hsc past 3unit exam

2001H 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

General Instructions

• Reading time – 5 minutes

• Working time – 2 hours

• Write using black or blue pen

• Board-approved calculators maybe used

• A table of standard integrals isprovided at the back of this paper

• All necessary working should beshown in every question

Total marks – 84

• Attempt Questions 1–7

• All questions are of equal value

Mathematics Extension 1

411

Total marks – 84Attempt Questions 1–7All questions are of equal value

Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.

MarksQuestion 1 (12 marks) Use a SEPARATE writing booklet.

(a) Use the table of standard integrals to find the exact value of

.

(b) Find .

(c) Evaluate .

(d) Let A be the point (–2, 7) and let B be the point (1, 5). Find the coordinates ofthe point P which divides the interval AB externally in the ratio 1 : 2.

(e) Is x + 3 a factor of x3 – 5x + 12? Give reasons for your answer.

(f) Use the substitution u = 1 + x to evaluate

.15 11

0

x x dx+⌠⌡−

3

2

2

1( )2 34

7

nn

+=∑

2ddx

x xsin2( )

dx

x16 20

2

−

⌠⌡

2

– 2 –

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

(a) Let ƒ(x) = 3x2 + x. Use the definition

to find the derivative of ƒ(x) at the point x = a .

(b) Find

(i)

(ii) .

(c) The letters A, E, I, O, and U are vowels.

(i) How many arrangements of the letters in the word ALGEBRAIC arepossible?

(ii) How many arrangements of the letters in the word ALGEBRAIC arepossible if the vowels must occupy the 2nd, 3rd, 5th and 8th positions?

(d) Find the term independent of x in the binomial expansion of

.xx

291

–

3

2

1

3cos2

03x dx

π⌠⌡

1e

edx

x

x1 +⌠⌡

′ = +( ) − ( )→f a

f a h f ahh( ) lim

0

2

– 3 –

Marks


(a) The function ƒ(x) = sinx + cosx – x has a zero near x = 1.2

Use one application of Newton’s method to find a second approximation to thezero. Write your answer correct to three significant figures.

Two circles, C1 and C2, intersect at points A and B. Circle C1 passes through thecentre O of circle C2. The point P lies on circle C2 so that the line PAT is tangentto circle C1 at point A. Let ∠APB = θ.

Copy or trace the diagram into your writing booklet.

(i) Find ∠AOB in terms of θ . Give a reason for your answer.

(ii) Explain why ∠TAB = 2θ .

(iii) Deduce that PA = BA.

(c) (i) Starting from the identity , andusing the double angle formulae, prove the identity

sin3θ = 3sinθ – 4sin3θ .

(ii) Hence solve the equation

sin3θ = 2sinθ for 0 ≤ θ ≤ 2π .

3

2sin sin cos cos sinθ θ θ θ θ θ+( ) = +2 2 2

2

1

1

C1

O

P TA

B

θ

C2

(b)

3

– 4 –

Marks


(a) Solve .

(b) An aircraft flying horizontally at V m s–1 releases a bomb that hits the ground4000 m away, measured horizontally. The bomb hits the ground at an angle of45° to the vertical.

Assume that, t seconds after release, the position of the bomb is given by

x = Vt, y = –5t2.

Find the speed V of the aircraft.

(c) A particle, whose displacement is x, moves in simple harmonic motion.

Find x as a function of t if

and if x = 3 and when t = 0.x = −6 3

˙x x= − 4

5

y

x

4

33

21

xx –

≤

– 5 –

Marks


The sketch shows the graph of the curve y = ƒ(x) where .

The area under the curve for 0 ≤ x ≤ 3 is shaded.

(i) Find the y intercept.

(ii) Determine the inverse function y = ƒ –1(x) , and write down the domainD of this inverse function.

(iii) Calculate the area of the shaded region.

(b) By using the binomial expansion, show that

What is the last term in the expansion when n is odd? What is the last term inthe expansion when n is even?

(c) A fair six-sided die is randomly tossed n times.

(i) Suppose 0 ≤ r ≤ n. What is the probability that exactly r ‘sixes’ appearin the uppermost position?

(ii) By using the result of part (b), or otherwise, show that the probabilitythat an odd number of ‘sixes’ appears is

.12

123

–

n

2

2

q p q pn

q pn

q pn n n n+( ) ( ) =

+

+ ⋅ ⋅ ⋅– – – –21

23

1 3 3

3

2

2

1

f xx( ) = 23

1cos–

0–3 3

y

x

y = 2cos–1 x3

(a)

– 6 –

Marks


(a) Prove by induction that

n3 + (n + 1)3 + (n + 2)3

is divisible by 9 for n = 1, 2, 3, …

Consider the variable point P(2at, at2) on the parabola x2 = 4ay .

(i) Prove that the equation of the normal at P is x + ty = at3 + 2at .

(ii) Find the coordinates of the point Q on the parabola such that the normalat Q is perpendicular to the normal at P.

(iii) Show that the two normals of part (ii) intersect at the point R, whosecoordinates are

.

(iv) Find the equation in Cartesian form of the locus of the point R given inpart (iii).

2

x a tt

y a tt

=

= + +

– ,

11

122

4

1

2

O

y

x

x2 = 4ay

QP

R

(b)

3

– 7 –

Marks


(a) A particle moves in a straight line so that its acceleration is given by

where v is its velocity and x is its displacement from the origin.

Initially, the particle is at the origin and has velocity v = 1.

(i) Show that v2 = (x – 1)2.

(ii) By finding an expression for , or otherwise, find x as a function of t.

Question 7 continues on page 9

2dtdx

2

dvdt

x= − 1

– 8 –

Marks

Question 7 (continued)

Consider the diagram, which shows a vertical tower OT of height h metres,

a fixed point A, and a variable point P that is constrained to move so that

angle AOP is radians. The angle of elevation of T from A is radians.

Let the angle of elevation of T from P be α radians and let angle ATP be

θ radians.

(i) By considering triangle AOP, show that

AP 2 = h2 + h2cot2α – h2cotα .

(ii) By finding a second expression for AP2, deduce that

.

(iii) Sketch a graph of θ for , identifying and classifying any

turning points. Discuss the behaviour of θ as α → 0 and as .

End of paper

α π→2

402

< <α π

cos sin cosθ α α= +1

2

1

2 2

3

1

π4

π3

T

π3

O

A

P

π4

α

θ

h

(b)

– 9 –

Marks

BLANK PAGE

– 10 –

BLANK PAGE

– 11 –

– 12 –

STANDARD INTEGRALS

x dxn

x n x n

xdx x x

e dxa

e a

ax dxa

ax a

ax dxa

ax a

ax dxa

ax a

ax ax dxa

ax

n n

ax ax

= + ≠ − ≠ <⌠⌡

= >⌠⌡

= ≠⌠⌡

= ≠⌠⌡

= − ≠⌠⌡

= ≠⌠⌡

=

+11

1 0 0

10

10

10

10

10

1

1

2

, ;

ln ,

,

cos sin ,

sin cos ,

sec tan ,

sec tan sec ,

, if

aa

a xdx

axa

a

a xdx

xa

a a x a

x adx x x a x a

x adx x x a

x x xe

≠⌠⌡

+= ≠⌠

⌡

−= > − < <⌠

⌡

−⌠⌡

= + −( ) > >

+⌠⌡

= + +( )=

−

−

0

1 10

10

10

1

2 21

2 21

2 2

2 2

2 2

2 2

tan ,

sin , ,

ln ,

ln

ln log ,NOTE : >> 0

© Board of Studies NSW 2001








Total marks – 84




411



BLANK PAGE

– 2 –




(a) Evaluate .

(b) Find for x > 0.

(c) Use the table of standard integrals to evaluate .

(d) State the domain and range of the function .

(e) The variable point (3t, 2t2) lies on a parabola. Find the Cartesian equation forthis parabola.

(f) Use the substitution u = 1 – x2 to evaluate . 32

1 2 2

2

3x

xdx

−( )⌠

⌡

2

2f xx

( ) sin=

−32

1

2sec tan2 20

6

x x dx

π⌠⌡

2ddx

x x3 2 ln( )

1lim sinx

xx→0

3

– 3 –


(a) Solve 2x = 3.

Express your answer correct to two decimal places.

(b) Find the general solution to .

Express your answer in terms of π.

(c) Suppose x3 – 2x2 + a ≡ (x + 2)Q(x) + 3 where Q(x) is a polynomial.

Find the value of a.

(d) Evaluate .

In the diagram the points A, B and C lie on the circle and CB produced meetsthe tangent from A at the point T. The bisector of the angle ATC intersects ABand AC at X and Y respectively. Let ∠TAB = β.


(i) Explain why ∠ACB = β.

(ii) Hence prove that triangle AXY is isosceles. 2

1

C

A

B

XY NOT TO

SCALE

αα

Tβ

(e)

32 42

0

4

sin x dx

π⌠⌡

2

22 3cos x =

2

– 4 –

Marks


(a) Seven people are to be seated at a round table.

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How manyseating arrangements are then possible?

(b) (i) Show that f(x) = ex − 3x2 has a root between x = 3.7 and x = 3.8 .

(ii) Starting with x = 3.8, use one application of Newton’s method to find abetter approximation for this root. Write your answer correct to threesignificant figures.

(c) A household iron is cooling in a room of constant temperature 22°C. At timet minutes its temperature T decreases according to the equation

where k is a positive constant.

The initial temperature of the iron is 80°C and it cools to 60°C after 10 minutes.

(i) Verify that T = 22 + Ae−kt is a solution of this equation, where A is aconstant.

(ii) Find the values of A and k.

(iii) How long will it take for the temperature of the iron to cool to 30°C?Give your answer to the nearest minute.

2

2

1

dTdt

k T= − −( )22

3

1

2

1

– 5 –

Marks


(a) Lyndal hits the target on average 2 out of every 3 shots in archery competitions.During a competition she has 10 shots at the target.

(i) What is the probability that Lyndal hits the target exactly 9 times? Leaveyour answer in unsimplified form.

(ii) What is the probability that Lyndal hits the target fewer than 9 times?Leave your answer in unsimplified form.

(b) The polynomial P(x) = x3 − 2x2 + kx + 24 has roots α, β, γ.

(i) Find the value of α + β + γ.

(ii) Find the value of αβγ.

(iii) It is known that two of the roots are equal in magnitude but oppositein sign.

Find the third root and hence find the value of k.

(c) A particle, whose displacement is x, moves in simple harmonic motion such that. At time t = 0, x = 1 and .

