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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 1

General Instructions ? Reading time – 5 minutes ? Working time – 2 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 – 84 ? Attempt Questions 1–7 ? All questions are of equal value

411

BLANK PAGE

– 2 –

Total marks – 84 Attempt Questions 1–7 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.

Marks Question 1 (12 marks) Use a SEPARATE writing booklet.

(a)

? dx Find ? . ? 49 + x 2

2

(b)

? Using the substitution u = x4 + 8, or otherwise, find ? x 3 x 4 + 8 dx . ?

3

(c)

Evaluate lim

sin 5 x . x →0 3 x

2

(d)

Using the sum of two cubes, simplify: sin3 θ + cos3 θ ? 1, sin θ + cosθ for 0 < θ <

2

π . 2

(e)

For what values of b is the line y = 12x + b tangent to y = x3 ?

3

– 3 –

Marks Question 2 (12 marks) Use a SEPARATE writing booklet.

(a)

Let ? (x) = sin–1 ( x + 5 ) . (i) (ii) (iii) State the domain and range of the function ? (x) . Find the gradient of the graph of y = ? (x) at the point where x = –5. Sketch the graph of y = ? (x) . 2 2 2

(b)

(i)

By applying the binomial theorem to show that n (1+ x )

n?1

(1 + x )n

and differentiating, ?n? + n ? ? x n ?1 . ?n?

1

?n? ?n? = ? ? + 2? ? x + ?1? ?2?

?n? + r ? ? x r ?1 + ?r ?

(ii)

Hence deduce that ?n? n3n?1 = ? ? + ?1? ?n? + r ? ? 2r?1 + ?r ? ?n? + n ? ? 2n ?1 . ?n?

1

Question 2 continues on page 5

– 4 –

Marks Question 2 (continued)

(c)

y

Q

R

T P

U x

The points P (2ap, ap2 ), Q (2aq, aq2 ) and R (2ar, ar2 ) lie on the parabola x2 = 4ay. The chord QR is perpendicular to the axis of the parabola. The chord PR meets the axis of the parabola at U. The equation of the chord PR is y = 1 ( p + r ) x ? apr . 2 (Do NOT prove this.) (Do NOT prove this.)

The equation of the tangent at P is y = px – ap2 . (i) (ii) Find the coordinates of U.

1 2

The tangents at P and Q meet at the point T. Show that the coordinates of T are (a ( p + q), apq) . Show that TU is perpendicular to the axis of the parabola.

(iii)

1

End of Question 2

– 5 –

Marks Question 3 (12 marks) Use a SEPARATE writing booklet.

(a)

?4 Find ? sin 2 x dx . ?0

π

2

(b)

(i)

By considering ?( x ) = 3loge x ? x , show that the curve y = 3loge x and the line y = x meet at a point P whose x-coordinate is between 1.5 and 2.

1

(ii) Use one application of Newton’s method, starting at x = 1.5, to find an approximation to the x-coordinate of P. Give your answer correct to two decimal places.

2

(c)

Sophie has five coloured blocks: one red, one blue, one green, one yellow and one white. She stacks two, three, four or five blocks on top of one another to form a vertical tower. (i) How many different towers are there that she could form that are three blocks high? (ii) How many different towers can she form in total? 1

2

Question 3 continues on page 7

– 6 –

Marks Question 3 (continued)

(d) P K

Q N

T

M

The points P, Q and T lie on a circle. The line MN is tangent to the circle at T with M chosen so that QM is perpendicular to MN. The point K on PQ is chosen so that TK is perpendicular to PQ as shown in the diagram.

(i) (ii) (iii)

Show that QKTM is a cyclic quadrilateral. Show that ∠KMT = ∠KQT . Hence, or otherwise, show that MK is parallel to TP.

1 1 2

End of Question 3

– 7 –

BLANK PAGE

– 8 –

Marks Question 4 (12 marks) Use a SEPARATE writing booklet.

(a)

The cubic polynomial P (x) = x3 + rx2 + sx + t , where r, s and t are real numbers, has three real zeros, 1, α and –α. (i) Find the value of r. (ii) Find the value of s + t. 1 2

(b) A particle is undergoing simple harmonic motion on the x-axis about the origin. It is initially at its extreme positive position. The amplitude of the motion is 18 and the particle returns to its initial position every 5 seconds. (i) Write down an equation for the position of the particle at time t seconds. (ii) How long does the particle take to move from a rest position to the point halfway between that rest position and the equilibrium position? 1 2

(c)

3 2 A particle is moving so that x = 18x + 27x + 9x .

Initially x = –2 and the velocity, v, is –6. (i) Show that v2 = 9x2 (1 + x)2 . (ii) Hence, or otherwise, show that 2 2

? 1 dx = ?3t . ? ? x (1+ x )

(iii) It can be shown that for some constant c, ? 1? loge ?1+ ? = 3t + c . ? x? (Do NOT prove this.) 2

Using this equation and the initial conditions, find x as a function of t.

