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2017

HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics

General Instructions

? ? ? ? ? ?

Reading time – 5 minutes Working time – 3 hours Write using black pen NESA approved calculators may be used A reference sheet is provided at the back of this paper In Questions 11–16, show relevant mathematical reasoning and/or calculations

Total marks: 100

Section I – 10 marks (pages 2–5) ? Attempt Questions 1–10 ? Allow about 15 minutes for this section Section II – 90 marks (pages 6–16) ? Attempt Questions 11–16 ? Allow about 2 hours and 45 minutes for this section

1250

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 What is the gradient of the line 2x + 3y + 4 = 0 ? A. B. C. D. ? 2 3

2

3

? 3

2

3

2

2

Which expression is equal to 3x2 ? x ? 2? A. B. C. D.

(3x ? 1) (x + 2) (3x + 1) (x ? 2) (3x ? 2) (x + 1) (3x + 2) (x ? 1)

3

What is the derivative of e x ? A. B. C. D. x 2e

x

2

2

?1

2 xe 2 x

2 xe x 2e x

2 2

– 2 –

4

The function? ( x ) is defined for a ≤ x ≤ b . On this interval, ? ′( x ) > 0 and ? ″( x ) < 0 . Which graph best represents y = ? ( x ) ? A. y B. y

a

b

x

a

b

x

C.

y

D.

y

a

b

x

a

b

x

5

It is given that ln a = ln b ? ln c, where a, b, c > 0. Which statement is true? A. a = b ? c B. a = C. D. b c b c ln b ln c

lna = lna =

– 3 –

6

The point P moves so that it is always equidistant from two fixed points, A and B. What best describes the locus of P ? A. B. C. D. A point A circle A parabola A straight line

7

Which expression is equivalent to tan q + cot q ? A. B. C. D. cosec q + sec q sec q cosec q 2 1

8

The region enclosed by y = 4 ? x , y = x and y = 2x + 1 is shaded in the diagram below. y y=x

4

O

4

x

y=4?x y = 2x + 1 Which of the following defines the shaded region? A. B. C. D. y ≤ 2x + 1, y ≥ 2x + 1, y ≤ 2x + 1, y ≥ 2x + 1, y ≤ 4 ? x, y ≤ 4 ? x, y ≥ 4 ? x, y ≥ 4 ? x, y≥x y≥x y≥x y≥x – 4 –

9

The graph of y = ? ′( x ) is shown. y

4 y = ? ′( x )

O

2

x

The curve y = ? ( x ) has a maximum value of 12. What is the equation of the curve y = ? ( x ) ? A. y = x2 ? 4x + 12 B. y = 4 + 4x ? x2 C. y = 8 + 4x ? x2 D. y = x2 ? 4x + 16

10

A particle is moving along a straight line. The graph shows the velocity, v, of the particle for time t ≥ 0 . v

O

t

How many times does the particle change direction? A. B. C. D. 1 2 3 4

– 5 –

Section II

90 marks Attempt Questions 11–16 Allow about 2 hours and 45 minutes for this section Answer each question in the appropriate 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 the Question 11 Writing Booklet. 2 . 5 ?1

(a)

Rationalise the denominator of

2

(b)

? 4 Find ? ( 2 x + 1) dx . ?

1

(c)

Differentiate

sin x . x

2

(d)

Differentiate x3 ln x .

2

(e)

In the diagram, OAB is a sector of the circle with centre O and radius 6 cm, where AOB = 30°. A

6 cm

NOT TO SCALE

30° O (i) (ii) 6 cm

B 1 1

Find the exact value of the area of the triangle OAB. Find the exact value of the area of the shaded segment. Question 11 continues on page 7 – 6 –

Question 11 (continued)

(f)

Determine the equation of the parabola shown. Write your answer in the form (x ? h)2 = 4a ( y ? k). y 4

2

1

O

2

x

(g)

Solve | 3x ? 1 | = 2.

2

(h)

Find the domain of the function ? ( x ) = 3 ? x .

2

End of Question 11

–7–

Question 12 (15 marks) Use the Question 12 Writing Booklet. (a) Find the equation of the tangent to the curve y = x2 + 4x ? 7 at the point (1, ?2).

2

(b)

The diagram shows the region bounded by y = 16 ? 4 x 2 and the x-axis. y

3

4

?2

O

2

x

The region is rotated about the x-axis to form a solid.

Find the exact volume of the solid formed.

(c) In an arithmetic series, the fifth term is 200 and the sum of the first four terms is 1200. Find the value of the tenth term.

3

Question 12 continues on page 9

– 8 –

Question 12 (continued) (d) The pointsA(?4, 0) and B (1, 5) lie on the line y = x + 4. The length of AB is 5 2 . The points C (0, ?2) and D (3, 1) lie on the line x ? y ? 2 = 0. The points A, B, D, C form a trapezium as shown. y B (1, 5) y=x+4 A(?4, 0) O C (0, ?2) (i) (ii) Find the perpendicular distance from point A(?4, 0) to the line x ? y ? 2 = 0. Calculate the area of the trapezium. 1 2 x?y?2=0 D (3, 1) x NOT TO SCALE

(e)

A spinner is marked with the numbers 1, 2, 3, 4 and 5. When it is spun, each of the five numbers is equally likely to occur. 1 5 4 3 2

The spinner is spun three times. (i) (ii) (iii) (iv) What is the probability that an even number occurs on the first spin? What is the probability that an even number occurs on at least one of the three spins? What is the probability that an even number occurs on the first spin and odd numbers occur on the second and third spins? What is the probability that an even number occurs on exactly one of the three spins? End of Question 12 – 9 –

1 1 1 1

Question 13 (15 marks) Use the Question 13 Writing Booklet.