(i) Show that, for all positions of the particle,

.

(ii) What is the particle’s greatest displacement?

(iii) Find x as a function of t. You may assume the general form for x. 2

1

x x= −4 2 2

2

x = 4˙x x= −16

2

1

1

2

1

– 6 –

Marks


(a) Use the principle of mathematical induction to show that

2 × 1! + 5 × 2! + 10 × 3! +…+ (n2 + 1)n! = n(n + 1)!

for all positive integers n.

The diagram shows a conical drinking cup of height 12 cm and radius 4 cm. Thecup is being filled with water at the rate of 3 cm3 per second. The height of waterat time t seconds is h cm and the radius of the water’s surface is r cm.

(i) Show that

(ii) Find the rate at which the height is increasing when the height of water

is 9 cm. (Volume of cone = .)

(c) Consider the function

.

(i) Show that for 0 < x < 1.

(ii) Sketch the graph of y = f(x). 2

3′( ) =f x 0

f x x x x( ) = − −( ) ≤ ≤− −2 2 1 0 11 1sin sin for

13

2π r h

3

1r h= 13

.

4 cm

12 cm

r

h

NOT TOSCALE

(b)

3

– 7 –

Marks


An angler casts a fishing line so that the sinker is projected with a speed V m s–1

from a point 5 metres above a flat sea. The angle of projection to the horizontalis θ, as shown.

Assume that the equations of motion of the sinker are

,

referred to the coordinate axes shown.

(i) Let (x,y) be the position of the sinker at time t seconds after the cast, andbefore the sinker hits the water.

It is known that x = Vt cosθ.

Show that .

(ii) Suppose the sinker hits the sea 60 metres away as shown in the diagram.

Find the value of V if .

(iii) For the cast described in part (ii), find the maximum height abovesea level that the sinker achieved.


2

θ = −tan 1 34

3

y Vt t= − +sinθ 5 52

2

˙ ˙x y= = −0 10 and

x

θ

y

O 60 m

NOT TOSCALE

5 m

V

(a)

– 8 –

Marks


(b) Let n be a positive integer.

(i) By considering the graph of y = show that

.

(ii) Hence deduce that

.

End of Question 6

Please turn over

11

11 1

+

< < +

+

ne

n

n n

3

11

11

ndxx n

n

n

+ < <⌠⌡

+

21x

– 9 –

Marks


(a) Let for all real values of x and let for x ≤ 0.

(i) Sketch the graph y = g(x) and explain why g(x) does not have an inversefunction.

(ii) On a separate diagram, sketch the graph of the inverse function y = ƒ–1(x).

(iii) Find an expression for y = ƒ–1(x) in terms of x.

(b) The coefficient of xk in (1 + x)n, where n is a positive integer, is denoted byck (so ck = nCk).

(i) Show that

.

(ii) Find the sum

.

Write your answer as a simple expression in terms of n.

End of paper

c c c c

n nn n0 1 2

1 2 2 3 3 41

1 2. . .− + − + −( )

+( ) +( )K

3

c c c n c nnn

0 1 212 3 1 2 2+ + + + +( ) = +( ) −K

3

3

1

2

f x ee

xx( ) = + 1

g x ee

xx( ) = + 1

– 10 –

Marks

BLANK PAGE

– 11 –

– 12 –

STANDARD INTEGRALS

x dxn

x n x n

xdx x x

e dxa

e a

ax dxa

ax a

ax dxa

ax a

ax dxa

ax a

ax ax dxa

ax

n n

ax ax

= + ≠ − ≠ <⌠⌡

= >⌠⌡

= ≠⌠⌡

= ≠⌠⌡

= − ≠⌠⌡

= ≠⌠⌡

=

+11

1 0 0

10

10

10

10

10

1

1

2

, ;

ln ,

,

cos sin ,

sin cos ,

sec tan ,

sec tan sec ,

, if

aa

a xdx

axa

a

a xdx

xa

a a x a

x adx x x a x a

x adx x x a

x x xe

≠⌠⌡

+= ≠⌠

⌡

−= > − < <⌠

⌡

−⌠⌡

= + −( ) > >

+⌠⌡

= + +( )=

−

−

0

1 10

10

10

1

2 21

2 21

2 2

2 2

2 2

2 2

tan ,

sin , ,

ln ,

ln

ln log ,NOTE : >> 0









Total marks – 84




411






(a) Find the coordinates of the point P that divides the interval joining (−3, 4) and(5, 6) internally in the ratio 1 : 3.

(b) Solve .

(c) Evaluate .

(d) A curve has parametric equations . Find the Cartesian equation

for this curve.

(e) Use the substitution u = x2 + 1 to evaluate

.x

xdx

2 3

0

2

1+( )⌠

⌡

3

2xt

y t= =2

3 2,

2limsinx

xx→0

32

33

21

x −≤

2

– 2 –


(a) Sketch the graph of y = 3cos−1 2x. Your graph must clearly indicate the domainand the range.

(b) Find .

(c) Evaluate .

(d) Find the coefficient of x4 in the expansion of .

(e) (i) Express cosx − sinx in the form Rcos (x + α), where α is in radians.

(ii) Hence, or otherwise, sketch the graph of y = cosx − sinx for 0 ≤ x ≤ 2π. 2

2

22 2 5+( )x

21

2 20

1

−

⌠⌡ x

dx

2ddx

x xtan−( )1

2

– 3 –

Marks


(a) How many nine-letter arrangements can be made using the letters of the wordISOSCELES?

(b) A particle moves in a straight line and its position at time t is given by

.

(i) Show that the particle is undergoing simple harmonic motion.

(ii) Find the amplitude of the motion.

(iii) When does the particle first reach maximum speed after time t = 0?

(c) (i) Explain why the probability of getting a sum of 5 when one pair of fair

dice is tossed is .

(ii) Find the probability of getting a sum of 5 at least twice when a pair ofdice is tossed 7 times.

(d) Use mathematical induction to prove that

for all positive integers n.

11 3

13 5

15 7

12 1 2 1 2 1× + × + × + + −( ) +( ) = +L

n nn

n

3

2

19

1

1

1

2

x t= +

4 2

3sin

π

2

– 4 –

Marks


(a) A committee of 6 is to be chosen from 14 candidates. In how many differentways can this be done?

(b) The function has a zero near x = 1.5. Taking x = 1.5 as a first

approximation, use one application of Newton’s method to find a second

approximation to the zero. Give your answer correct to three decimal places.

(c) It is known that two of the roots of the equation 2x3 + x2 − kx + 6 = 0 arereciprocals of each other. Find the value of k.

In the diagram, CQ and BP are altitudes of the triangle ABC. The lines CQ andBP intersect at T, and AT is produced to meet CB at R.


(i) Explain why CPQB is a cyclic quadrilateral.

(ii) Explain why PAQT is a cyclic quadrilateral.

(iii) Prove that ∠TAQ = ∠QCB.

(iv) Prove that AR ⊥ CB. 2

2

1

1

A

R

Q

BC

PT

(d)

2

3f x xx( ) = −sin

23

1

– 5 –

Marks


(a) Find .

(b) The graph of f(x) = x2 − 4x + 5 is shown in the diagram.

(i) Explain why f(x) does not have an inverse function.

(ii) Sketch the graph of the inverse function, g−1(x), of g(x), whereg(x) = x2 − 4x + 5, x ≤ 2.

(iii) State the domain of g−1(x).

(iv) Find an expression for y = g−1(x) in terms of x.

(c) Dr Kool wishes to find the temperature of a very hot substance using histhermometer, which only measures up to 100°C. Dr Kool takes a sample of thesubstance and places it in a room with a surrounding air temperature of 20°C,and allows it to cool.

After 6 minutes the temperature of the substance is 80°C, and after a further2 minutes it is 50°C. If T(t) is the temperature of the substance after t minutes,then Newton’s law of cooling states that T satisfies the equation

,

where k is a constant and A is the surrounding air temperature.

(i) Verify that T = A + Bekt satisfies the above equation.

(ii) Show that , and find the value of B.

(iii) Hence find the initial temperature of the substance. 1

3k e= −log 2

2

1

dTdt

k T A= −( )

2

1

1

1

y

x

(2, 1)

O

2cos2 3x dx⌠⌡

– 6 –

Marks

– 7 –


(a) The acceleration of a particle P is given by the equation

,

where x metres is the displacement of P from a fixed point O after t seconds.

Initially the particle is at O and has velocity 8 ms−1 in the positive direction.

(i) Show that the speed at any position x is given by 2(x2 + 4) ms−1.

(ii) Hence find the time taken for the particle to travel 2 metres from O.

In the diagram, ABCD is a unit square. Points E and F are chosen on AD and DCrespectively, such that ∠AEG = ∠FHC, where G and H are the points at whichBE and BF respectively cut the diagonal AC.

Let AE = p, FC = q, ∠AEG = α and ∠AGE = β.

(i) Express α in terms of p, and β in terms of q.

(ii) Prove that p + q = 1 − pq.

(iii) Show that the area of the quadrilateral EBFD is given by

.

(iv) What is the maximum value of the area of EBFD? 2

12

12 1

− + −+( )

p pp

1

2

2

A

G

H

B

CD

E

F q

1

1

pβ

α

α

(b)

2

3

d x

dtx x

2

228 4= +( )


(a) David is in a life raft and Anna is in a cabin cruiser searching for him. They arein contact by mobile telephone. David tells Anna that he can see Mt Hope. FromDavid’s position the mountain has a bearing of 109°, and the angle of elevationto the top of the mountain is 16°.

Anna can also see Mt Hope. From her position it has a bearing of 139°, and thetop of the mountain has an angle of elevation of 23°.

The top of Mt Hope is 1500 m above sea level.

Find the distance and bearing of the life raft from Anna’s position.


1500 m

23°

16°

H

B

A

D

4

– 8 –

Marks


(b) A particle is projected from the origin with velocity v ms−1 at an angle α to thehorizontal. The position of the particle at time t seconds is given by the parametricequations

,

where g ms−2 is the acceleration due to gravity. (You are NOT required to derivethese.)

(i) Show that the maximum height reached, h metres, is given by

.

(ii) Show that it returns to the initial height at .

(iii) Chris and Sandy are tossing a ball to each other in a long hallway. Theceiling height is H metres and the ball is thrown and caught at shoulderheight, which is S metres for both Chris and Sandy.

