3 Unit Maths-TRial HSC Test 7 Trial...Pre HSC ReVISION MATHEMATICS EXTENSION 1 (d) The concentration...

1 Which expression is a correct factorisation of x3

x 2 − 3x + 9) (B) (x − 3)(x 2 )

2 + 3x ++ 9) x 2 + 6x + 9)

3 Unit Maths-TRial HSC Test 7

+ 27?

(A) (x +3)(+ 6x + 9

(C) (x +3)(x (D) (x +3)(

2 The point P divides the interval from A(–2, 2) to B(8, –3) externally in the ratio 2:3

What is the x-coordinate of P?

(A) 4

(B) 2 8

C) 28/5

(D) –28/5

page1

Pre HSC ReVISION MATHEMATICS EXTENSION 1

3 The points A, B and C lie on a circle with centre O, as shown in the diagram.

radians.

A

B C

O NOT TO SCALE

5

π

π

The size of

X

2π∠ ABC is3

2p 3

What is the size of ∠ XCB in radians?

(A) π 7

(B) π

(C) 3

(D) 6

(XC is a tangent to a circle)

page2


hung.phanLine

hung.phanLine

2 − 4xy + 11y2 = 64 and 16x2 − 9xy + 11y2

(A)

Q4. The quadratic equation passing through the intersection point of 12x = 78.,

and the Origin is

2 − 388 xy + 187 y 2 = 1984

(B) 2 − 388 xy + 187 y 2

460x2 − 388 xy + 187 y 2(C) = 2015

460x2 − 388 xy + 187 y 2(D) = 2012

460x

5 What is a probability if 7 Men and 8 women are sitting in a circle such that no two 2 Women are sit next to each other?

(D) 0.00000194

page3


(A) 0.0000194

(B) 0.000291

(C) 0.000194

= 0 88x

x)3 (3 – x)2?

1 3

(A) y

xO

1 3

(B) y

xO

1 3

(C) y

xO

1 3

(D) y

xO

6 Which diagram best represents the graph y = x (1 –

page4


θ satisfies 5 sinθ = 13

and π θ π .< <2

What is the value of sin 2θ ?

10(A)

13

10

(B) −

13

120

(C)

169

120

(D) −

169

7 The angle

page5


y

p

p 2

O 1 x–1

Which function does the graph represent?

(A)

π

y = cos −1 x

(B) y = + sin −1 x 2

(C) y = – cos −1 x

(D) π 1y = – – s iin− x 2

(A)

1 1

5 ≥ 1 3 – x

(B) − ≤ 0 x − 3 x + 2

(C) x2 – – 6 0x ≤

(D) 2x 1− ≥ 5

8 The diagram shows the graph of a function.

9 Which inequality has the same solution as x + 2 + x − 3 = 5?

page6


2 3x dx?⎮⌡

1 ⎛ 1 ⎞(A) x − sin3x C

⎝⎜ ⎠⎟+

2 3

1 ⎛ 1 ⎞(B) x + sin3x C

⎝⎜ ⎠⎟+

2 3

11 ⎛ 1 ⎞(C) x −⎜ sin6x C⎝

+⎟ 2 6 ⎠

1 ⎛ 1 ⎞(D) x +⎜ sin6x C⎝

+⎟ 2 6 ⎠

⌠Q10. which expression is equal to sin

page7


(a) x −( )2 ( ) = + +2 3 . Find the value of a.

(b) Find the acute angle between the lines 2 4x y+ = and x y− = 2, to the nearestdegree.

(c) Let A −( )1 2, and B 3 5,( ) be points in the plane. Find the coordinates of thepoint C which divides the interval AB externally in the ratio 3 : 1.

(e) Solve the inequality 21

1x −

≤ .

(f) Using the substitution u ex= , find ee

dxx

x1 2+⌠⌡

.

1

2

2

3

3

QUESTION 11A

is a factor of the polynomial P x x x a

Marks

(x + 6)

Find the remainder When the polynomial P(x) is divided by (x + 1)(x – 3) the remainder is 2x + 7. Q,11d (x + 6) is a factor of p(x). When the polynomial P(x) is divided by (x + 1)(x – 3),

5

page8


(a) The function f x x x( ) = − +3 1ln( ) has one root between 0·5 and 1.

(i) Show that the root lies between 0·8 and 0·9.