– 9 –

Marks Question 5 (12 marks) Use a SEPARATE writing booklet.

(a)

Show that y = 10 e–0.7t + 3 is a solution of

dy = ?0.7 ( y ? 3) . dt

2

(b)

Let ?( x ) = loge 1+ e x

(

)

for all x. Show that ?(x) has an inverse.

2

(c)

x cm

A hemispherical bowl of radius r cm is initially empty. Water is poured into it at a constant rate of k cm3 per minute. When the depth of water in the bowl is x cm, the volume, V cm3 , of water in the bowl is given by V=

π 2 x (3r ? x ) . 3

(Do NOT prove this.)

(i)

Show that

dx k . = dt π x ( 2r ? x )

2

(ii)

Hence, or otherwise, show that it takes 3.5 times as long to fill the bowl 2 to the point where x = r as it does to fill the bowl to the point where 3

1

x= r. 3

2

Question 5 continues on page 11

– 10 –

Marks Question 5 (continued)

(d)

(i)

Use the fact that tan (α ? β ) =

tan α ? tan β to show that 1+ tan α tan β

1

1+ tan nθ tan ( n +1)θ = cot θ ( tan ( n +1)θ ? tan nθ ) . Use mathematical induction to prove that, for all integers n ≥ 1, + tan nθ tan ( n +1)θ = ? ( n +1) + cot θ tan ( n +1)θ .

(ii)

3

tan θ tan 2θ + tan 2θ tan 3θ +

End of Question 5

– 11 –

Question 6 (12 marks) Use a SEPARATE writing booklet. Two particles are fired simultaneously from the ground at time t = 0.

(a)

π Particle 1 is projected from the origin at an angle θ, 0 < θ < , with an initial 2 velocity V.

Particle 2 is projected vertically upward from the point A, at a distance a to the right of the origin, also with an initial velocity of V. y V V

θ

O A x

It can be shown that while both particles are in flight, Particle 1 has equations of motion: x = Vt cosθ y = Vt sin θ ? and Particle 2 has equations of motion: x=a y = Vt ? 1 2 gt . 2 1 2 gt , 2

Do NOT prove these equations of motion. Let L be the distance between the particles at time t.

Question 6 continues on page 13

– 12 –

Marks Question 6 (continued)

(i) Show that, while both particles are in flight,

L2 = 2V 2 t 2 (1 ? sin θ ) ? 2aVt cosθ + a 2 .

2

(ii) An observer notices that the distance between the particles in flight first decreases, then increases. Show that the distance between the particles in flight is smallest when t= a cosθ 1 ? sin θ and that this smallest distance is a . 2V ( ? sin θ ) 1 2

3

(iii) Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if V > ag cosθ . 2sin θ (1 ? sin θ )

1

(b) In an endurance event, the probability that a competitor will complete the course is p and the probability that a competitor will not complete the course is q = 1 – p. Teams consist of either two or four competitors. A team scores points if at least half its members complete the course. (i) Show that the probability that a four-member team will have at least three of its members not complete the course is 4pq3 + q4. (ii) Hence, or otherwise, find an expression in terms of q only for the probability that a four-member team will score points. (iii) Find an expression in terms of q only for the probability that a two-member team will score points. (iv) Hence, or otherwise, find the range of values of q for which a two-member team is more likely than a four-member team to score points. 1

2

1

2

End of Question 6

– 13 –

Marks Question 7 (12 marks) Use a SEPARATE writing booklet.

A gutter is to be formed by bending a long rectangular metal strip of width w so that the cross-section is an arc of a circle. Let r be the radius of the arc and 2θ the angle at the centre, O, so that the cross-sectional area, A, of the gutter is the area of the shaded region in the diagram on the right. O B Gutter B C C 2θ r r CROSS-SECTION

w

(a)

Show that, when 0 < θ ≤

π , the cross-sectional area is 2

A = r 2 (θ ? si θ cos

2

).

(b)

The formula in part (a) for A is true for 0 < θ < π .

(Do NOT prove this.)

3

By first expressing r in terms of w and θ, and then differentiating, show that dA dθ for 0 < θ < π . = w 2 cosθ (sin θ ? θ cosθ ) 2θ 3

Question 7 continues on page 15

– 14 –

Marks Question 7 (continued) Let g (θ ) = sin θ ? θ cosθ . By considering g′ (θ ) , show that g (θ ) > 0 for 0 < θ < π .

(c)

3

(d)

Show that there is exactly one value of θ in the interval 0 < θ < π for which dA = 0. dθ

2

(e)

Show that the value of θ for which area. Find this area in terms of w.

dA = 0 gives the maximum cross-sectional dθ

2

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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