The ball is thrown with a velocity v ms−1. Show that the maximumseparation, d metres, that Chris and Sandy can have and still catch the ballis given by

End of paper

d H Svg

H S v g H S

dvg

v g H S

= × −( )

− −( ) ≥ −( )

= ≤ −( )

42

4

4

22 2

22

, ,

, .

if and

if

S

d

H

4

2xvg

=2

2sin α

hv

g=

2 2

2sin α

2

x vt

y vt gt

=

= −

cos

sin

α

α 12

2

– 9 –

Marks

BLANK PAGE

– 10 –

BLANK PAGE

– 11 –

– 12 –

STANDARD INTEGRALS

x dxn

x n x n

xdx x x

e dxa

e a

ax dxa

ax a

ax dxa

ax a

ax dxa

ax a

ax ax dxa

ax

n n

ax ax

= + ≠ − ≠ <⌠⌡

= >⌠⌡

= ≠⌠⌡

= ≠⌠⌡

= − ≠⌠⌡

= ≠⌠⌡

=

+11

1 0 0

10

10

10

10

10

1

1

2

, ;

ln ,

,

cos sin ,

sin cos ,

sec tan ,

sec tan sec ,

, if

aa

a xdx

axa

a

a xdx

xa

a a x a

x adx x x a x a

x adx x x a

x x xe

≠⌠⌡

+= ≠⌠

⌡

−= > − < <⌠

⌡

−⌠⌡

= + −( ) > >

+⌠⌡

= + +( )=

−

−

0

1 10

10

10

1

2 21

2 21

2 2

2 2

2 2

2 2

tan ,

sin , ,

ln ,

ln

ln log ,NOTE : >> 0



411










Total marks – 84






(a) Indicate the region on the number plane satisfied by .

(b) Solve .

(c) Let A be the point (3, −1) and B be the point (9, 2).

Find the coordinates of the point P which divides the interval AB externally inthe ratio 5 :2.

(d) Find .

(e) Use the substitution u = x − 3 to evaluate

.x x dx−⌠⌡

33

4

3

2dx

x4 20

1

−

⌠⌡

2

34

13

x +<

2y x≥ + 1

– 2 –


(a) Evaluate .

(b) Find .

The line AT is the tangent to the circle at A, and BT is a secant meeting thecircle at B and C.

Given that AT = 12, BC = 7 and CT = x, find the value of x.

(d) (i) Write 8cosx + 6sinx in the form Acos(x − α), where A > 0 and 0 ≤ α ≤ π–2

.

(ii) Hence, or otherwise, solve the equation 8cosx + 6sinx = 5 for 0 ≤ x ≤ 2π.Give your answers correct to three decimal places.

(e) A four-person team is to be chosen at random from nine women and seven men.

(i) In how many ways can this team be chosen?

(ii) What is the probability that the team will consist of four women? 1

1

2

2

2

x

C

B

AT

7

12

(c)

2ddx

xcos− ( )1 23

2limsin

x

x

x→

0

52

– 3 –

Marks


(a) Find .

(b) Let P(x) = (x + 1)(x − 3)Q(x) + a (x + 1) + b, where Q(x) is a polynomialand a and b are real numbers.

When P(x) is divided by (x + 1) the remainder is −11.

When P(x) is divided by (x − 3) the remainder is 1.

(i) What is the value of b?

(ii) What is the remainder when P(x) is divided by (x + 1)(x − 3)?

(c) A ferry wharf consists of a floating pontoon linked to a jetty by a 4 metre longwalkway. Let h metres be the difference in height between the top of thepontoon and the top of the jetty and let x metres be the horizontal distancebetween the pontoon and the jetty.

(i) Find an expression for x in terms of h.

(ii) When the top of the pontoon is 1 metre lower than the top of the jetty, thetide is rising at a rate of 0.3 metres per hour.

At what rate is the pontoon moving away from the jetty?


3

1

4

xh

Jetty

Pontoon

2

1

2cos2 4x dx⌠⌡

– 4 –

Marks


The length of each edge of the cube ABCDEFGH is 2 metres. A circle is drawnon the face ABCD so that it touches all four edges of the face. The centre of thecircle is O and the diagonal AC meets the circle at X and Y.

(i) Explain why ∠FAC = 60°.

(ii) Show that .

(iii) Calculate the size of ∠XFY to the nearest degree.

End of Question 3

1

1FO = 6 metres

1

B

X

A

CD

FE

O

Y

H G

(d)

– 5 –

Marks


(a) Use mathematical induction to prove that for all integers n ≥ 3,

.

(b) The two points are on the parabola x2 = 4ay.

(i) The equation of the tangent to x2 = 4ay at an arbitrary point onthe parabola is y = tx − at2. (Do not prove this.)

Show that the tangents at the points P and Q meet at R, where R is thepoint .

(ii) As P varies, the point Q is always chosen so that ∠POQ is a right angle,where O is the origin.

Find the locus of R.

(c) Katie is one of ten members of a social club. Each week one member is selectedat random to win a prize.

(i) What is the probability that in the first 7 weeks Katie will win at least1 prize?

(ii) Show that in the first 20 weeks Katie has a greater chance of winningexactly 2 prizes than of winning exactly 1 prize.

(iii) For how many weeks must Katie participate in the prize drawing so thatshe has a greater chance of winning exactly 3 prizes than of winningexactly 2 prizes?

2

2

1

2

a p q apq+( )( ),

22 2at at,( )P ap ap Q aq aq2 22 2, and , ( ) ( )

123

124

125

12 2

1−

−

−

−

=

−( )Kn n n

3

– 6 –

Marks


(a) A particle is moving along the x-axis, starting from a position 2 metres to theright of the origin (that is, x = 2 when t = 0) with an initial velocity of 5 ms−1

and an acceleration given by

.

(i) Show that .

(ii) Hence find an expression for x in terms of t.

(b) The diagram below shows a sketch of the graph of y = f(x), wherefor x ≥ 0.

(i) Copy or trace this diagram into your writing booklet.

On the same set of axes, sketch the graph of the inverse function, y = f −1(x).

(ii) State the domain of f −1(x).

(iii) Find an expression for y = f −1(x) in terms of x.

(iv) The graphs of y = f(x) and y = f −1(x) meet at exactly one point P. Let α be the x-coordinate of P. Explain why α is a root of the equation

x3 + x − 1 = 0 .

(v) Take 0.5 as a first approximation for α. Use one application of Newton’smethod to find a second approximation for α.

2

1

2

1

1

y

x1

1

O

f xx

( ) =+1

1 2

3

2x x= +2 1

˙x x x= +2 23

– 7 –

Marks


The points A, B, C and D are placed on a circle of radius r such that AC and BDmeet at E. The lines AB and DC are produced to meet at F, and BECF is a cyclicquadrilateral.

Copy or trace this diagram into your writing booklet.

(i) Find the size of ∠DBF, giving reasons for your answer.

(ii) Find an expression for the length of AD in terms of r.


1

2

A

B

CD

E

F

NOT TOSCALE

(a)

– 8 –

Marks


(b) A fire hose is at ground level on a horizontal plane. Water is projected from thehose. The angle of projection, θ, is allowed to vary. The speed of the water as itleaves the hose, v metres per second, remains constant. You may assume that ifthe origin is taken to be the point of projection, the path of the water is given bythe parametric equations

x = vtcosθ

y = vtsinθ − 1–2

gt2

where g ms−2 is the acceleration due to gravity. (Do NOT prove this.)

(i) Show that the water returns to ground level at a distance metresfrom the point of projection.

This fire hose is now aimed at a 20 metre high thin wall from a point ofprojection at ground level 40 metres from the base of the wall. It is known thatwhen the angle θ is 15°, the water just reaches the base of the wall.

(ii) Show that v2 = 80g.

(iii) Show that the cartesian equation of the path of the water is given by

.

(iv) Show that the water just clears the top of the wall if

tan2θ − 4tanθ + 3 = 0.

(v) Find all values of θ for which the water hits the front of the wall.

End of Question 6

2

2

y xx= −tan

secθ θ2 2

160

2

1

40

v

θ

20

Wall

2v

g

2 2sin θ

– 9 –

Marks


(a) The rise and fall of the tide is assumed to be simple harmonic, with the timebetween successive high tides being 12.5 hours. A ship is to sail from a wharf tothe harbour entrance and then out to sea. On the morning the ship is to sail, hightide at the wharf occurs at 2 am. The water depths at the wharf at high tide andlow tide are 10 metres and 4 metres respectively.

(i) Show that the water depth, y metres, at the wharf is given by

, where t is the number of hours after high tide.

(ii) An overhead power cable obstructs the ship’s exit from the wharf. Theship can only leave if the water depth at the wharf is 8.5 metres or less.

Show that the earliest possible time that the ship can leave the wharfis 4:05 am.

(iii) At the harbour entrance, the difference between the water level at hightide and low tide is also 6 metres. However, tides at the harbour entranceoccur 1 hour earlier than at the wharf. In order for the ship to be able tosail through the shallow harbour entrance, the water level must be at least2 metres above the low tide level.

The ship takes 20 minutes to sail from the wharf to the harbour entranceand it must be out to sea by 7 am. What is the latest time the ship can leavethe wharf?

(b) (i) Show that for all positive integers n,

.

(ii) Hence show that for 1 ≤ k ≤ n,

.

(iii) Show that .

(iv) By differentiating both sides of the identity in (i), show that for 1 ≤ k < n,

.

End of paper

nn

kn

n

kk

k

kk

n

k−( )

−−

+ −( )−−

+ +−−

=+

12

12

3

1

1

1 1L

3

1nn

kk

n

k

−

= +( )+

11

1

n

k

n

k

n

k

k

k

n

k

−−

+−−

+−−

+ +−−

=

1

1

2

1

3

1

1

1L

1

x x x x x xn n n1 1 1 1 1 1 11 2 2+( ) + +( ) + + +( ) + +( ) +[ ] = +( ) −− − L

1

2

2

yt= +

7 3

425

cosπ

2

– 10 –

Marks

BLANK PAGE

– 11 –

– 12 –

STANDARD INTEGRALS

x dxn

x n x n

xdx x x

e dxa

e a

ax dxa

ax a

ax dxa

ax a

ax dxa

ax a

ax ax dxa

ax

n n

ax ax

= + ≠ − ≠ <⌠⌡

= >⌠⌡

= ≠⌠⌡

= ≠⌠⌡

= − ≠⌠⌡

= ≠⌠⌡

=

+11

1 0 0

10

10

10

10

10

1

1

2

, ;

ln ,

,

cos sin ,

sin cos ,

sec tan ,

sec tan sec ,

, if

aa

a xdx

axa

a

a xdx

xa

a a x a

x adx x x a x a

x adx x x a

x x xe

≠⌠⌡

+= ≠⌠

⌡

−= > − < <⌠

⌡

−⌠⌡

= + −( ) > >

+⌠⌡

= + +( )=

−

−

0

1 10

10

10

1

2 21

2 21

2 2

2 2

2 2

2 2

tan ,

sin , ,

ln ,

ln

ln log ,NOTE : >> 0



411










Total marks – 84



BLANK PAGE

– 2 –




(a) Find .

