Girraween High School

2018 Year 12 Trial Higher School Certificate

Mathematics Extension 1

Time allowed: Two (2) Hours

(Plus 5 minutes reading time)

Instructions

• Attempt all questions.

• For Questions 1-10, shade the circle for the letter corresponding to the correct answer on

your answer sheet.

• For Questions 11-15, start each question on a new page. Each question should be clearly

labelled.

• All necessary working must be shown for Questions 11-15.

• Marks may be deducted for careless or badly arranged work.

• Board approved calculators may be used.

• A Mathematics reference sheet is provided.

• All diagrams are NOT TO SCALE.

Section l

10 marks

Attempt Questions 1-10

Allow about 15 minutes for this section

Question l

The point C which divides the interval between A(14, -20) and B(-4,1)

in the ratio 4: -1 is:

(A) (-10,8)

Question 2

2x Jim .

3 =

x~o sin x 3

(A)-2

Question 3

(B) (20, -27)

2 (B) -

3

(C) (2, -6)

1 (C)-

3

(D) (8,-13)

(D) 3

The number of different ways of arranging the letters of the word MOORABOOL

in a circle is

(A)362880 (B) 40 320 (C) 15 120 (D) 1680

Question 4

The remainder when ax3 + x 2 + x + 11 is divided by (2x + 3) is 5. The value

of a is:

(A) 1 (B) 2 (C) 3 (D) 4

Question 5

The co-efficient of x2 in the expansion of [ 2x + :2f1 is

Question 6

cos5xcos2x - sin5xsin2x =

(A) cos3x (B) sin3x (C) cos?x (D) sin?x

Exa111i11atio11 co11ti1111es 011 the followi11g page

Question 7

J sin23x.dx =

(A) ;sin3 3x + C 1 1 1 1

(B) - x - - cos6x + C (C) - x - - cos6x + C 2 2 2 12

(D) !:.x _ 2.sin6x + C 2 12

Question 8

Which of the graphs below is ofy = x 3 - 6x2 + 9x - 4?

(A)

y

x

(C)

y

x

Question 9

The general solution of sin2x = - !:. is 2

(B)

(D)

(A)x=nrr+c-1rx~ 6

(B) x = nrr + (-1r x !!_ 2 12

(D) x = nrr + c-1r+1 x !!_ 2 12

Question 10

Ify = sin-1 (Sx) dy = 'dx

1 5 (A) v1-2sx2 (B) V1-25x2

y

x

y

x

(C) x = nrr + (-1r+1 x ~ 6

Examination continues 011 the following page

Section II

69 marks

Attempt Questions 11-15

Allow about 1 hour and 45 minutes for this section

Start all answers on a separate page in your answer booklet.

In Questions 11-15 your responses should include all relevant mathematical reasoning and/ or

calculations.

Question 11 (16 Marks)

5 1 (a) Solve for x: -- > - -

3x-z 4

(b) Find the acute angle between the lines y = 2x - 1 and x + 3y = 2

(c) Find the exact value of cos 75°.

(d) If a:, fJ and y are the roots of the polynomial equation

1 1 1 2x3 - x 2 + 3x - 4 = 0, find the value of-+ - + -

a{3 ay {3y

n: .

(e) Use the substitution u = cosx to find J.04 sm x . dx

l+cosx

(f) Use the substitution x = u 2 to find J 1 r;::;. dx

l+vx

Question 12 (12 Marks)

(a) Use the method of mathematical induction to prove

Marks

3

3

2

2

3

3

2 + 6 + 20+ ... +(n2 + n) = ~ (n + l)(n + 2) for all positive integers n. 4

Question 12 continues 011 the followi11g page

Question 12 (continued)

(b) In the diagram below, line PTQ is a tangent to the circle TABC at T.

BT bisects AC at D and AT= CT. LQTC =a and LCTB = /J.

(see diagram)

(i) Copy the diagram in to your answer booklet and state why LT AC = a.

(ii) Prove LATP =a.

(iii) Prove LATE= /J.

(iv) Prove TB is a diameter of the circle TABC.

Question 13 (14 Marks)

(a) Sketch the graph ofy = Zcos-1 G), stating its domain and range.

(b) The probability that it will rain on a certain day in August in Sydney

is~. Assuming that the probability that it will rain on one day is independent 9

of the probability that it will rain on the next day:

Marks

1

2

2

3

Marks

3

(i) Find the probability that it will rain on 3 days in the next week. 2

(ii) Find the probability that it will rain on at least 2 days in the next week. 2

Question 13 continues 011 the following page

Question 13 (continued)

(c) A particle is moving so that its position at time tis given by

x = 2sin3t + 2../3cos3t.

Marks

(i) By showing that x = -n2x, n a real number, prove that the particle is l

moving with simple harmonic motion.

(ii) By expressing the particle's position at time tin the fonn 3

x = Rcos(3t - a), find the period and amplitude of the motion.

(iii) Find the first time that the particle reaches x = 0 and find its velocity 3

and acceleration then.

Question 14 (14 Marks)

(a) A particle is moving with simple hannonic motion about x = 0 with

acceleration x = -n2x and amplitude a. (Note that in this question a refers

to the amplitude of the motion).

(i) Show that v 2 = n2 (a2 - x2).

(ii) For this particle, v = 10 when x = 5 and v = 5 when x = 7.

Find the period of the motion.

Question 14 co11ti11ues 011 the .following page

2

3

Question 14 (continued)

(b) A howitzer with a muzzle velocity of200m/s is located lOOm

up the side of a hill. It is firing shells at an angle of a to the horizontal.

(see diagram)

DIAGRAM NOT TO SCALE

HILL 100m

WALL

(i) Given that x = 0 and ji = -g, show that x = 200tcosa and

y = - ~ gt 2 + 200tsina + 100.

2

(ii) Show that y = - --2.!.._ sec 2 a + xtana + 100. 80 000

(iii) The shell is being aimed at the base of a wall 3000m away horizontally.

At which two angles must the shell be fired in order to hit this target?

(Let g = 10m/s2).

Exami11atio11 continues 011 the followi11g page

Marks

4

2

3

Question 15 (13 Marks)

(a) P(Zap, ap 2) and Q (Zaq, aq 2

) are variable points on the parabola

x 2 = 4ay. A is the intersection of the tangent to the parabola at P and the

line through Qparallel to the axis of the parabola, while B is the intersection

of the tangent to the parabola at Q and the line through P parallel to the axis

of the parabola. (see diagram)

The equation of the tangent to the parabola at P is y = px - ap 2

(Do NOT prove this!).

(i) Prove that PQAB is a parallelogram.

(ii) lfp > q, find the area of the parallelogram in tenns ofp and q.

Question 15 continues on the following page

Marks

3

2

Question 15 (continued)

(b) An aeroplane flying due East at 900km/h at a constant altitude

is seen by an observer on the ground on a bearing of 290°T at an

angle of elevation of a. One minute later it is sighted by the observer

at an angle of elevation of 35° and on a bearing of OS0°T. If the

aeroplane flies from A to B, the observer is at P and C and D are on

the ground directly beneath A and B (see diagram)

A-----------fi>B

D

p

(i) Show that AB = 15000m and LCPD = 120°.

(ii) Show that the height of the aeroplane above the ground is given by

BD = lSOOOsin20°tan35°

sin60°

(iii) Find the angle of elevation (a) when the aeroplane is first seen

(answer to the nearest minute).

END OF EXAMINATION

Marks

2

3

3

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