(ii) Hence use the halving-the-interval method to find the value of the root,correct to one decimal place.

(b) Use the table of standard integrals on page 12 to show that

dx

x26

15

642

+=⌠

⌡ln .

(c) The points O and P in the plane are d cm apart. A circle centre O is drawn topass through P, and another circle centre P is drawn to pass through O. The twocircles meet at S and T, as in the diagram.

S

T

PO

(i) Show that triangle SOP is equilateral.

(ii) Show that the size of angle SOT is 23π .

(iii) Hence find the area of the shaded region in terms of d.

4

3

5

QUESTION 11B

page9


(a) Evaluate . 1

9 2 0

3

+

⌠

⌡⎮

x dx 3

(b) Differentiate x2 tan x with respect to x. 2

(c) Solve x

x − 3 .< 2 3

(d) Use the substitution u = 2 – x to evaluate .x x dx 2 5

1

2

−( )⌠ ⌡⎮ 3

(e) In how many ways can a committee of 3 men and 4 women be selected from a group of 8 men and 10 women?

1

(f) (i) Use the binomial theorem to find an expression for the constant term

in the expansion of .2 13

12

x x

−⎛ ⎝⎜

⎞ ⎠⎟

2

(ii) For what values of n does ⎛ ⎝⎜2x3 − 1

⎞⎠⎟x

n

have a non-zero constant term? 1

Question 11 C

page10


(a)

y = f (x)

y

xO– 4 4

14

– 2 2

Let f xx

( ) .=−

1

16 2 The graph of y f x= ( ) is sketched above.

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

(ii) Find the area enclosed by y f x= ( ), the x axis, x = −2, and x = 2 .

(b) Show that cos .2

48

14

2x dx = −⌠

⌡π

π

π

(c) The function g x( ) is given by g x x( ) cos .= +2 The graph y g x= ( ) forπ π4 2

≤ ≤x is rotated about the x axis.

Find the volume of the solid generated. (You may use the result of part (b).)

(d) The function h x( ) is given by

h x x x x( ) sin cos , .= + ≤ ≤− −1 1 0 1

(i) Find ′h x( ).

(ii) Sketch the graph of y h x= ( ).

4

2

3

3

QUESTION12A

page11


(a) Use mathematical induction to prove that 23n – 3n is divisible by 5 for n ≥ 1. 3

(b) Let ƒ ( )x = 4 x − 3 .

(i) Find the domain of ( ). ƒ x 1

(ii) −1 Find an expression for the inverse function ƒ x ( ) . 2

(iii) Find the points where the graphs y = x ƒ ( ) and y = x intersect. 1

(iv) = ƒ −1On the same set of axes, sketch the graphs y = ƒ ( )x and y ( )x showing the information found in part (iii).

2

Question 12B

page12


(d) Let A(0, –k) be a fixed point on the y-axis with k > 0. The point C(t, 0) is on the x-axis. The point B(0, y) is on the y-axis so that ABC is right-angled with the right angle at C. The point P is chosen so that OBPC is a rectangle as shown in the diagram.

y

B (0, y) P

O C(t, 0) x

A(0, –k)

(i) Show that P lies on the parabola given parametrically by 2

t 2x = t and y = .

k

(ii) Write down the coordinates of the focus of the parabola in terms of k. 1

Question 12B (continued)

page13


(a) Prove that sinsin

coscos

sin , cos .3 3 2 0 0θθ

θθ

θ θ− = ≠ ≠( )for

(b)

r

h

Grain is poured at a constant rate of 0·5 cubic metres per second. It forms aconical pile, with the angle at the apex of the cone equal to 60°. The height of thepile is h metres, and the radius of the base is r metres.

(i) Show that r h=3

.

(ii) Show that V, the volume of the pile, is given by

V h= π3

9.

(iii) Hence find the rate at which the height of the pile is increasing when theheight of the pile is 3 metres.

3

4

QUESTION 13A

page14


(c) Q

S

D C

A B

P RX Z Y

The figure shows the net of an oblique pyramid with a rectangular base. In thisfigure, PXZYR is a straight line, PX = 15 cm, RY = 20 cm, AB = 25 cm, andBC = 10 cm. Further, AP = PD and BR = RC.

When the net is folded, points P, Q, R, and S all meet at the apex T, which liesvertically above the point Z in the horizontal base, as shown below.