(b) Sketch the region in the plane defined by .

(c) State the domain and range of .

(d) Using the substitution u = 2x2 + 1, or otherwise, find .

(e) The point P(1, 4) divides the line segment joining A(–1, 8) and B (x, y)internally in the ratio 2 : 3. Find the coordinates of the point B .

(f) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45°. Find thetwo possible values of m .

2

2

3x x dx2 1254+( )⌠

⌡

2yx=

−cos 1

4

2y x≤ +2 3

11

492xdx

+⌠⌡

– 3 –


(a) Find .

(b) Use the binomial theorem to find the term independent of x in the expansion

of .

(c) (i) Differentiate e3x(cosx – 3sinx) .

(ii) Hence, or otherwise, find .

(d) A salad, which is initially at a temperature of 25°C, is placed in a refrigeratorthat has a constant temperature of 3°C. The cooling rate of the salad isproportional to the difference between the temperature of the refrigerator andthe temperature, T, of the salad. That is, T satisfies the equation

= – k(T – 3),

where t is the number of minutes after the salad is placed in the refrigerator.

(i) Show that T = 3 + Ae–kt satisfies this equation.

(ii) The temperature of the salad is 11°C after 10 minutes. Find thetemperature of the salad after 15 minutes.

3

1

dT

dt

1e x dxx3 sin⌠⌡

2

212

12

xx

−

3

2ddx

x2 51sin–( )

– 4 –

Marks


(a) (i) Show that the function g(x) = x2 – loge(x + 1) has a zero between0.7 and 0.9.

(ii) Use the method of halving the interval to find an approximation to thiszero of g(x) , correct to one decimal place.

(b) (i) By expanding the left-hand side, show that

sin(5x + 4x) + sin(5x – 4x) = 2sin5x cos4x.

(ii) Hence find .

(c) Use the definition of the derivative, , to find

when .

In the circle centred at O the chord AB has length 7. The point E lies on AB andAE has length 4. The chord CD passes through E.

Let the length of CD be l and the length of DE be x .

(i) Show that x2 – lx + 12 = 0.

(ii) Find the length of the shortest chord that passes through E. 2

2

A

B

E

C

x4

NOT TOSCALE

D

O

(d)

f x x x( ) = +2 5

2′( )f x′( ) =+( ) − ( )

→f x

f x h f x

hhlim

0

2sin cos5 4x x dx⌠⌡

1

2

1

– 5 –

Marks


(a) Evaluate .

(b) By making the substitution , or otherwise, show that

.

(c) The points P(2ap, ap2) and Q(2aq, aq2) lie on the parabola x2 = 4ay .The equation of the normal to the parabola at P is x + py = 2ap + ap3 andthe equation of the normal at Q is similarly given by x + qy = 2aq + aq3.

(i) Show that the normals at P and Q intersect at the point R whosecoordinates are

(–apq[p + q], a[p2 + pq + q2 + 2]).

(ii) The equation of the chord PQ is . (Do NOT show this.)

If the chord PQ passes through (0, a), show that pq = –1.

(iii) Find the equation of the locus of R if the chord PQ passes through (0, a).

(d) Use the principle of mathematical induction to show that 4n – 1 – 7n > 0 for allintegers n ≥ 2.

3

2

1y p q x apq= +( ) −12

2

cosecθ θ θ+ =cot cot2

2t = tanθ2

2cos sinx x dx2

0

4π

⌠⌡

– 6 –

Marks


(a) Find the exact value of the volume of the solid of revolution formed when the

region bounded by the curve y = sin2x , the x-axis and the line x = is rotated

about the x-axis.

(b) Two chords of a circle, AB and CD, intersect at E. The perpendiculars to AB at Aand CD at D intersect at P. The line PE meets BC at Q, as shown in the diagram.

(i) Explain why DPAE is a cyclic quadrilateral.

(ii) Prove that ∠APE = ∠ABC.

(iii) Deduce that PQ is perpendicular to BC .

(c) A particle moves in a straight line and its position at time t is given by

.

(i) Express in the form R sin(3t – α) , where α isin radians.

(ii) The particle is undergoing simple harmonic motion. Find the amplitudeand the centre of the motion.

(iii) When does the particle first reach its maximum speed after time t = 0? 1

2

23 3 3sin cost t−

x t t= + −5 3 3 3sin cos

1

2

1

D

P

BA

C Q

E

π8

3

– 7 –

Marks


(a) There are five matches on each weekend of a football season. Megan takes part

in a competition in which she earns one point if she picks more than half of the

winning teams for a weekend, and zero points otherwise. The probability that

Megan correctly picks the team that wins any given match is .

(i) Show that the probability that Megan earns one point for a givenweekend is 0.7901, correct to four decimal places.

(ii) Hence find the probability that Megan earns one point every week of theeighteen-week season. Give your answer correct to two decimal places.

(iii) Find the probability that Megan earns at most 16 points during theeighteen-week season. Give your answer correct to two decimal places.


2

1

2

23

– 8 –

Marks


(b) An experimental rocket is at a height of 5000 m, ascending with a velocity of

m s–1 at an angle of 45° to the horizontal, when its engine stops.

After this time, the equations of motion of the rocket are:

x = 200t

y = –4.9t2 + 200t + 5000,

where t is measured in seconds after the engine stops. (Do NOT show this.)

(i) What is the maximum height the rocket will reach, and when will it reachthis height?

(ii) The pilot can only operate the ejection seat when the rocket isdescending at an angle between 45° and 60° to the horizontal. What arethe earliest and latest times that the pilot can operate the ejection seat?

(iii) For the parachute to open safely, the pilot must eject when the speed ofthe rocket is no more than 350 m s –1. What is the latest time at which thepilot can eject safely?

End of Question 6

2

3

2

xO

y

5000 m

200 2

– 9 –

Marks


(a) An oil tanker at T is leaking oil which forms a circular oil slick. An observer ismeasuring the oil slick from a position P, 450 metres above sea level and2 kilometres horizontally from the centre of the oil slick.

(i) At a certain time the observer measures the angle, α , subtended bythe diameter of the oil slick, to be 0.1 radians. What is the radius, r,at this time?

(ii) At this time, = 0.02 radians per hour. Find the rate at which the radius

of the oil slick is growing.

(b) Let ƒ (x) = Ax3 – Ax + 1, where A > 0.

(i) Show that ƒ (x) has stationary points at .

(ii) Show that ƒ (x) has exactly one zero when .

(iii) By observing that ƒ (–1) = 1, deduce that ƒ (x) does not have a zero in

the interval –1 ≤ x ≤ 1 when .

(iv) Let g(θ) = 2cosθ + tanθ, where .

By calculating g′(θ) and applying the result in part (iii), or otherwise,

show that g(θ) does not have any stationary points.

(v) Hence, or otherwise, deduce that g(θ) has an inverse function.

End of paper

1

3− < <π θ π2 2

03 3

2< <A

1

2A < 3 32

1x = ± 33

2dαdt

2

2 kmr

T

P

r

450 m

α

– 10 –

Marks

BLANK PAGE

– 11 –

– 12 –

STANDARD INTEGRALS

x dxn

x n x n

xdx x x

e dxa

e a

ax dxa

ax a

ax dxa

ax a

ax dxa

ax a

ax ax dxa

ax

n n

ax ax

= + ≠ − ≠ <⌠⌡

= >⌠⌡

= ≠⌠⌡

= ≠⌠⌡

= − ≠⌠⌡

= ≠⌠⌡

=

+11

1 0 0

10

10

10

10

10

1

1

2

, ;

ln ,

,

cos sin ,

sin cos ,

sec tan ,

sec tan sec ,

, if

aa

a xdx

axa

a

a xdx

xa

a a x a

x adx x x a x a

x adx x x a

x x xe

≠⌠⌡

+= ≠⌠

⌡

−= > − < <⌠

⌡

−⌠⌡

= + −( ) > >

+⌠⌡

= + +( )=

−

−

0

1 10

10

10

1

2 21

2 21

2 2

2 2

2 2

2 2

tan ,

sin , ,

ln ,

ln

ln log ,NOTE : >> 0









• 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



411

BLANK PAGE

– 2 –

Total marks – 84 Attempt Questions 1–7 All questions are of equal value


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

2

3

2

2

3

(a) Find . dx

x49 + 2

⌠

⌡⎮

(b) Using the substitution u = x4 + 8, or otherwise, find x3⌠ ⌡⎮ x4 8+ .dx

(c) Evaluate .lim sin

x

x

x→0

5 3

(d) Using the sum of two cubes, simplify:

,

for .0 2

< <θ π

sin cos sin cos

3 3 1

θ θ θ θ

+ +

−

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

– 3 –

Marks

2

2

2

1

1


(a) Let ƒ(x) = sin–1(x + 5).

(i) State the domain and range of the function ƒ(x).

(ii) Find the gradient of the graph of y = ƒ(x) at the point where x = –5.

(iii) Sketch the graph of y = ƒ(x).

(b) (i) By applying the binomial theorem to (1 + x)n and differentiating,show that

n n−1 ⎛n⎞ ⎛n⎞ ⎛n⎞r 1

⎛n⎞(1+ x) = ⎜ ⎟ + 2 ⎜ ⎟ x + � + −− −r ⎜ ⎟ x + � + n ⎜ ⎟ xn 1 .

1 2 r n⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(ii) Hence deduce that

n n n n3n−1

⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ + � + r ⎜ ⎟ 2r−1

⎛ ⎞ + � + n ⎜ ⎟ 2nn−1 .

⎝1⎠ ⎝r ⎠ ⎝n⎠


– 4 –

Marks

1

2

1


(c)

x

UT

y

P

Q R

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 p r x apr= +( ) −1 2

(Do NOT prove this.)

The equation of the tangent at P is y = px – ap2 . (Do NOT prove this.)

(i)

(ii)

(iii)

Find the coordinates of U.

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.

End of Question 2

– 5 –

π⌠ 4

⎮ sin2 x dx . ⌡0


(a) Find

(b) (i) By considering ƒ( )x = 3log x − xe , show that the curve y = 3log x e and

the line y = x meet at a point P whose x-coordinate is between 1.5 and 2.

