2013澳大利亚高考数学试题答案

2013

HIGHER SCHOOL CERTIFICATE

EXAMINATION

Mathematics Extension 2

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 In Questions 11–16, show

relevant mathematical reasoning and/or calculations

Total marks – 100 Section I

Pages 2–6

10 marks

Attempt Questions 1–10

Allow about 15 minutes for this section Section II

Pages 7–17

90 marks

Attempt Questions 11–16

Allow about 2 hours and 45 minutes for this section

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

Which expression is equal to tan xdx?

(A) sec2 x+c

(B) –ln(cos x)+c

2 tan x(C) +c

2

(D) ln ( sec x+tann x)+ c

2

Which pair of equations gives the directrices of 4x2 – 25y2= 100? (A)

x =(B) x =±1 29

(C) x =±29 (D) x =±29

25

3 The Argand diagram below shows the complex number z.

Which diagram best represents z2?

(A) (B)

(C) (D)

4

The polynomial equation 4x3 + x2 3x + 5 = 0 has roots α, β and γ . Which polynomial equation has roots α+ 1, β+ 1 and γ+ 1? (A) 4x3 11x2 + 7x + 5 = 0 (B) 4x3 + x2 3x + 6 = 0 (C) 4x3 + 13x2 + 11x + 7 = 0 (D) 4x3 2x2 2x + 8 = 0

5 Which region on the Argand diagram is defined by

π π ≤ z 1≤ ? 4 3

6

1

dx ?Which expression is equal to

x 2 6x +5 x 3

(A) sin 1 +C 2 x 3 (B) cos 1 +C

2

(C) ln x 3 +

(D) ln x 3 +

+Cx 3 )2 +4 2

x 3 4 +C )

7 The angular speed of a disc of radius 5 cm is 10 revolutions per minute. What is the speed of a mark on the circumference of the disc? (A) 50 cm min–1 (B)

1

cm min–1 2

(C) 100π cm min–1 (D)

1

cm min–1 4π

8

The base of a solid is the region bounded by the circle x2 + y2= 16. Vertical

Which integral represents the volume of the solid?

(A) 4x2 dx

4

(B) 4π x2 dx

4

(C) 416 x2dx

4

(D) 4π16 x2dx

4

44 4

4

()

()

9 Which diagram best represents the graph y =

sin x

?

10 A hostel has four vacant rooms. Each room can accommodate a maximum of four people. In how many different ways can six people be accommodated in the four rooms? (A) 4020 (B) 4068 (C) 4080 (D) 4096

Section II

90 marks

Attempt Questions 11–16

Allow about 2 hours and 45 minutes for this section

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.

w =1 +i3.(a) z =2 i3 (i) z+ .

(ii) Express w in modulus–argument form. (iii) Write w24 in its simplest form.

12 2

(b) Find numbers A, B and C such that 2

= A x 3

+ Bx +C x+2

2

x 3 x+2 2

x 2+8x +11

.

(c)

Factorise z2 + 4iz + 5.

2

32

(d) Evaluate x 1 xdx.

0

1

3

(e) Sketch the region on the Argand diagram defined by z2 +2 ≤8.

3

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

(a) Using the substitution t = tanx π

2

1

2 , or otherwise, evaluate

dx . 0 4+5cos x

(b) The equation logx

e y log e

(1000 y)= 50

log e3 implicitly defines y as a function of x.

Show that y satisfies the differential equation

dy dx = y 50 y 1 1000

. (c) The diagram shows the region bounded by the graph y = ex, the x-axis and the

lines x = 1 and x = 3. The region is rotated about the line x = 4 to form a solid.

Find the volume of the solid.

Question 12 continues on page 9

4

2

4

Question 12 (continued)

(d) The points P cp, c and Q

cq, c

, where p ≠q , lie on the rectangular

hyperbola with equation xy = c2.

The tangent to the hyperbola at P intersects the x-axis at A and the y-axis at B. Similarly, the tangent to the hyperbola at Q intersects the x-axis at C and the y- axis at D.

(i) Show that the equation of the tangent at P is x + p2y = 2cp. (ii) Show that A, B and O are on a circle with centre P. (iii) Prove that BC is parallel to PQ.

End of Question 12

2 2 1

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

1

n

(a) Let I

n = 1 x2 )2 dx,

where n ≥ 0 is an integer. 0

((i) Show that I n = nn + 1 I n 2

for every integer n ≥ 2. (ii) Evaluate I5.

(b) The diagram shows the graph of a function (x).

Sketch the following curves on separate half-page diagrams. (i) y2 = (x ) (ii) y =

1 1 x

Question 13 continues on page 11

3 2

2 3

Question 13 (continued)

(c) The points A, B, C and D lie on a circle of radius r, forming a cyclic

quadrilateral. The side AB is a diameter of the circle. The point E is chosen on the diagonal AC so that DE ⊥AC.Letα=∠DAC and β=∠ACD.

(i) Show that AC =2r sin (α+β).

(ii) By considering ABD, or otherwise, show that AE =2r cosαsinβ. (iii) Hence, show that sin (α+β) =sinαcosβ+sinβcosα.