B

CA

D

X

Y

Z

T

(i) Show that triangle TXY is right-angled.

(ii) Hence show that T is 12 cm above the base.

(iii) Hence find the angle that the face DCT makes with the base.

5

QUESTION 13A (continued)

page15


(a) A cup of hot coffee at temperature T°C loses heat when placed in a coolerenvironment. It cools according to the law

dTdt

k T T= −( )0 ,

where t is time elapsed in minutes, and T0 is the temperature of the environmentin degrees Celsius.

(i) A cup of coffee at 100°C is placed in an environment at –20°C for3 minutes, and cools to 70°C. Find k.

(ii) The same cup of coffee, at 70°C, is then placed in an environment at20°C. Assuming k remains the same, find the temperature of the coffeeafter a further 15 minutes.

(b) A particle is moving along the x axis. Its velocity v at position x is given by

v x x= −10 2 .

Find the acceleration of the particle when x = 4.

5

2

QUESTION 13B

page16


(a) ⎛ ⎛ −1 Write sin 2cos ⎝⎜ ⎝⎜

2 ⎞ ⎞ ⎠⎟ 3 ⎠⎟

in the form a b , where a and b are rational. 2

(b) (i)

(ii)

2 Find the horizontal asymptote of the graph y =

2 x

Without the use of calculus, sketch the graph y =

2 x

, showing the

1

2

. + 9

2 2 x 2 x + 9

asymptote found in part (i).

(c) A particle is moving in a straight line according to the equation

x = 5 + 6 cos 2t + 8 sin 2t,

where x is the displacement in metres and t is the time in seconds.

(i) Prove that the particle is moving in simple harmonic motion by showing 2 2 that x satisfies an equation of the form ��x = − n ( x − c ) .

(ii) When is the displacement of the particle zero for the first time? 3

Question 13 C

page17


(d) The concentration of a drug in the blood of a patient t hours after it was administered is given by

C(t) = 1.4te–0.2t ,

where C(t) is measured in mg/L.

(i) Initially the concentration of the drug in the blood of the patient increases until it reaches a maximum, and then it decreases.

Find the time when this maximum occurs.

3

(ii) Taking t = 20 as a first approximation, use one application of Newton’s method to find approximately when the concentration of the drug in the blood of the patient reaches 0.3 mg/L.

2

Question 13 C (continued)

page18


O

x

A trolley is moving in simple harmonic motion about the origin O. The displacement,x metres, of the centre of the trolley from O at time t seconds is given by

x t= +6 2 4

sin .π

(a) State the period and amplitude of the motion.

(b) Sketch the graph of x t= +6 2 4

sin π for 0 2≤ ≤t π.

(c) Find the velocity of the trolley when t = 0.

(d) Find the first time after t = 0 when the centre of the trolley is at x = 3.

(e) wP

A particle P, on top of the trolley, is moving in simple harmonic motion aboutthe centre of the trolley. Its displacement, w metres, from the centre of thetrolley at time t seconds, is given by

w t= sin( ).2

The displacement, y metres, of P from the origin is the sum of the twodisplacements x and w, so that

y t t= + + ( )6 2 4 2sin sin .

π

(i) Show that P is moving in simple harmonic motion about O.

(ii) Find the amplitude of this motion.

2

2

2

2

4

QUESTION 14A

page19


(a) The diagram shows a large semicircle with diameter AB and two smaller semicircles with diameters AC and BC, respectively, where C is a point on the diameter AB. The point M is the centre of the semicircle with diameter AC.

The line perpendicular to AB through C meets the largest semicircle at the point D. The points S and T are the intersections of the lines AD and BD with the smaller semicircles. The point X is the intersection of the lines CD and ST.

A M C

D

X

S

T

B

Copy or trace the diagram into your writing booklet.

(i) Explain why CTDS is a rectangle.

(ii) Show that MXS and MXC are congruent.

(iii) Show that the line ST is a tangent to the semicircle with diameter AC.

1

2

1

Question 14 B

page20


(b) A firework is fired from O, on level ground, with velocity 70 metres per second at an angle of inclination θ. The equations of motion of the firework are

x = 70t cosθ and y = 70t sinθ – 4.9t2. (Do NOT prove this.)