(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.

(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?

Marks

2

1

2

1

2


– 6 –

Marks

1

1

2


(d)

Q

T

M

K

N

P

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) Show that QKTM is a cyclic quadrilateral.

(ii) Show that ∠KMT = ∠KQT .

(iii) Hence, or otherwise, show that MK is parallel to TP.

End of Question 3

– 7 –

BLANK PAGE

– 8 –

Marks

1

2

1

2

2

2

2


(a) The cubic polynomial , 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 .

P(x) = x3 + rx2 + sx + t

(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?

– 9 –

(c) A particle is moving so that x�� = 18x3 + 27x2 + 9x .

Initially x = –2 and the velocity, v, is –6.

(i) Show that v2 = 9x2(1 + x)2.

(ii) Hence, or otherwise, show that

(iii) It can be shown that for some constant c,


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

⌠ 1⎮ dx = −3t . ⌡ x (1+ x)

⎛ 1 ⎞loge ⎜1+ ⎟ = 3t c+ .

⎝ x ⎠

Marks

2

2

2

2


(a) Show that

(b) Let

= 10 e–0.7t + 3 is a solution of = −0 7. (y − 3). dt

ydy

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

(c)

ƒ( )x = log exe 1+

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


(i) Show that

(ii) Hence, or otherwise, show that it takes 3.5 times as long to fill the bowl

to the point where as it does to fill the bowl to the point where


π V = x2 (3r − x) .

3

dx k= . dt πx r x(2 − )

2x = r

31

x = r . 3

– 10 –

tan (α β) tanα − tan β− = 1+ tanα tan β

Marks

1

3


(d) (i) Use the fact that to show that

1+ tan nθ tan (n +1)θ = cotθ (tan (n +1)θ − tan nθ) .

(ii) Use mathematical induction to prove that, for all integers n ≥ 1,

tanθ tan 2θ + tan 2θ tan 3θ + � + tan nθ tan (n +1)θ = − (n +1)) + cotθ tan (n +1)θ .

End of Question 5

– 11 –


(a) Two particles are fired simultaneously from the ground at time t = 0.

Particle 1 is projected from the origin at an angle θ, 0 θ π< < , 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.

x

V

θ

V

O A

y

It can be shown that while both particles are in flight, Particle 1 has equations of motion:

x Vt θ= cos

2y Vt sin θ − 1 gt= ,

2

and Particle 2 has equations of motion:

x a =

y Vt − 1 gt2= .

2

Do NOT prove these equations of motion.

Let L be the distance between the particles at time t.


– 12 –

Marks

2

3

1

1

2

1

2


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

L 2 = 2V 2t 2 1 − sin θ − 2aVt cos θ + a . ( ) 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

a cos θ t =

2 1V ( − sin θ) . 1 − sinθ

2

(iii) Show that the smallest distance between the two particles in flight occurs

while Particle 1 is ascending if ag cos θ

V > .2sin θ (1− sin θ)

s aand that this smallest distance i

(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 isq = 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.

End of Question 6

(

(

– 13 –

w 2 dA cos θ ( sin θ − θ cos θ )= d θ 2 θ 3


π(a) Show that, when 0 < ≤θ , the cross-sectional area is

2

A r= 2 ( θ − si θ cos ) .

(b) The formula in part (a) for A is true for 0 < θ < π . (Do NOT prove this.)

By first expressing r in terms of w and θ, and then differentiating, show that

for 0 < θ < π .


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

CROSS-SECTION 2θ

B C r r

Gutter

B C

ww

Marks

2

3

– 14 –


(c) Let g ( )θ = sin θ − θ cos θ .

By considering g ′( )θ , show that g ( )θ > 0 for 0 < θ < π .

Marks

3

2

2

– 15 –

(d) Show that there is exactly one value of θ in the interval 0 < θ < π for which

dA = 0 . dθ

dA (e) Show that the value of θ for which = 0 gives the maximum cross-sectional

dθ area. Find this area in terms of w.

End of paper

STANDARD INTEGRALS

⌠ x dn 1 x = x n+1

⎮ , n ≠ −1; x ≠ 0, if n < 0 ⌡ n + 1

⌠ 1 ⎮ dx = ln x, x > 0 ⌡ x

⌠ ax 1 ax⎮ e dx = e , a ≠ 0 ⌡ a

⌠ 1 ⎮ cosax dx = sin ax, a ≠ 0 ⌡ a

⌠ 1 ⎮ sin ax dx = − cosax, a ≠ 0 ⌡ a

⌠ 1 ⎮ sec2 ax dx = tan ax, a ≠ 0⌡ a

⌠ 1 ⎮ sec ax tan ax dx = sec ax, aa ≠ 0 ⌡ a

⌠ 1 1 ⎮ = tan −

xdx 1 , a ≠ 0

⌡ a2 + x2 a a

⌠ 1 − x dx = sin 1 , a > 0, − a < x < a

a

⌠ 1 ⎮ dx = ln⌡ x2 − a2 (x + x 2 − a 2 ) , x > >a 0

⌠ 1 dx = ln(x + ⎮ x2 + a2

⌡ x2 + a2 )NOTE : ln x = log x x e , >> 0

⎮ ⌡ a 2 − x2

– 16 –


2007 H I G H E R S C H O O L C E R T I F I C AT E










Total marks – 84



411



Marks

2

2

2

3

3


3(a) Write ( 1 + 5 ) in the form a b+ 5 , where a and b are integers.

(b) The interval AB, where A is (4, 5) and B is (19, –5), is divided internally in the

ratio 2 : 3 by the point P(x, y). Find the values of x and y.

(c) Differentiate tan–1(x4) with respect to x.

(d) The graphs of the line x – 2y + 3 = 0 and the curve y = x3 + 1 intersect at (1, 2). Find the exact value, in radians, of the acute angle between the line and the tangent to the curve at the point of intersection.

2 ⌠ 4 2 x (e) Use the substitution u = 25 – x to evaluate ⎮ dx .

⌡3 25 − x2

– 2 –

Marks

2

2

1

3

2

2


θ 1 − cos θ θ(a) By using the substitution t = tan , or otherwise, show that = tan .

2 sin θ 2

−(b) Let ƒ ( )x = 2cos 1 x .

(i) Sketch the graph of y = ƒ ( ) x , indicating clearly the coordinates of the endpoints of the graph.

(ii) State the range of ƒ ( )x .

(c) The polynomial P(x) = x2 + ax + b has a zero at x = 2. When P(x) is divided by x + 1, the remainder is 18.

Find the values of a and b.

(d) A skydiver jumps from a hot air balloon which is 2000 metres above the ground. The velocity, v metres per second, at which she is falling t seconds after jumping is given by v = 50(1 – e–0.2t).

(i) Find her acceleration ten seconds after she jumps. Give your answer correct to one decimal place.

(ii) Find the distance that she has fallen in the first ten seconds. Give your answer correct to the nearest metre.

– 3 –

Marks

3

3

2

2

2


(a) Find the volume of the solid of revolution formed when the region bounded

1by the curve y = , the x-axis, the y-axis and the line x = 3, is rotated

9 + x 2

about the x-axis.

x − 2(b) (i) Find the vertical and horizontal asymptotes of the hyperbola y =

x − 4 x − 2

and hence sketch the graph of y = . x − 4

x − 2(ii) Hence, or otherwise, find the values of x for which ≤ 3 .

x − 4

(c) A particle is moving in a straight line with its acceleration as a function of x

given by x�� = − −e 2 x . It is initially at the origin and is travelling with a velocity

of 1 metre per second.

(i) Show that x e� = −x .

(ii) Hence show that x = log (te + 1).

– 4 –

Marks

1

1

2

3

3

2


(a) In a large city, 10% of the population has green eyes.

(i) What is the probability that two randomly chosen people both have green eyes?

(ii) What is the probability that exactly two of a group of 20 randomly chosen people have green eyes? Give your answer correct to three decimal places.

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

(b) Use mathematical induction to prove that 72n–1 + 5 is divisible by 12, for all integers n ≥ 1.

(c) A B

P NOT

X Q TO SCALE

C D

The diagram shows points A, B, C and D on a circle. The lines AC and BD are perpendicular and intersect at X. The perpendicular to AD through X meets AD at P and BC at Q.

Copy or trace this diagram into your writing booklet.

(i) Prove that ∠QXB = ∠QBX.

(ii) Prove that Q bisects BC.

– 5 –

Marks

2

2

2

3


(a)

O

θ Q r

T P

The points P and Q lie on the circle with centre O and radius r. The arc PQ subtends an angle θ at O. The tangent at P and the line OQ intersect at T, as shown in the diagram.

(i) The arc PQ divides triangle TPO into two regions of equal area.

Show that tan θ = 2θ.

(ii) A first approximation to the solution of the equation 2θ – tan θ = 0 is θ = 1.15 radians. Use one application of Newton’s method to find a better approximation. Give your answer correct to four decimal places.

(b) Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

(c) Find the exact values of x and y which satisfy the simultaneous equations

−1 1 sin os −x + c 1 π

y = and 2 3

1 2π3sin −1 x − cos −1 y = .

2 3


– 6 –

Marks

1

2


(d) y

Q (2aq, aq2)

x2 = 4ay

P (2ap, ap2)

O x

The diagram shows a point P (2ap, ap2) on the parabola x2 = 4ay. The normal

to the parabola at P intersects the parabola again at Q(2aq, aq2).

The equation of PQ is x + py – 2ap – ap3 = 0. (Do NOT prove this.)

(i) Prove that p2 + pq + 2 = 0.

(ii) If the chords OP and OQ are perpendicular, show that p2 = 2.

End of Question 5

– 7 –

BLANK PAGE

– 8 –

Marks

2

1

2

2

1

3

1


(a) A particle moves in a straight line. Its displacement, x metres, after t seconds is given by

x = 3 sin 2t − cos 2t + 3.

(i) Prove that the particle is moving in simple harmonic motion about

x = 3 by showing that x�� = −4(x − 3) .

(ii) What is the period of the motion?

(iii) Express the velocity of the particle in the form x� = Acos (2 t − α) , where α is in radians.

(iv) Hence, or otherwise, find all times within the first π seconds when the particle is moving at 2 metres per second in either direction.

(b) Consider the function ƒ ( ) = x − − x e e x .

(i) Show that ƒ ( ) x is increasing for all values of x.