End of Question 13

2 2 1

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

(a) The diagram shows the graph y = ln x.

= ln x

By comparing relevant areas in the diagram, or otherwise, show that

lnt > 2 t 1 t + 1

, for t > 1.(b) Let z i

2 = 1 + i and, for n > 2, let z n= z n 1 1 + z

n 1

. Use mathematical induction to prove that zn = n for all integers n ≥ 2.

Question 14 continues on page 13

3

3

Question 14 (continued)

(c) (i) Given a positive integer n, show that sec 2n

θ=

k∑n

=0

n k

tan2 k

θ. (ii) Hence, by writing sec8θ as sec6θ sec2θ, find

sec 8θdθ. (d) A triangle has vertices A, B and C. The point D lies on the interval AB such that

AD = 3 and DB = 5. The point E lies on the interval AC such that AE = 4, DE = 3 and EC = 2.

B5

D 3

A 3 4

NOT TO SCALE

2 C

(i) Prove that ABC and AED are similar. (ii) Prove that BCED is a cyclic quadrilateral. (iii) Show that CD = 21 .

(iv) Find the exact value of the radius of the circle passing through the points

B, C, E and D.

End of Question 14

1

2

1 1 2 2

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

(a) The Argand diagram shows complex numbers w and z with arguments φ and θ

respectively, where φ<θ. The area of the triangle formed by 0, w and z is A.

Show that = 4 iA.

(b) The polynomial P(x) = ax4 + bx3 + cx2 + e has remainder –3 when divided by x – 1. The polynomial has a double root at x = –1.

(i) Show that 4a

+2c = 92

. (ii) Hence, or otherwise, find the slope of the tangent to the graph y = P(x)

when x = 1.

(c) Eight cars participate in a competition that lasts for four days. The probability

that a car completes a day is 0.7. Cars that do not complete a day are eliminated.

(i) Find the probability that a car completes all four days of the competition. (ii) Find an expression for the probability that at least three cars complete all

four days of the competition.

Question 15 continues on page 15

3

2 1

1 2

Question 15 (continued)

(d) A ball of mass m is projected vertically into the air from the ground with initial

velocity u. After reaching the maximum height H it falls back to the ground. While in the air, the ball experiences a resistive force kv2, where v is the velocity of the ball and k is a constant.

The equation of motion when the ball falls can be written as

mv = mg kv2

.

(Do NOT prove this.)

(i) Show that the terminal velocity vT

(ii) Show that when the ball goes up, the maximum height H is

2H = v 2

T

u

2 g ln 1+v 2 T

.

(iii) When the ball falls from height H it hits the ground with velocity w.

Show that

1

1

w2

=

u2 + 1 v 2. T

End of Question 15

1 3

2

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

(a)

(i) Find the minimum value of P(x) =2x3 15x2 +24x +16, for x ≥0. (ii) Hence, or otherwise, show that for x ≥0,

( x +1 ) x 2 + ( x +4 )2

≥25x2 .(iii) Hence, or otherwise, show that for m ≥0 and n ≥0,

( m+n)2

+ ( m+n+ 4 )2

≥

100mn

m+n+1 .

(b) A small bead P of mass m can freely move along a string. The ends of the string

S. The bead undergoes uniform circular motion with radius r and constant angular velocity ω in a horizontal plane.

The forces acting on the bead are the gravitational force and the tension forces along the string. The tension forces along PS and PS′ have the same magnitude T.

The length of the string is 2a and SS′=2ae, where 0 <e <1. The horizontal plane through P meets SS′at Q. The midpoint of SS′is O and β=∠S′PQ.Theparameter θ is chosen so that OQ =a cosθ.

Question 16 continues on page 17

2 1

2

Question 16 (continued)

(i) What information indicates that P lies on an ellipse with foci S and S′,

and with eccentricity e? (ii) Using the focus–directrix definition of an ellipse, or otherwise, show that

SP = a(1 e cosθ ).

(iii) Show that sin β =

e + cos θ

1 + ecos .

(iv) By considering the forces acting on P in the vertical direction, show that

2T mg1 e 2

=

cosθ

1 e 2 cos 2

θ

.

(v) Show that the force acting on P in the horizontal direction is

mrω 2 =

2T 1 e2 sin θ

1 e 2 cos 2

θ

.

tanθ =(vi)

Show that End of paper

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1

2

2

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BLANK PAGE

BLANK PAGE

STANDARD INTEGRALS

n

xdx 1 dx ax edx

=

1n+1

x , n ≠ 1; x ≠0, if n <0n +1

=ln x, x >0

1ax= e , a ≠0 1

= sinax, a ≠01

= cosax, a ≠0

1

= tanax, a ≠0

cosax dx

sin ax dx

2

sec ax dx

1 secax tanax dx = secax, a ≠0

1

dx 22 a +x 1

dx

2 a x 2 1

dx

x2 a2

dx 1 x

= tan 1, a ≠0 1 x =sin , a >0, a <x <a

2

=lnx + x 2 a , x >a>0

((

) )

2=lnx + x 2+a

NOTE : ln x =loge x , x >0

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