The firework explodes when it reaches its maximum height.

x

y

O

150

125 180

q

(i)

(ii)

(iii)

Show that the firework explodes at a height of 250 sin2θ metres.

Show that the firework explodes at a horizontal distance of 250 sin 2θ metres from O.

For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from O, and at least 150 m above the ground.

For what values of θ will this occur?

2

3

1

Question 14 B (continued)

page21


(c) A plane P takes off from a point B. It flies due north at a constant angle α to the horizontal. An observer is located at A, 1 km from B, at a bearing 060° from B. Let u km be the distance from B to the plane and let r km be the distance from the observer to the plane. The point G is on the ground directly below the plane.

A

G

P

60°

1

B

r

u

a

N

(i)

(ii)

Show that .

The plane is travelling at a constant speed of 360 km/h.

At what rate, in terms of α, is the distance of the plane from the observer changing 5 minutes after take-off?

r u u= + −1 2 cosα 3

2

Question 14 B(continued)

page22


(a) Using the fact that 1 1 14 9 13+( ) +( ) = +( )x x x , show that

40

94

419

34

29

24

39

14

49

013

4C C C C C C C C C C C+ + + + = .

(b) Consider the function f x x( ) = −( ) +[ ]14 1 72 .(i) Sketch the parabola y f x= ( ) , showing clearly any intercepts with the

axes, and the coordinates of its vertex. Use the same scale on both axes.

(ii) What is the largest domain containing the value x = 3, for which thefunction has an inverse function f x− ( )1 ?

(iii) Sketch the graph of y f x= ( )−1 on the same set of axes as your graph inpart (i). Label the two graphs clearly.

(iv) What is the domain of the inverse function?

(v) Let a be a real number not in the domain found in part (ii). Findf f a− ( )( )1 .

(vi) Find the coordinates of any points of intersection of the two curvesy f x= ( ) and y f x= ( )−1 .

2

1 0

QUESTION 14C

page23


Question 13 (15 marks) Use a SEPARATE writing booklet.

(a) A spherical raindrop of radius r metres loses water through evaporation at a rate that depends on its surface area. The rate of change of the volume V of the raindrop is given by

dV = −10−4 A ,dt

where t is time in seconds and A is the surface area of the raindrop. The surface

area and the volume of the raindrop are given by 2 A = 4πr and V = 4 πr3 3

respectively.

(i) Show thatdr

dt is constant. 1

(ii) How long does it take for a raindrop of volume 10–6 m3 to completely evaporate?

2

(b) The point P (2ap, ap2) lies on the parabola .2x = 4ay The tangent to the parabola at P meets the x-axis at T ap, 0( ). The normal to the tangent at P meets the y-axis at N(0, 2a + ap2).

x

y

P(2ap, ap2)

N (0, 2a + ap2)

T (ap, 0)

x2 = 4ay

O

The point G divides NT externally in the ratio 2 :1.

(i) Show that the coordinates of G are (2ap, –2a – ap2). 2

(ii) Show that G lies on a parabola with the same directrix and focal length as the original parabola.

2

page24


Question 13 (continued)

(c) Points A and B are located d metres apart on a horizontal plane. A projectile is fired from A towards B with initial velocity u m s–1 at angle to the horizontal.α

At the same time, another projectile is fired from B towards A with initial velocity w m s–1 at angle β to the horizontal, as shown on the diagram.

The projectiles collide when they both reach their maximum height.

u w

a b

A d B

The equations of motion of a projectile fired from the origin with initial velocity V m s–1 at angle θ to the horizontal are

x = Vt cosθ and y Vt = −sinθ .g

t 2

2 (Do NOT prove this.)

(i) How long does the projectile fired from A take to reach its maximum height?

2

(ii) Show that u sinα = w sinβ. 1

(iii) Show that .d uw g

= +sin( )α β 2

page25



(d) The circles C1 and C2 touch at the point T. The points A and P are on C1. The line AT intersects C2 at B. The point Q on C2 is chosen so that BQ is parallel to PA.