(ii) Show that the inverse function is given by

⎛−1( ) x + x 2 4 ⎞+ ƒ x = log e ⎜ ⎟ .

⎝ 2 ⎠

(iii) Hence, or otherwise, solve ex – e – x = 5. Give your answer correct to two decimal places.

– 9 –

Marks

1

2


a

y = loge x

y = kxn

common tangent

O

y

x

(a)

The graphs of the functions y = kxn and y = log xe have a common tangent at x = a, as shown in the diagram.

(i) By considering gradients, show that a n 1 = . nk

(ii) Express k as a function of n by eliminating a.


– 10 –

y

8 metres e

θ O 1010 metres metres x

Marks

2

2

2

3


(b) A small paintball is fired from the origin with initial velocity 14 metres per second towards an eight-metre high barrier. The origin is at ground level, 10 metres from the base of the barrier.

The equations of motion are

x = 14t cos θ

y = 14t sin θ – 4.9t2

where θ is the angle to the horizontal at which the paintball is fired and t is the time in seconds. (Do NOT prove these equations of motion.)

(i) Show that the equation of trajectory of the paintball is

⎛ 1 + m 2 ⎞ y m= x − x2 , where . ⎜ ⎟ m = tan θ

⎝ 40 ⎠

(ii) Show that the paintball hits the barrier at height h metres when

m = ± 2 3 − 0 . 4h .

Hence determine the maximum value of h.

(iii) There is a large hole in the barrier. The bottom of the hole is 3.9 metres above the ground and the top of the hole is 5.9 metres above the ground. The paintball passes through the hole if m is in one of two intervals. One interval is 2.8 ≤ m ≤ 3.2.

Find the other interval.

(iv) Show that, if the paintball passes through the hole, the range is

40 m metres.

1 + m2

Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier.

End of paper – 11 –

STANDARD INTEGRALS

⌠ 1⎮ +x dn x = x n 1 , n 1; x ⎮ ≠ − ≠ 0, if n < 0⌡ n + 1

⌠ 1⎮ dx = ln x , xx > 0⎮ x⌡

⌠ ⎮ e dax 1

x = e ax , a ≠ 0⎮ a⌡

⌠ 1⎮ cosax dx = sinax , aa⎮ ≠ 0 a⌡

⌠ 1⎮ sinax dx = − cosax , a ≠ 0⎮ a⌡

⌠ ⎮ 2 1

sec ax dx = ttan ax , a ⎮ ≠ 0 a⌡

⌠ 1⎮ secax tanax dx ⎮ = secax , a ≠ 0a⌡

⌠ 1 1 −1 x ⎮ dx = tan , a ≠ 0⎮ ⌡ a aaa2 + x 2

⌠ 1 − x⎮ dx sin 1 , aa 0, a x a⎮ = > − < < 2 2 a ⌡ a − x

⌠ 1 ⎮ dx = ln(x + x 2 − a 2 ), x > a >> 0⎮⌡ x2 − a2

⌠ 1 ⎮ dx = ln(x + x 2 + a 2⎮ ⌡ x 2 + a 2

)NOTE : ln x = log xx , xe > 0

– 12 –












Total marks – 84



411

BLANK PAGE

– 2 –

π ⌠ 4

⎮ cosθ sin 2 θ d θ .⌡0

⌠ 1 1 dx .⎮

⌡− 1 4 − x 2




(a) The polynomial x3 is divided by x + 3. Calculate the remainder.

(b) Differentiate cos–1 (3x) with respect to x.

(c) Evaluate

(d) Find an expression for the coefficient of x8y4 in the expansion of (2x + 3y)12 .

(e) Evaluate

(f) Let . ƒ ( )x = log e ⎣ ⎡(x − 3 )(5 − x)⎤⎦What is the domain of ƒ ( ) x ?

Marks

2

2

2

2

2

2

– 3 –

d ⎛ 1 ⎞⎜ v 2⎟dx ⎝ 2 ⎠

e2

⌠ 1⎮ dx . ⎮ 2 ⌡ e x (log xe )

Marks

3

3

3

3


(a) Use the substitution u = log x to evaluate e

(b) A particle moves on the x-axis with velocity v. The particle is initially at rest at x = 1. Its acceleration is given by x�� = +x 4.

Using the fact that x�� = , find the speed of the particle at x = 2.

(c) The polynomial p(x) is given by p(x) = ax3 + 16x2 + cx – 120, where a and c are constants.

The three zeros of p(x) are –2, 3 and α.

Find the value of α.

(d) The function ƒ ( )x = tan x − log e x has a zero near x = 4.

Use one application of Newton’s method to obtain another approximation to this zero. Give your answer correct to two decimal places.

– 4 –

A

θ

�

P O C Q xx

Marks

1

3

3

2

1

2


(a) (i) Sketch the graph of y = 2x − 1 .

(ii) Hence, or otherwise, solve 2x − 1 ≤ x − 3 .

(b) Use mathematical induction to prove that, for integers n ≥ 1,

n1 × +3 2 4 3 × + × +5 � + n n ( + 2 ) = ( n + 1 2)( n + 7) .

6

(c)

A race car is travelling on the x-axis from P to Q at a constant velocity, v.

A spectator is at A which is directly opposite O, and OA = � metres. When the

car is at C, its displacement from O is x metres and ∠OAC = θ, with

π π− < θ .2 2

dθ v�(i) Show that = .

dt �2 + x 2

dθ(ii) Let m be the maximum value of .

dt

Find the value of m in terms of v and � .

dθ m(iii) There are two values of θ for which = .

dt 4 Find these two values of θ.

– 5 –

Marks

2

3

1

1


(a) A turkey is taken from the refrigerator. Its temperature is 5°C when it is placed in an oven preheated to 190°C.

Its temperature, T°C, after t hours in the oven satisfies the equation

dT = −k T 190)( − . dt

(i) Show that T = 190 – 185e –kt satisfies both this equation and the initial condition.

(ii) The turkey is placed into the oven at 9 am. At 10 am the turkey reaches a temperature of 29°C. The turkey will be cooked when it reaches a temperature of 80°C.

At what time (to the nearest minute) will it be cooked?

(b) Barbara and John and six other people go through a doorway one at a time.

(i) In how many ways can the eight people go through the doorway if John goes through the doorway after Barbara with no-one in between?

(ii) Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.


– 6 –

Marks

2

1

2


y

O

M

x K

T

L

P (2ap, ap2)

x2 = 4ay

Q(2aq, aq2)

(c)

The points P (2ap, ap2), Q (2aq, aq2) lie on the parabola x2 = 4ay. The tangents to the parabola at P and Q intersect at T. The chord QO produced meets PT at K, and ∠PKQ is a right angle.

(i) Find the gradient of QO, and hence show that pq = –2.

(ii) The chord PO produced meets QT at L. Show that ∠PLQ is a right angle.

(iii) Let M be the midpoint of the chord PQ. By considering the quadrilateral PQLK, or otherwise, show that MK = ML .

End of Question 4

– 7 –

Marks

2

3

1

3

3


(a) Let ƒ ( ) 1 x = −x x 2 for x ≤ 1. This function has an inverse, ƒ −1 ( )x .

2

(i) Sketch the graphs of y = ƒ ( )x and y = ƒ −1 ( )x on the same set of axes. (Use the same scale on both axes.)

(ii) Find an expression for ƒ −1 ( ) x .

ƒ −1 ⎛ 3⎞(iii) Evaluate ⎜ ⎟ . ⎝ 8⎠

(b) A particle is moving in simple harmonic motion in a straight line. Its maximum speed is 2 m s–1 and its maximum acceleration is 6 m s–2.

Find the amplitude and the period of the motion.

(c) T C2

L P M

Q C1

K

Two circles C1 and C2 intersect at P and Q as shown in the diagram. The tangent TP to C2 at P meets C1 at K. The line KQ meets C2 at M. The line MP meets C1 at L.


Prove that ΔPKL is isosceles.

– 8 –

T

North

O 120°

B A

Marks

1

3

3

2

3


(a) From a point A due south of a tower, the angle of elevation of the top of the tower T, is 23°. From another point B, on a bearing of 120° from the tower, the angle of elevation of T is 32°. The distance AB is 200 metres.

(i) Copy or trace the diagram into your writing booklet, adding the given information to your diagram.

(ii) Hence find the height of the tower.

(b) It can be shown that sin 3θ = 3 sinθ – 4 sin3θ for all values of θ. (Do NOT prove this.)

Use this result to solve sin 3θ + sin 2θ = sinθ for 0 ≤ θ ≤ 2π .

(c) Let p and q be positive integers with p ≤ q .

(i) Use the binomial theorem to expand (1 + )p + x q, and hence write down

( p q1 + x) + the term of which is independent of x.

x q

( p q1 + x) + q

(ii) Given that = ( + ) p ⎛ 1 ⎞1 x ⎜ 1 , apply the binomial theorem q + ⎟ x ⎝ x ⎠

and the result of part (i) to find a simpler expression for

⎛ p q⎞ ⎛ ⎞ ⎛ p q ⎞ ⎛ ⎞ ⎛ p⎞⎞ ⎛ q⎞1 + + + + . ⎜ 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ � ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ 2 2⎠ ⎝ ⎠ ⎝ p⎠ ⎝ p ⎠

– 9 –

Marks

2


β

y

w

O P Q N

θ

α

x

h

V

L M

rr

A projectile is fired from O with velocity V at an angle of inclination θ across level ground. The projectile passes through the points L and M, which are both h metres above the ground, at times t1 and t2 respectively. The projectile returns to the ground at N.

The equations of motion of the projectile are

1x Vt cosθ and y = Vt sinθ − gt2 . (Do NOT prove this.)=

2

2V 2h(a) Show that t + t = sinθ AND t t = . 1 2 1 2g g


– 10 –


Let ∠LON = α and ∠LNO = β. It can be shown that

h h tan α = and tan β = . (Do NOT prove this.)

Vt1 cos θ Vt2 cos θ

(b) Show that tan α + tan β = tan θ .

gh (c) Show that tanα tan β = .

2V 2 cos 2 θ

Let ON = r and LM = w .

(d) Show that r = h(cot α + cot β ) and w = h(cot β – cot α ).

Let the gradient of the parabola at L be tan φ.

