b) In the circle below, the chord AC is a diameter, B is a point anywhere on thediameter. O is the centre of the circle.

The chord DE passes through B at an angle 45 degrees to the chord AC, asshown below.

The chord D0E 0 is a reflection of DE along the chord AC.

A C

D

E

D0

E 0

B

45�

O

(i) 2Show that angle DOE 0 is a right angle.

(ii) 3Hence or otherwise show that AC2 = 2DB2 + 2BE2

Question 14 Continues on Next Page

9

(c) It can be easily shown that

(z � 1)�1 + z + z2 + z3 + · · ·+ zn�1

�= zn � 1

(i) 1By making the appropriate substitution or otherwise show that

xn�1X

k=0

(1 + x)k =

"nX

k=0

✓n

k

◆xk#� 1

(ii) 1Hence show that for 1 k n✓k � 1k � 1

◆+

✓k

k � 1

◆+

✓k + 1

k � 1

◆+ · · ·+

✓n� 1k � 1

◆=

✓n

k

◆

(iii) 1Explain why the expression in part (ii) is only valid for 1 k n

(iv) 1By expressing the binomial coe�cients in factorials or otherwise, show that

n

✓n� 1k

◆= (k + 1)

✓n

k + 1

◆

(v) 2By di↵erentiating both sides of the identity in part (i), show that

k

✓k � 1k � 1

◆+ (k + 1)

✓k

k � 1

◆+ · · ·++(n� 1)

✓n� 2k � 1

◆= k

✓n

k + 1

◆

End of Exam Paper

10

– 2 –

Mathematics

– 3 –

Mathematics (continued)

– 4 – © 2016 Board of Studies, Teaching and Educational Standards NSW

Mathematics Extension 1