3

C2

B

T

Q A

C1

P


Prove that the points Q, T and P are collinear.

page26



(a) (i) Show that for ,1 1 1− + < 0 .

k + 1 k k + 1( )2 k > 0 1

(ii) Use mathematical induction to prove that for all integers n ≥ 2, 1 1 1 1 1+ + + · · · + < 2 − .2 2 2 21 2 3 n n

3

(b) (i) Write down the coefficient of x2n in the binomial expansion of (1 + x)4n . 1

(ii) Show that 2n 2n ⎛ −2n⎞ 2n k2 2n k1 + x x + 2( + x 2 ) = ∑ x − ( ) . ⎜ ⎟⎝ k ⎠k=0 2

(iii) It is known that

− ⎛ 1n k− 2 2n k 2 − 2n k ⎛ 2n k− ⎞ 2n k− − 2n kx2 (x + 2) n k = − ⎞ 2 n k x − + 2 1 x − + ⎜ ⎟ ⎜ ⎟⎟⎝ 0 ⎠ ⎝ 1 ⎠n k⎞⎛ 2 − 0 4n−2k+ · · · + 2 x . ⎜ ⎟⎝ 2 −n k⎠

(Do NOTprove this.)

Show that n4n⎞ n− k ⎛ 2n⎞ ⎛ 2n k⎞⎛ 2 2 − = ∑ 2 .⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ 2n⎠ ⎝ k ⎠ ⎝ k ⎠k =0

3

(c) The equation t1

e = t

has an approximate solution t0 = 0.5.

(i) Use one application of Newton’s method to show that t1 = 0.56 is 1

another approximate solution of te = .t

2

(ii) Hence, or otherwise, find an approximation to the value of r for which rxthe graphs y = e and y = log x have a common tangent at their pointe

of intersection.

3

page27



(a) The diagram shows a large semicircle with diameter AB and two smaller semicircles with diameters AC and BC, respectively, where C is a point on the diameter AB. The point M is the centre of the semicircle with diameter AC.

The line perpendicular to AB through C meets the largest semicircle at the point D. The points S and T are the intersections of the lines AD and BD with the smaller semicircles. The point X is the intersection of the lines CD and ST.

A M C

D

X

S

T

B


(i) Explain why CTDS is a rectangle.

(ii) Show that MXS and MXC are congruent.

(iii) Show that the line ST is a tangent to the semicircle with diameter AC.

1

2

1

page28



(b) A firework is fired from O, on level ground, with velocity 70 metres per second at an angle of inclination θ. The equations of motion of the firework are

x = 70t cosθ and y = 70t sinθ – 4.9t2. (Do NOT prove this.)

The firework explodes when it reaches its maximum height.

x

y

O

150

125 180

q

(i)

(ii)

(iii)

Show that the firework explodes at a height of 250 sin2θ metres.

Show that the firework explodes at a horizontal distance of 250 sin 2θ metres from O.

For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from O, and at least 150 m above the ground.

For what values of θ will this occur?

2

3

1

page29



(c) A plane P takes off from a point B. It flies due north at a constant angle α to the horizontal. An observer is located at A, 1 km from B, at a bearing 060° from B. Let u km be the distance from B to the plane and let r km be the distance from the observer to the plane. The point G is on the ground directly below the plane.

A

G

P

60°

1

B

r

u

a

N

(i)

(ii)

Show that .

The plane is travelling at a constant speed of 360 km/h.

At what rate, in terms of α, is the distance of the plane from the observer changing 5 minutes after take-off?

r u u= + −1 2 cosα 3

2

page30


John draws 2 balls together from the urn. The probability that they have the same colour is .

Peter adopts a different procedure. He draw one ball from the urn, notes its colour and replaces it.He then draws a second ball from the urn and notes it colour. The probability that both balls have

the same colour is now .

How many red balls and white ball are there in the urn.

(a) between what times the ship can enter the harbour on Monday.

(b). The bigger ship can enter the harbour only when the minimum depth of water is at 12.3 m.Between what times the bigger ship can enter on the following next Monday.

An urn contains r red balls and w white balls.

QUESTION 1 Marks

6

6

The depth of water in a harbour is 7.2m at low water and 13.6m at high water. OnMonday, low water is at 2:05 pm and high water at 8:20 pm . The ship can enter the harbour if thedepth of water Is minimum at 8.8m .

QUESTION 2

page31


(i) In how many ways can six cards be selected without replacement so thatexactly two are spades and four are clubs? (Assume that the order ofselection of the six cards is not important.)

(ii) In how many ways can six cards be selected without replacement ifat least five must be of the same suit? (Assume that the order of selectionof the six cards is not important.)