(e) Show that tan φ = tan α – tan β .

w r (f) Show that = . tanφ tan θ

End of paper

Marks

2

1

2

3

2

– 11 –

⌠ n 1⎮ n+x dx = x 1 , n 1; x ⎮ ≠ − ≠ 0, if n < 0

⌡ n + 1

⌠ 1⎮ dx = ln x , xx > 0⎮ x⌡

⌠ ⎮ e dax 1

x = e ax , a ≠ 0⎮ a⌡

⌠ 1⎮ cosax dx ⎮ = sinax , aa ≠ 0 a⌡


⌠ ⎮ sec 2 1

ax dx = ttan ax , a ⎮ ≠ 0 a⌡

⌠ 1⎮ secax tanax dx secax , a ⎮ = ≠ 0a⌡

⌠ 1 1 − x ⎮ dx = tan 1 , a ≠ 0⎮

aa2⌡ + a ax 2

⌠ 1 x⎮ dx sin −1 , aa 0, a x a⎮ = > − < < 2 2 a ⌡ a − x

⌠ 1 ⎮ dx = ln(x + x 2 − a 2⎮ ), x > a >> 0⌡ x2 − a2

⌠ 1 ⎮ dx ⎮ = ln(x + x 2 + a 2⌡ x 2 + a 2

)NOTE : ln x = log xx , x e > 0

STANDARD INTEGRALS

– 12 –












Total marks – 84



411




2

1

1

3

2

3

(a) Factorise 8x3 + 27.

(b) Let ƒ x = ln(x − 3 . What is the domain of ( ) ( ) ) ƒ x ?

(c) Find sin 2x

lim . x→0 x

(d) Solve the inequality x + 3> 1. 2x

(e) Differentiate x cos2 x .

(f) Using the substitution u = x3 + 1, or otherwise, evaluate⌠ 2

2 x3 +1 x e dx . ⎮⌡0

– 2 –


3

2

2

2

2

1

(a) The polynomial p(x) = x3 – ax + b has a remainder of 2 when divided by

(x – 1) and a remainder of 5 when divided by (x + 2).

Find the values of a and b.

π(b) (i) Express 3 sin x + 4 cos x in the form A sin(x + α) where 0 ≤ α ≤ .

2

(ii) Hence, or otherwise, solve 3 sin x + 4 cos x = 5 for 0 ≤ x ≤ 2π. Give your answer, or answers, correct to two decimal places.

(c) The diagram shows points P(2t, t2) and Q(4t, 4t2) which move along the parabola x2 = 4y. The tangents to the parabola at P and Q meet at R.

P (2t, t2)

y

x

x2 = 4y

Q (4t, 4t2)

R

NOT TO SCALE

(i) Show that the equation of the tangent at P is y = tx – t2 .

(ii) Write down the equation of the tangent at Q, and find the coordinates of the point R in terms of t.

(iii) Find the Cartesian equation of the locus of R.

– 3 –

1 − cos 2 θtan 2 θ =

1 + cos 2 θ

π tan .

8

3 + e 2 x .

4

x + 1 ,y =

2


( )(a) Let ƒ x =

(i) Find the range of ƒ ( ) x .

(ii) Find the inverse function ƒ −1( )x .

(b) (i) On the same set of axes, sketch the graphs of y = cos 2x and for –π ≤ x ≤ π .

(ii) Use your graph to determine how many solutions there are to the equation 2 cos 2x = x + 1 for –π ≤ x ≤ π .

(iii) One solution of the equation 2 cos 2x = x + 1 is close to x = 0.4. Use one application of Newton’s method to find another approximation to this solution. Give your answer correct to three decimal places.

(c) (i) Prove that provided that cos 2θ ≠ –1.

(ii) Hence find the exact value of

1

2

2

1

3

2

1

– 4 –

4 2

ƒ ( ) x + 3xx = .

x 4 + 3


(a) A test consists of five multiple-choice questions. Each question has four alternative answers. For each question only one of the alternative answers is correct.

Huong randomly selects an answer to each of the five questions.

(i) What is the probability that Huong selects three correct and two incorrect answers?

(ii) What is the probability that Huong selects three or more correct answers?

(iii) What is the probability that Huong selects at least one incorrect answer?

(b) Consider the function

(i) Show that ƒ ( ) x is an even function.

(ii) What is the equation of the horizontal asymptote to the graph

y = ƒ ( )x ?

(iii) Find the x-coordinates of all stationary points for the graph y = ƒ ( )x .

(iv) Sketch the graph y = ƒ ( )x . You are not required to find any points

of inflexion.

2

2

1

1

1

3

2

– 5 –

d x2 = − n x2 dt 2

dx v =

dt


(a) The equation of motion for a particle moving in simple harmonic motion is given by

where n is a positive constant, x is the displacement of the particle and t is time.

(i) Show that the square of the velocity of the particle is given by

v2 = n2(a2 – x2)

where and a is the amplitude of the motion.

(ii) Find the maximum speed of the particle.

(iii) Find the maximum acceleration of the particle.

(iv) The particle is initially at the origin. Write down a formula for x as a function of t, and hence find the first time that the particle’s speed is half its maximum speed.


3

1

1

2

– 6 –


1

1

2

1

(b) The cross-section of a 10 metre long tank is an isosceles triangle, as shown in the diagram. The top of the tank is horizontal.

h m3 m

120°120°h m 3 m

10 m10 m

NOT TO

SCALE

When the tank is full, the depth of water is 3 m. The depth of water at time t days is h metres.

(i) Find the volume, V, of water in the tank when the depth of water is h metres.

(ii) Show that the area, A, of the top surface of the water is given by

.A h= 20 3

(iii) The rate of evaporation of the water is given by

, dV dt

= −kA

where k is a positive constant.

Find the rate at which the depth of water is changing at time t.

(iv) It takes 100 days for the depth to fall from 3 m to 2 m. Find the time taken for the depth to fall from 2 m to 1 m.

End of Question 5

– 7 –


1

2

1

(a) Two points, A and B, are on cliff tops on either side of a deep valley. Let h and R

be the vertical and horizontal distances between A and B as shown in the

diagram. The angle of elevation of B from A is θ, so that θ = tan−1 ⎛ h ⎞ .⎝⎜ R ⎠⎟

hh

A

y

x

B

θ

θ

RR

At time t = 0, projectiles are fired simultaneously from A and B. The projectile

from A is aimed at B, and has initial speed U at an angle θ above the horizontal.

The projectile from B is aimed at A and has initial speed V at an angle θ below

the horizontal.

The equations for the motion of the projectile from A are

xl = Ut cos θ and ,y Ut gt1 21

2 = −sinθ

and the equations for the motion of the projectile from B are

x2 = R – Vt cos θ and y h Vt2 = − −sinθ .gt21 2

(Do NOT prove these equations.)

(i) Let T be the time at which x1 = x2.

Show that .T R

U V =

+( )cosθ

(ii) Show that the projectiles collide.

(iii) If the projectiles collide on the line x = λR, where 0 < λ < 1, show that

.V U= −⎛ ⎝⎜

⎞ ⎠⎟

1 1

λ


– 8 –

� + = .


(b) (i) Sum the geometric series 3

1

1

3

(1 + x)r + (1 + x)r + 1 + · · · + (1 + x)n

and hence show that

⎛ r⎞ ⎛ r + 1⎞ ⎛ n⎞ ⎛ n + 1⎞ ⎝⎜ r⎠⎟

+ ⎝⎜ r ⎠⎟

+ ⎝⎜ r ⎠⎟ ⎝⎜ r + 1⎠⎟

(ii) Consider a square grid with n rows and n columns of equally spaced points.

y

x

The diagram illustrates such a grid. Several intervals of gradient 1, whose endpoints are a pair of points in the grid, are shown.

(1) Explain why the number of such intervals on the line y = x is

equal to .n

2

⎛ ⎝⎜

⎞ ⎠⎟

(2) Explain why the total number, Sn, of such intervals in the grid is given by

.S n n

n = ⎛ ⎝⎜

⎞ ⎠⎟ + ⎛ ⎝⎜

⎞ ⎠⎟ + +

−⎛ ⎝⎜

⎞ ⎠⎟ + ⎛ ⎝⎜

⎞ ⎠⎟ +

2

2

3

2

1

2 2�

nn −⎛ ⎝⎜

⎞ ⎠⎟ + +

⎛ ⎝⎜

⎞ ⎠⎟ + ⎛ ⎝⎜

⎞ ⎠⎟

1

2

3

2

2

2�

(iii) Using the result in part (i), show that

.S n n n

n = ( − ) −( )1 2 1

6

End of Question 6

– 9 –

2


d ( ) 1(a) (i) Use differentiation from first principles to show that x = 1. dx

(ii) Use mathematical induction and the product rule for differentiation d n n−1to prove that x = nx for all positive integers n. dx

( )

(b) A billboard of height a metres is mounted on the side of a building, with its bottom edge h metres above street level. The billboard subtends an angle θ at the point P, x metres from the building.

aa

hh

P xx

θ

(i) Use the identity to show that

.θ = + ( + )

⎡

⎣ ⎢

⎤

⎦ ⎥tan−1

2

ax

x h a h

tan tan tan

tan tan A B

A B A B

( − ) = − 1 +

2

(ii) The maximum value of θ occurs when

Find the value of x for which θ is a maximum.

d dx θ = 0 and x is positive. 3


– 10 –

a

h


(c) Consider the billboard in part (b). There is a unique circle that passes through the top and bottom of the billboard (points Q and R respectively) and is tangent to the street at T.

Let φ be the angle subtended by the billboard at S, the point where PQ intersects the circle.

QQ

RR

P

S

T

θ

φ a

h

Copy the diagram into your writing booklet.

(i) Show that θ < φ when P and T are different points, and hence show that θ is a maximum when P and T are the same point.

(ii) Using circle properties, find the distance of T from the building.

3

1

End of paper

– 11 –

STANDARD INTEGRALS

⌠ 1n n+1⎮ x dx = x , n ≠ −1; x ≠ 0, if n < 0⎮ n + 1⌡

⌠ 1⎮ dx = ln x , xx > 0⎮ x⌡

⌠ 1ax ax⎮ e dx = e , a ≠ 0⎮ a⌡

⌠ ⎮ ⎮ cosax dx = 1

sinax , aa ≠ 0 a⌡


⌠ ⎮ 2 ⎮ sec ax dx = 1

ttan ax , a ≠ 0 a⌡

⌠ 1⎮ secax tan ax dx = secax , a ≠ 0⎮ a⌡

⌠ 1 1 x ⎮ dx = tan−1 , a ≠ 0⎮ 2 2 a a⌡ aa + x

⌠ 1 x−1dx = sin , aa > 0, a x a⎮ − < < a

a x2 2−⌡⎮

1 2 2⎮ ⌠

dx = ln(x + x − a ), x > a >> 0⎮⌡ x2 − a2

1 2 2⎮ ⌠

dx = ln x + x + a ⎮ 2 2⌡ x + a ( )

NOTE : ln x = log xx , xe > 0

– 12 –












Total marks – 84



3370




(a) Use the table of standard integrals to find ⌠ 1

dx . 4 2−⌡

⎮ x

1

(d) Solve3 < 4 .

x + 2 3

1

3

3

1

⎛ x ⎞−1(b) Let ƒ x = . What is the domain of ƒ (x)?( ) cos ⎜ ⎟⎝ 2 ⎠

(c) Solve ln(x + 6) = 2 ln x .

