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2006

H I G H E R S C H O O L C E R T I F I C AT E

E X A M I N AT I O N

Mathematics Extension 2

General Instructions ? Reading time – 5 minutes ? Working time – 3 hours ? Write using black or blue pen ? Board-approved calculators may be used ? A table of standard integrals is provided at the back of this paper ? All necessary working should be shown in every question

Total marks – 120 ? Attempt Questions 1–8 ? All questions are of equal value

412

Total marks – 120 Attempt Questions 1–8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.

Marks Question 1 (15 marks) Use a SEPARATE writing booklet.

(a)

? x dx . Find ? 2 ? 9 ? 4x

2

? dx (b) By completing the square, find ? 2 . ? x ? 6x +13

2

(c)

(i)

Given that

16x ? 43

( x ? 3) ( x + 2)

2

can be written as a b x ?3 c x+2

3

16x ? 43

( x ? 3) ( x + 2)

2

=

( x ? 3)

2

+

+

,

where a, b and c are real numbers, find a, b and c.

? 16x ? 43 dx . (ii) Hence find ? 2 ? ( x ? 3) ( x + 2)

2

? (d) Evaluate ? te? t dt . ?0

2

3

(e)

Use the substitution t = tan

θ to show that 2

? 3 dθ 1 = log 3 . ? 2 ?π sin θ

2 2π

3

– 2 –

Marks Question 2 (15 marks) Use a SEPARATE writing booklet. Let z = 3 + i and w = 2 – 5i. Find, in the form x + iy, (i) z2 (ii) z w (iii) w . z 1

1

1

(a)

(b)

(i)

Express

3 ? i in modulus-argument form.

2

2

1

(ii) Express

(

3 ?i

)

7

in modulus-argument form. 3 ?i

(iii) Hence express

(

)

7

in the form x + iy.

(c)

Find, in modulus-argument form, all solutions of z3 = –1.

2

i (d) The equation z ? 1 ? 3i

+ z ? 9 ? 3

= 10 corresponds to an ellipse in the Argand diagram. (i) Write down the complex number corresponding to the centre of the ellipse.

(ii) Sketch the ellipse, and state the lengths of the major and minor axes. (iii) Write down the range of values of arg(z) for complex numbers z corresponding to points on the ellipse.

1

3

1

– 3 –

Marks Question 3 (15 marks) Use a SEPARATE writing booklet.

(a)

The diagram shows the graph of y = ?(x). The graph has a horizontal asymptote at y = 2. y y = ? (x) 2

O

3

x

Draw separate one-third page sketches of the graphs of the following: (i) y = ( ? ( x )) y= 1 ?( x )

2

2

(ii) (iii)

2 2

y = x ?( x ) .

Question 3 continues on page 5

– 4 –

Marks Question 3 (continued)

(b)

The diagram shows the graph of the hyperbola x2 y2 ? =1. 144 25 y

O

x

(i)

Find the coordinates of the points where the hyperbola intersects the x-axis. Find the coordinates of the foci of the hyperbola. Find the equations of the directrices and the asymptotes of the hyperbola.

1

(ii) (iii)

2 2

(c)

Two of the zeros of P (x) = x4 – 12x3 + 59x2 – 138x + 130 are a + ib and a + 2ib, where a and b are real and b > 0. (i) (ii) Find the values of a and b. Hence, or otherwise, express P (x) as the product of quadratic factors with real coefficients. 3 1

End of Question 3

– 5 –

Marks Question 4 (15 marks) Use a SEPARATE writing booklet.

(a)

The polynomial p(x) = ax3 + bx + c has a multiple zero at 1 and has remainder 4 when divided by x + 1. Find a, b and c.

3

(b) The base of a solid is the parabolic region x2 ≤ y ≤ 1 shaded in the diagram. y

3

y = x2 1

O

x

Vertical cross-sections of the solid perpendicular to the y-axis are squares. Find the volume of the solid.