4

QUESTION 3 Marks

A standard pack of 52 cards consists of 13 cards of each of the four suits: spades,hearts, clubs and diamonds.

page32


(a) If V >√

gd, prove that the water can reach the wall above ground level.(b) Suppose that V = 2

√gd. Show that the portion of the wall that can be sprayed with

water is a parabolic segment of height 158 d and area52 d

2√

15 .

6

QUESTION 4

A tall building stands on level ground. The nozzle of a water sprinkler is positioned ata point P on the ground at a distance d from a wall of the building. Water sprays fromthenozzle with speed V and the nozzle can be pointed in any direction from P .

page33


ẍ = −n2x, ÿ = −n2y, x(0) = y(0), and ẋ(0) = ẏ(0),where n is a positive constant, and x(0) and ẋ(0) are not both zero.

(a) Show thatd

dt(ẋy − xẏ) = 0, and hence that ẋy = xẏ, for all t.

(b) Show thatd

dt

(x

y

)= 0, and hence that y = x, for all t.

(c) Hence show that x = a cos nt + b sinnt, where a = x(0) and b =ẋ(0)n

.

6

Suppose that x and y are functions of t satisfyingQUESTION 5

page34


(a) The amount of fuel F in litres required per hour to propel a plane in level flightat constant speed u km/h is given by

,

where A and B are positive constants.

(i) Show that a pilot wishing to remain in level flight for as long a period aspossible should fly at

km/h .

(ii) Show that a pilot wishing to fly as far as possible in level flight shouldfly approximately 32% faster than the speed given in part (i).

The diagram shows an inclined plane that makes an angle of α radians with thehorizontal. A projectile is fired from O, at the bottom of the incline, with a speedof V m s–1 at an angle of elevation θ to the horizontal, as shown.

With the above axes, you may assume that the position of the projectile isgiven by

,

where t is the time, in seconds, after firing, and g is the acceleration due togravity.

x Vt

y Vt gt

=

= −

cos

sin

θ

θ 12

2

8

θ

y

xO

T

α

V

r

(b)

BA3

14

F AuBu

= +3

4

QUESTION 6 Marks

page35


For simplicity we assume that the unit of length has been chosen so that

.

(i) Show that the path of the trajectory of the projectile is

.

(ii) Show that the range of the projectile, r = OT metres, up the inclinedplane is given by

.

(iii) Hence, or otherwise, deduce that the maximum range, R metres, up theincline is

.

(iv) Consider the trajectory of the projectile for which the maximum range Ris achieved.

Show that, for this trajectory, the initial direction is perpendicular to thedirection at which the projectile hits the inclined plane.

R =+( )

1

2 1 sinα

r = −( )sin coscos

θ α θα2

y x x= tan – secθ θ2 2

21

2Vg

=

QUESTION 6 b (continue) Marks

page36


t (in seconds) 0 7 9 11 18

x (in metres) 0 2 0

(a) Complete the table if the particle is moving with constant acceleration.(b) Complete the table if the particle is moving in simple harmonic motion with centre

the origin and period 12 seconds.

◦ at 4:00 am and 19◦ at 4:00pm. Find, correct to the nearest minute,the times between 4:00 am and 4:00pm when the temperature is:(a) 14◦ (b) 11◦

A particle moves on a line, and the table below shows some observations of its positionsat certain times:

QUESTION 7

QUESTION 8

The temperature at each instant of a day can be modelled by a simple harmonic functionoscillating between 9

4

4

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QUESTION 9

The rise and fall in sea level due to tides can be modelled by simple harmonic motion. Ona certain day, a channel is 10 metres deep at 9:00 am when it is low tide, and 16 metresdeep at 4:00pm when it is high tide. If a ship needs 12 metres of water to sail down achannel safely, at what times (correct to the nearest minute) between 9:00 am and 9:00pmcan the ship pass through?

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sinnt + sin(nt + π) ≡ 0.(b) Show that sinnt + sin(nt + α) + sin(nt + 2α) ≡ (1 + 2 cos α) sin(nt + α), and hence

sinnt + sin(nt + 2π3 ) + sin(nt +4π3 ) ≡ 0.

(c) Prove sinnt+sin(nt+α)+sin(nt+2α)+sin(nt+3α) ≡ (2 cos 12 α+2 cos 32 α) sin(nt+32 α).Hence show that

sinnt + sin(nt + π2 ) + sin(nt + π) + sin(nt +3π2 ) ≡ 0.