(e) Use the substitution u = 1 − x to evaluate⌠1

x 1 − x dx. ⎮⌡

0

(f) Five ordinary six-sided dice are thrown.

What is the probability that exactly two of the dice land showing a four? Leave your answer in unsimplified form.

– 2 –

2


(a) The derivative of a function ƒ (x) is given by

ƒ ′(x) = sin2 x.

Find ƒ (x), given that ƒ (0) = 2.

(b) The mass M of a whale is modelled by

M = 36 − 35.5e−kt,

where M is measured in tonnes, t is the age of the whale in years and k is a positive constant.

(i) Show that the rate of growth of the mass of the whale is given by the 1

2

1

differential equation

dM = k (36 − M ).dt

(ii) When the whale is 10 years old its mass is 20 tonnes.

Find the value of k, correct to three decimal places.

(iii) According to this model, what is the limiting mass of the whale?


– 3 –


(c) Let P(x) = (x + 1)(x − 3) Q(x) + ax + b,

where Q(x) is a polynomial and a and b are real numbers. The polynomial P(x) has a factor of x − 3. When P(x) is divided by x + 1 the remainder is 8 .

(i) Find the values of a and b. 2

1

3

(ii) Find the remainder when P(x) is divided by (x + 1)(x − 3).

(d) A radio transmitter M is situated 6 km from a straight road. The closest point on the road to the transmitter is S.

A car is travelling away from S along the road at a speed of 100 km h−1. The distance from the car to S is x km and from the car to M is r km.

M

6 r

S x

drFind an expression in terms of x for , where t is time in hours.

dt

End of Question 2

– 4 –

y

x


(a) At the front of a building there are five garage doors. Two of the doors are to be painted red, one is to be painted green, one blue and one orange.

(i) How many possible arrangements are there for the colours on the doors?

(ii) How many possible arrangements are there for the colours on the doors if the two red doors are next to each other?

1

1

ƒ2

(b) Let ( )x = −e x . The diagram shows the graph y = ƒ ( )x .

(i) The graph has two points of inflexion.

Find the x coordinates of these points.

(ii) Explain why the domain of ƒ (x) must be restricted if ƒ (x) is to have an inverse function.

(iii) Find a formula for ƒ −1 ( )x if the domain of ƒ (x) is restricted to x ≥ 0.

(iv) State the domain of ƒ −1 ( )x .

(v) Sketch the curve y = ƒ −1 ( )x .

2 (vi) (1) Show that there is a solution to the equation = −x e x between

x = 0.6 and x = 0.7.

(2) By halving the interval, find the solution correct to one decimal place.

3

1

2

1

1

1

1

– 5 –


(a) A particle is moving in simple harmonic motion along the x-axis.

Its velocity v, at x, is given by v2 = 24 − 8x − 2x2 .

(i) Find all values of x for which the particle is at rest.

(ii) Find an expression for the acceleration of the particle, in terms of x.

(iii) Find the maximum speed of the particle.

1

1

2

3

2

(b) (i) Express

where R > 0 and .

2 2 3

cos cos θ θ π+ +⎛

⎝⎜ ⎞ ⎠⎟

0 2

< α < π

in the form R cos (θ + α ),

(ii) Hence, or otherwise, solve

for 0 < θ < 2π .

2 cos θ 2 cos θ+ ⎛ ⎝⎜ 3

π+ ⎞ ⎠⎟ ,3=


– 6 –

3


2(c) The diagram shows the parabola x = 4ay . The point P(2ap, ap2) , where p ≠ 0, is on the parabola.

y

O x

L

P (2ap, ap2)

S (0, a)

y = −aM

The tangent to the parabola at P, y = px − ap2, meets the y-axis at L.

The point M is on the directrix, such that PM is perpendicular to the directrix.

Show that SLMP is a rhombus.

End of Question 4

– 7 –


(a) A boat is sailing due north from a point A towards a point P on the shore line. The shore line runs from west to east.

In the diagram, T represents a tree on a cliff vertically above P, and L represents a landmark on the shore. The distance PL is 1 km.

From A the point L is on a bearing of 020°, and the angle of elevation to T is 3°.

After sailing for some time the boat reaches a point B, from which the angle of elevation to T is 30°.

1 km1 km E

N

20°

3°

30°

W P

NOT TO SCALE

B

A

L

T

(i) Show that BP = .° °

3 3 20

tan tan

3

1 (ii) Find the distance AB.


– 8 –

1

1

2


(b) Let ƒ x x x

( ) = + ⎛ ⎝⎜

⎞ ⎠⎟

− −tan tan 1 1 1 for x ≠ 0.

(i)

(ii)

By differentiating ƒ (x), or otherwise, show that for

Given that ƒ (x) is an odd function, sketch the graph y = ƒ (x).

ƒ x( ) = π 2

x > 0. 3

1

(c) In the diagram, ST is tangent to both the circles at A.

The points B and C are on the larger circle, and the line BC is tangent to the smaller circle at D. The line AB intersects the smaller circle at X.

S

C

A

D

X

B

T


(i) Explain why ∠AXD = ∠ABD + ∠XDB.

(ii) Explain why ∠AXD = ∠TAC + ∠CAD.

(iii) Hence show that AD bisects ∠BAC.

End of Question 5

– 9 –


(a) (i) Show that cos(A − B) = cos A cos B(1 + tan A tan B) . 1

1 (ii) Suppose that and .

Deduce that if , then

0 2

< <Bπ

tan tanA B = − 1 A B−

B A< < π

= . π 2

(b) A basketball player throws a ball with an initial velocity v m s−1 at an angle θ

to the horizontal. At the time the ball is released its centre is at (0, 0), and the

player is aiming for the point (d, h) as shown on the diagram. The line joining

(0, 0) and (d, h) makes an angle α with the horizontal, where .0 2

< <α θ < π

y

x d

(d, h)

θ

α

Assume that at time t seconds after the ball is thrown its centre is at the point (x, y) , where

x vt θ= cos

y vt sin θ − 5 2 .= t

(You are NOT required to prove these equations.)


– 10 –


(i) If the centre of the ball passes through (d, h) show that

.v d2

2

5 = −cos sin cos tan θ θ θ α

3

1

1

2

1

2

(ii) (1) What happens to v as

(2) What happens to v as

θ → α ?

θ → ?π 2

(iii) For a fixed value of α , let

Show that when .

F θ θ θ( ) = −cos sin

tan tan 2 1θ α = −′( ) =F θ 0

.θ αcos tan 2

(iv) Using part (a) (ii) or otherwise show that ′( )F θ = 0 when θ = α 2

.π+ 4

(v) Explain why v2 is a minimum when θ = α 2

. π+ 4

End of Question 6

– 11 –


(a) Prove by induction that

47n + 53 × 147n − 1

is divisible by 100 for all integers n ≥ 1.

3

1

1

2

(b) The binomial theorem states that

1 0 1

+( ) = ⎛

⎝⎜ ⎞

⎠⎟ + ⎛

⎝⎜ ⎞

⎠⎟x

n n x n

2 2+

⎛

⎝⎜ ⎞

⎠⎟ n

x 3

3+ ⎛

⎝⎜ ⎞

⎠⎟ n

x ++ +… .⎛

⎝⎜ ⎞

⎠⎟ n

n xn

(i)

(ii)

Show that .

Hence, or otherwise, find the value of

.

2 0

n

k

n n

k =

⎛

⎝⎜ ⎞

⎠⎟ = ∑

100

0

100

1

100

2

100

100

⎛

⎝⎜ ⎞

⎠⎟ + ⎛

⎝⎜ ⎞

⎠⎟ + ⎛

⎝⎜ ⎞

⎠⎟ + + ⎛

⎝⎜ ⎞

⎠⎟…

(iii) Show that n n2 1− = k n

kk

n

1=

⎛

⎝⎜ ⎞

⎠⎟∑ .


– 12 –


(c) (i) A box contains n identical red balls and n identical blue balls. A selection 1

1

3

of r balls is made from the box, where 0 ≤ r ≤ n.

Explain why the number of possible colour combinations is r + 1.

(ii) Another box contains n white balls labelled consecutively from 1 to n. A selection of n − r balls is made from the box, where 0 ≤ r ≤ n.

Explain why the number of different selections is .n

r

⎛

⎝⎜ ⎞

⎠⎟

(iii) The n red balls, the n blue balls and the n white labelled balls are all placed into one box, and a selection of n balls is made.

Using part (b), or otherwise, show that the number of different selections is (n + 2)2n − 1 .

End of paper

– 13 –

BLANK PAGE

– 14 –

BLANK PAGE

– 15 –

STANDARD INTEGRALS

⌠ ⎮ n n+1 ⎮

x dx = 1

x , n ≠ −1; x ≠ 0, if n < 0 n + 1⌡

⌠ 1⎮ ⎮

dx = ln x , xx > 0 x⌡

⌠ 1ax ax⎮ e dx = e , a ≠ 0⎮ a⌡

⌠ 1⎮ cosax dx = sinax , aa ≠ 0⎮ a⌡

⌠ 1⎮ sin ax dx = − cosax , a ≠ 0⎮ a⌡

⌠ ⎮ 2 ⎮

sec ax dx = 1

tant ax , a ≠ 0 a⌡

⌠ 1⎮ secax tanax dx = secax , a ≠ 0⎮ a⌡

⌠ 1 1 x ⎮ dx = tan−1 , a ≠ 0⎮ 2 2 a a⌡ aa + x

⌠ 1 x⎮ dx = sin−1 , aa > 0 , − a < x < a a

a x2 2−⌡⎮

1 2 2⎮ ⌠

dx = ln(x + x − a ), x > a >> 0⎮ ⌡ x2 − a2

1 2 2⎮ ⌠

dx = ln(x + x + a )⎮ 2 2⌡ x + a

NOTE : ln x = log xx , x > 0e

– 16 –


hsc past 3unit exam

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