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2014

HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Extension 1

General Instructions ? Reading time – 5 minutes ? Working time – 2 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 11–14, show relevant mathematical reasoning and/or calculations

Total marks – 70 Section I Pages 2–5

10 marks ? Attempt Questions 1–10 ? Allow about 15 minutes for this section Section II Pages 6–13

60 marks ? Attempt Questions 11–14 ? Allow about 1 hour and 45 minutes for this section

2620

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

The points A, B and C lie on a circle with centre O, as shown in the diagram. The size of ∠ ACB is 40°. C 40° O NOT TO SCALE

A B What is the size of ∠ AOB ? (A) 20° (B) (C) 40° 70°

(D) 80°

2

Which expression is equal to cos x ? sin x ? (A) ? π? 2 cos ? x + ? ? 4? ? π? 2 cos ? x ? ? ? 4?

(B)

? π? (C) 2 cos ? x + ? ? 4? ? π? (D) 2cos ? x ? ? ? 4?

–2–

3

? 5? What is the constant term in the binomial expansion of ? 2 x ? ? ? x3 ? ? 12? (A) ? ? 29 53 ? 3? ? 12? (B) ? ? 23 59 ? 9? ? 12? (C) ? ? ? 29 53 ? 3? ? 12? (D) ? ? ? 23 59 ? 9?

12

?

4

The acute angle between the lines 2x + 2y = 5 and y = 3x + 1 is θ . What is the value of tan θ ? (A) 1 7 1 2 1

(B) (C)

(D) 2

5

Which group of three numbers could be the roots of the polynomial equation x 3 + ax 2 ? 41x + 42 = 0 ? (A) 2, 3, 7 (B) (C) 1, ? 6, 7 ? 1, ? 2, 21

(D) ? 1, ? 3, ? 14

–3–

6

What is the derivative of 3sin ?1 6 4 ? x2 (B) 3 4 ? x2 (C) 3 2 4 ? x2 (D) 3 4 4 ? x2

x ? 2

(A)

7

A particle is moving in simple harmonic motion with period 6 and amplitude 5. Which is a possible expression for the velocity, v, of the particle? ?π ? 5π cos ? t ? 3 ?3 ?

(A) v =

(B)

?π ? v = 5cos ? t ? ?3 ? v= ?π ? 5π co s ? t ? 6 ?6 ?

(C)

?π ? (D) v = 5cos ? t ? ?6 ?

–4–

8

In how many ways can 6 people from a group of 15 people be chosen and then arranged in a circle? (A) (B) (C) (D) 14! 8! 14! 8!6 15! 9! 15! 9!6

9

The remainder when the polynomial P ( x ) = x 4 ? 8x 3 ? 7x 2 + 3 is divided by x 2 + x is ax + 3. What is the value of a ? (A) –14 (B) (C) –11 –2

(D) 5

10

Which equation describes the locus of points ( x, y) which are equidistant from the distinct points ( a + b, b ? a) and ( a ? b, b + a)? (A) bx + ay = 0 (B) (C) bx + ay = 2ab bx ? ay = 0

(D) bx ? ay = 2ab

–5–

Section II

60 marks Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11–14, your responses should include relevant mathematical reasoning and/or calculations.

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

(a)

? ? 2? 2? Solve ? x + ? ? 6 ? x + ? + 9 = 0 . ? ? x? x?

2

3

(b)

The probability that it rains on any particular day during the 30 days of November is 0.1. Write an expression for the probability that it rains on fewer than 3 days in November.

2

(c)

Sketch the graph y = 6 tan–1x , clearly indicating the range.

2

(d)

? Evaluate ? ?2

5

x x –1

dx using the substitution x = u2 + 1.

3

(e)

Solve

x2 + 5 >6. x

3

(f)

e x ln x Differentiate x .

2

–6–

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

(a)

A particle is moving in simple harmonic motion about the origin, with displacement x metres. The displacement is given by x = 2 sin 3t, where t is time in seconds. The motion starts when t = 0. (i) What is the total distance travelled by the particle when it first returns to the origin? What is the acceleration of the particle when it is first at rest? 1

(ii)

2

(b)

π The region bounded by y = cos 4x and the x-axis, between x = 0 and x = , 8 is rotated about the x-axis to form a solid.

y

3

y = cos 4x p 8

NOT TO SCALE x

O

Find the volume of the solid.

(c)

A particle moves along a straight line with displacement x m and velocity v m s–1. The acceleration of the particle is given by = 2 ? e x

? x 2.

3

Given that v = 4 when x = 0, express v 2 in terms of x.

Question 12 continues on page 8

–7–

Question 12 (continued)

(d)

Use the binomial theorem to show that ? n? ? n? ? n? n ? n? 0 = ? ? ? ? ? + ? ? ? + ( ? 1) ? ? . ? 0? ? 1 ? ? 2? ? n?

2

(e)

The diagram shows the graph of a function f ( x ) . The equation f ( x ) = 0 has a root at x = α . The value x1 , as shown in the diagram, is chosen as a first approximation of α . y y = f ( x)

1

O

a

x1

x

A second approximation, x2 , of α is obtained by applying Newton’s method once, using x1 as the first approximation. Using a diagram, or otherwise, explain why x1 is a closer approximation of α than x2 .