(a)

Using the cosine rule, find the value of x in the following diagram.

3

x ?4 60° x + 4

13

NOT TO SCALE

(b)

Consider the curve y = 2x3 + 3x2 ? 12x + 7. (i) (ii) (iii) Find the stationary points of the curve and determine their nature. Sketch the curve, labelling the stationary points. Hence, or otherwise, find the values of x for which dy is positive. dx 4 2 1

(c)

By letting m = t

3 , or otherwise, solve t 3 + t

3 ? 6 = 0 .

1

2

1

2

(d) The rate at which water flows into a tank is given by dV 2t = , dt 1 + t 2 where V is the volume of water in the tank in litres and t is the time in seconds. Initially the tank is empty. Find the exact amount of water in the tank after 10 seconds.

3

– 10 –

Question 14 (15 marks) Use the Question 14 Writing Booklet.

(a) Sketch the curve y = 4 + 3 sin 2x for 0 ≤ x ≤ 2p.

3

(b)

(i)

?3 Find the exact value of ? cos x dx .

?0

π

1

(ii) Using Simpson’s rule with one application, find an approximation to the integral

π ?3

2

? cos x dx ,

?0

leaving your answer in terms of p and 3 .

(iii) Using parts (i) and (ii), show that

1

π≈

18 3 . 3+ 4 3

Question 14 continues on page 12

– 11 –

Question 14 (continued)

(c) Carbon-14 is a radioactive substance that decays over time. The amount of carbon-14 present in a kangaroo bone is given by C ( t ) = Aekt, where A and k are constants, and t is the number of years since the kangaroo died. (i) Show that C ( t ) satisfies dC = kC. dt 1 2

(ii) After 5730 years, half of the original amount of carbon-14 is present. Show that the value of k, correct to 2 significant figures, is ?0.00012. (iii) The amount of carbon-14 now present in a kangaroo bone is 90% of the original amount. Find the number of years since the kangaroo died. Give your answer correct to 2 significant figures.

2

(d)

The shaded region shown is enclosed by two parabolas, each with x-intercepts at x = ?1 and x = 1. The parabolas have equations y = 2k x 2 ? 1 and y = k 1 ? x 2 , where k > 0. y y = 2k x ? 1

3

(

)

(

)

(

2

)

k ?1 ?2k y = k 1 ? x2 1 x

(

)

Given that the area of the shaded region is 8, find the value of k.

End of Question 14

– 12 –

Question 15 (15 marks) Use the Question 15 Writing Booklet.

(a)

The triangle ABC is isosceles with AB = AC and the size of BAC is x°. Points D and E are chosen so that rABC, rACD and rADE are congruent, as shown in the diagram. E A x° D

3

C B Find the value of x for which AB is parallel to ED, giving reasons.

(b)

Anita opens a savings account. At the start of each month she deposits $X into the savings account. At the end of each month, after interest is added into the savings account, the bank withdraws $2500 from the savings account as a loan repayment. Let Mn be the amount in the savings account after the nth withdrawal. The savings account pays interest of 4.2% per annum compounded monthly. (i) Show that after the second withdrawal the amount in the savings account is given by M2 = X 1.00352 + 1.0035 ? 2500 1.0035 + 1 . (ii) Find the value of X so that the amount in the savings account is $80 000 after the last withdrawal of the fourth year. 3 2

(

)

(

)

Question 15 continues on page 14

– 13 –

Question 15 (continued) (c) Two particles move along the x-axis. When t = 0, particle P1 is at the origin and moving with velocity 3. For t ≥ 0, particle P1 has acceleration given by a1 = 6t + e?t. (i) Show that the velocity of particle P1 is given by v1 = 3t2 + 4 ? e?t. 2

When t = 0, particle P2 is also at the origin. For t ≥ 0, particle P2 has velocity given by v2 = 6t + 1 ? e?t. (ii) (iii) When do the two particles have the same velocity? Show that the two particles do not meet for t > 0. 2 3

End of Question 15

– 14 –

Question 16 (15 marks) Use the Question 16 Writing Booklet.

(a) John’s home is at point A and his school is at point B. A straight river runs nearby. The point on the river closest to A is point C, which is 5 km from A. The point on the river closest to B is point D, which is 7 km from B. The distance from C to D is 9 km.

To get some exercise, John cycles from home directly to point E on the river,

x km from C, before cycling directly to school at B, as shown in the diagram.

b A a 7 km B

5 km

C x km

E 9 km

D River

The total distance John cycles from home to school is L km. (i) (ii) (iii) (iv) Show that L = Show that if x 2 + 25 + 49 + ( 9 ? x )2 . 1 3 2 1

dL = 0, then sin a = sin b. dx

Find the value of x that makes sin a = sin b. Explain why this value of x gives a minimum for L.

Question 16 continues on page 16

– 15 –

Question 16 (continued)

(b) A geometric series has first term a and limiting sum 2. Find all possible values for a.

3

(c)

In the triangleABC, the point M is the mid-point of BC. The point D lies on AB and BD = DA + AC. The line that passes through the point C and is parallel to MD meets BA produced at E. E A

D

B

M

C

Copy or trace this diagram into your writing booklet. (i) (ii) Prove that rACE is isosceles. The point F is chosen on BC so that AF is parallel to DM. E A 3 2

D

B

Show that AF bisects

M

BAC.

F

C

End of paper

– 16 –

? 2017 NSW Education Standards Authority

2017

HIGHER SCHOOL CERTIFICATE EXAMINATION

REFERENCE SHEET

– Mathematics –

– Mathematics Extension 1 –

– Mathematics Extension 2 –

1251

Mathematics

– 2 –

Mathematics (continued)

– 3 –

Mathematics Extension 1

–4–

? 2017 NSW Education Standards Authority

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