(c)

? 1? ? 1? ? 1? Let P ? p, ? , Q ? q, ? and R ?r , ? be three distinct points on the hyperbola ? r? ? p? ? q? xy = 1. (i) Show that the equation of the line, , through R, perpendicular to PQ, 1 is y = pqx ? pqr + . r Write down the equation of the line, m, through P, perpendicular to QR. The lines and m intersect at T. Show that T lies on the hyperbola. 2

(ii) (iii)

1 2

Question 4 continues on page 7

– 6 –

Marks Question 4 (continued)

(d)

A

K

M

B

P

L

C

In the acute-angled triangle ABC, K is the midpoint of AB, L is the midpoint of BC and M is the midpoint of CA. The circle through K, L and M also cuts BC at P as shown in the diagram. Copy or trace the diagram into your writing booklet. (i) (ii) (iii) Prove that KMLB is a parallelogram. Prove that ∠KPB = ∠KML . Prove that AP ⊥ BC. 1 1 2

End of Question 4

– 7 –

Marks Question 5 (15 marks) Use a SEPARATE writing booklet.

(a)

A solid is formed by rotating the region bounded by the curve y = x ( x – 1 )2 and the line y = 0 about the y-axis. Use the method of cylindrical shells to find the volume of this solid. Show that cos (α + β ) + cos (α ? β ) = 2 cos α cos β .

3

(b)

(i)

1

(ii) Hence, or otherwise, solve the equation cosθ + cos 2θ + cos 3θ + cos 4θ = 0 for 0 ≤ θ ≤ 2π .

3

(c)

A particle, P, of mass m is attached by two strings, each of length , to two fixed points, A and B, which lie on a vertical line as shown in the diagram.

A

α

NOT TO SCALE

P

B

The system revolves with constant angular velocity ω about AB. The string AP makes an angle α with the vertical. The tension in the string AP is T1 and the tension in the string BP is T2 where T1 ≥ 0 and T2 ≥ 0. The particle is also subject to a downward force, mg, due to gravity. (i) Resolve the forces on P in the horizontal and vertical directions. 2 1

(ii) If T2 = 0, find the value of ω in terms of , g and α.

Question 5 continues on page 9 – 8 –

Marks Question 5 (continued)

(d) In a chess match between the Home team and the Away team, a game is played on each of board 1, board 2, board 3 and board 4. On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2. The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 1, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD. (i) How many different recordings are possible? 1 1 3

(ii) Calculate the probability of the result which is recorded as WDLD. (iii) Teams score 1 point for each game won, and 0 points for each game lost.

What is the probability that the Home team scores more points than the Away team? 1 a point for each game drawn 2

End of Question 5

– 9 –

Marks Question 6 (15 marks) Use a SEPARATE writing booklet. (a) In Δ ABC, ∠CAB = α, ∠ABC = β and ∠BCA = γ . The point O is chosen inside Δ ABC so that ∠OAB = ∠OBC = ∠OCA = θ , as shown in the diagram. A

θ

O B

θ

θ

C

(i)

Show that

OA sin ( β ? θ ) = . OB sin θ

1

(ii) (iii)

Hence show that sin3 θ = sin (α ? θ ) sin ( β ? θ ) sin (γ ? θ ) . Prove the identity cot x ? cot y = Hence show that sin ( y ? x ) . sin x sin y

2 1

(iv)

1

(cot θ

(v)

? cot α ) ( cot θ ? cot β ) ( cot θ ? cot γ ) = cosec α cosec β cosec γ . e 2

Hence find the value of θ when Δ ABC is an isosceles right triangle.

Question 6 continues on page 11

– 10 –

Marks Question 6 (continued)

(b) In an alien universe, the gravitational attraction between two bodies is proportional to x –3, where x is the distance between their centres. A particle is projected upward from the surface of a planet with velocity u at time t = 0. Its distance x from the centre of the planet satisfies the equation k x=? . x3 (i) Show that k = gR3 , where g is the magnitude of the acceleration due to gravity at the surface of the planet and R is the radius of the planet.