(f)

Milk taken out of a refrigerator has a temperature of 2°C. It is placed in a room of constant temperature 23°C. After t minutes the temperature, T °C, of the milk is given by T = A – Be –0.03t , where A and B are positive constants. How long does it take for the milk to reach a temperature of 10°C?

3

End of Question 12

–8–

Question 13 (15 marks) Use a SEPARATE writing booklet. Use mathematical induction to prove that 2n + ( ? 1) integers n ≥ 1.

n +1

(a)

is divisible by 3 for all

3

(b)

One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of 40 m above the point where the rope is attached to the truck. The distance from the truck to the small wheel is L m, and the horizontal distance between them is x m. The rope makes an angle θ with the horizontal at the point where it is attached to the truck. The truck moves to the right at a constant speed of 3 m s?1, as shown in the diagram.

40 m

L

x

q

3 m s –1

(i)

Using Pythagoras’ Theorem, or otherwise, show that dL = 3cos θ . dt

dL = cos θ . dx

2

(ii)

Show that

1

Question 13 continues on page 10

–9–

Question 13 (continued)

(c)

The point P 2at , at 2 lies on the parabola x 2 = 4 ay with focus S. The point Q divides the interval PS internally in the ratio t 2 : 1.

(

)

y x 2 = 4ay

P 2at , at 2

Q S

(

)

O

x

(i) (ii) (iii)

Show that the coordinates of Q are x = Express the slope of OQ in terms of t.

2at 1 + t2

and y =

2at 2 1 + t2

.

2 1 3

Using the result from part (ii), or otherwise, show that Q lies on a fixed circle of radius a.

Question 13 continues on page 11

– 10 –

Question 13 (continued)

(d)

In the diagram, AB is a diameter of a circle with centre O. The point C is chosen such that ? ABC is acute-angled. The circle intersects AC and BC at P and Q respectively. C

Q P B

O

NOT TO SCALE

A

Copy or trace the diagram into your writing booklet.

(i) (ii)

Why is ∠BAC = ∠CQP ? Show that the line OP is a tangent to the circle through P, Q and C.

1 2

End of Question 13

– 11 –

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

(a)

The take-off point O on a ski jump is located at the top of a downslope. The angle between the downslope and the horizontal is with velocity V m s?1

π . A skier takes off from O 4 π at an angle θ to the horizontal, where 0 ≤ θ < . The 2

skier lands on the downslope at some point P, a distance D metres from O. y V q O

p 4 D

x

P The flight path of the skier is given by x = Vt cos θ , 1 y = ? gt 2 + Vt sin θ , 2 (Do NOT prove this.)

where t is the time in seconds after take-off.

(i)

Show that the cartesian equation of the flight path of the skier is given by y = x tan θ ? gx 2 2V

2

2

sec 2 θ .

(ii)

Show that D = 2 2 Show that

V2 cos θ ( cos θ + sin θ ). g

3

(iii) (iv)

dD V2 =2 2 ( cos 2θ ? sin 2θ ). dθ g

2 3

Show that D has a maximum value and find the value of θ for which this occurs. Question 14 continues on page 13 – 12 –

Question 14 (continued)

(b)

Two players A and B play a game that consists of taking turns until a winner is determined. Each turn consists of spinning the arrow on a spinner once. The spinner has three sectors P, Q and R. The probabilities that the arrow stops in sectors P, Q and R are p, q and r respectively.

P R Q

The rules of the game are as follows: ? If the arrow stops in sector P, then the player having the turn wins. ? If the arrow stops in sector Q, then the player having the turn loses and the other player wins. ? If the arrow stops in sector R, then the other player takes a turn. Player A takes the first turn.

(i)

Show that the probability of player A winning on the first or the second turn of the game is (1 ? r )( p + r ) . Show that the probability that player A eventually wins the game is p+r . 1+ r

2

(ii)

3

End of paper

– 13 –

BLANK PAGE

– 14 –

BLANK PAGE

– 15 –

STANDARD INTEGRALS

? n ? x dx ? ? 1 ? x dx ? ? ax ? e dx ? ? ? cos ax dx ? ? ? sin ax d x ? ? 2 ? sec ax dx ?

=

1 n +1 x , n ≠ ? 1; x ≠ 0 , if n < 0 n +1

= ln x , x > 0

=

1 ax e , a≠0 a 1 sin ax , a ≠ 0 a

=

1 = ? cos ax , a ≠ 0 a 1 tan ax , a ≠ 0 a

=

? 1 ? sec ax tan ax dx = a sec ax , a ≠ 0 ? ? 1 dx ? 2 ? a + x2 ? ? ? ? ? ? ? ? ?

1 a2 ? x 2 1 x 2 ? a2 1 x 2 + a2 = 1 x tan ?1 , a ≠ 0 a a

dx

x = sin ?1 , a > 0 , ? a < x < a a

dx

= ln x + x 2 ? a 2 , x > a > 0

( (

) )

dx

= ln x + x 2 + a 2

NOTE : ln x = loge x , x > 0

– 16 –

? 2014 Board of Studies, Teaching and Educational Standards NSW

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