1

(ii) Show that v, the velocity of the particle, is given by v2 = gR3 ? gR ? u 2 . x2

3

(

)

(iii) It can be shown that x = (Do NOT prove this.) Show that if u ≥

R 2 + 2uRt ? gR ? u 2 t 2 .

(

)

2

gR the particle will not return to the planet.

(iv) If u < gR the particle reaches a point whose distance from the centre of the planet is D, and then falls back. (1) Use the formula in part (ii) to find D in terms of u, R and g. (2) Use the formula in part (iii) to find the time taken for the particle to return to the surface of the planet in terms of u, R and g. 1 1

End of Question 6

– 11 –

Marks Question 7 (15 marks) Use a SEPARATE writing booklet. The curves y = cos x and y = tan x intersect at a point P whose x-coordinate is α. (i) Show that the curves intersect at right angles at P. 1+ 5 . 2 3 2

(a)

(ii) Show that sec 2 α =

(b)

(i)

? π Let I n = ? sec n t dt , where 0 ≤ x < . Show that 2 ?0

In = sec n?2 x tan x n ? 2

+ I .

n ?1 n ? 1 n? 2

x

3

(ii) Hence find the exact value of

2

π

?3 4 ? sec t dt . ?0

Question 7 continues on page 13

– 12 –

Marks Question 7 (continued)

(c)

The sequence { xn } is given by x1 = 1 and xn+1 = 4 + xn for n ≥ 1. 1 + xn 4

(i) Prove by induction that for n ≥ 1 ?1 + α n ? ?, xn = 2 ? ? n? ?1 ? α ? 1 where α = ? . 3 (ii) Hence find the limiting value of xn as n → ∞.

1

End of Question 7

– 13 –

Marks Question 8 (15 marks) Use a SEPARATE writing booklet.

(a)

Suppose 0 ≤ t ≤

1 2

.

(i) Show that 0 ≤

2t 2 1? t

2

≤ 4t 2 .

2

(ii)

Hence show that 0 ≤

1 1 + ? 2 ≤ 4t 2 . 1+ t 1 ? t

1

(iii)

By integrating the expressions in the inequality in part (ii) with respect 1 to t from t = 0 to t = x (where 0 ≤ x ≤ ) , show that 2 ? 1+ x ? 4x 3 0 ≤ loge ? . ? ? 2x ≤ ?1 ? x ? 3

2

(iv)

Hence show that for 0 ≤ x ≤

1 2

3

1

4x ? 1+ x ? ?2x ≤e 3 . 1≤? ?e ?1 ? x ?

Question 8 continues on page 15

– 14 –

Marks Question 8 (continued)

(b)

For x > 0, let ? (x) = xne–x , where n is an integer and n ≥ 2. y

O (i)

a

b

x 4

The two points of inflexion of ? (x) occur at x = a and x = b, where 0 < a < b. Find a and b in terms of n. Show that ? ?1 + ?( b ) ? = ?( a ) ? 1 ? ? ? 1 ? ? n ? ?2 e 1 ? ? n?

n

(ii)

2

n

.

(iii)

Using the result of part (a) (iv), show that ?( b ) 1≤ ≤ e3 n . ?( a )

4

2

(iv)

What can be said about the ratio

?( b ) as n → ∞ ? ?( a )

1

End of paper

– 15 –

STANDARD INTEGRALS ? n ? x dx ? ?1 ? x dx ? ? ax ? e dx ? ? ? cos ax dx ? ? ? sin ax dx ? ? 2 ? sec ax dx ? ? ? sec ax tan ax dx ? 1 ? dx ? 2 ? a + x2 1 ? dx ? 2 ? a ? x2 1 ? dx ? 2 ? x ? a2 1 ? dx ? 2 ? x + a2 = 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, a ≠ 0 a 1 x tan ?1 , a ≠ 0 a a

=

=

=

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

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

( (

)

= ln x + x 2 + a 2

)

x>0

NOTE : ln x = loge x,

– 16 – ? Board of Studies NSW 2006

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