(d) Generalise these results to sin nt+sin(nt+α)+sin(nt+2α)+ · · ·+sin (nt+(k−1)α),and show that if α =

2πk

, then

sinnt + sin(nt + α) + sin(nt + 2α) + · · · + sin (nt + (k − 1)α) ≡ 0.

A particle is moving in simple harmonic motion with period 2π/n , centre the origin,initial position x(0) and initial velocity v(0). Find its displacement–time equation in theformx= b sinnt + c cos nt, and write down its amplitude.

b. Show that for any particle moving in simple harmonic motion, the ratio of the averagespeed over one oscillation to the maximum speed is 2 : π.

a.

QUESTION 10

QUESTION 11

[Sums to products and products to sums are useful in this question.](a) Express sin nt + s i n ( nt + α) in the form a sin(nt + ε), andhenceshowthat

6

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v = where A is a constant.)

Let x km be the distance the snow plough has cleared and let t be the timein hours from the beginning of the snowfall. Let t = T correspond to 6 am.

(i) Explain carefully why, for t ≥ T,

, where k is a constant.

(ii) In the period from 6 am to 8 am the snow plough clears 1 km of road, butit takes a further 3·5 hours to clear the next kilometre.

At what time did it begin snowing?

dxdt

kt

=

A–h

QUESTION 12 Marks

6 The first snow of the season begins to fall during the night. The depth of the

snow, h, increases at a constant rate through the night and the following day.

At 6 am a snow plough begins to clear the road of snow. The speed, v km/h,of the snow plough is inversely proportional to the depth of snow. (This means

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1 of one town is growing exponentially, with P1 = Aet , and thepopulation P2 of another town is growing at a constant rate, with P2 = Bt+C, whereA, B and C are constants. When the first population reaches P1 = Ae, it is foundthat P1 = P2 , and also that both populations are increasing at the same rate.(i) Show that the second population was initially zero (that is, that C = 0).(ii) Draw a graph showing this information.(iii) Show that the result in part (i) does not change if P1 = Aat , for some a > 1.

[Hint: You may want to use the identity at = et log a .](b) Two graphs are drawn on the same axes, one being y = log x and the other y = mx+b.

It is found that the straight line is tangent to the logarithmic graph at x = e.(i) Show that b = 0, and draw a graph showing this information.(ii) Show that the result in part (i) does not change if y = loga x, for some a > 1.

(c) Explain the effect of the change of base in parts (a) and (b) in terms of stretching.(d) Explain in terms of a reflection why the questions in parts (a) and (b) are equivalent.

QUESTION 13

(a) The population P

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0 e12 t . When the population reaches a critical value, N = Nc ,

the model changes to N =B

C + e−t, with the constants B and C chosen so that both

models predict the same rate of growth at that time .

c and N0 .(b) Show that the population reaches a limit, and find that limit in terms of Nc .

The growing population of rabbits on Brair Island can initially be modelled by the law ofnatural growth, with N = N

QUESTION 14

(a) Find the values of B and C in terms of N

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Salt water

Water andsalt watermixture

1000 Ltank

(a) If the salt water entering the tank contains 2 grams of salt per litre, and is flowing inat the constant rate of w litres/min, how much salt is entering the tank per minute?

(b) If there are Q grams of salt in the tank at time t, how much salt is in 1 litre at time t?(c) Hence write down the amount of salt leaving the tank per minute.

(d) Use the previous parts to show thatdQ

dt= − w

1000(Q − 2000).

(e) Show that Q = 2000 + Ae−w t

1 0 0 0 is a solution of this differential equation.(f) Determine the value of A. (g) What happens to Q as t → ∞?(h) If there is 1 kg of salt in the tank after 534 hours, find w.

QUESTION 15

In the diagram, a tank initially contains 1000 litresof pure water. Salt water begins pouring into thetank from a pipe and a stirring blade ensures thatit is completely mixed with the pure water. A secondpipe draws the water and salt water mixture off at thesame rate, so that there is always a total of 1000 litresin the tank.

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that is,dP

dt= k(P − B).

(a) Take reciprocals, integrate, and hence show that log(P − B) = kt + C.(b) Take exponentials and finally show that P − B = Aekt .

ideal population I, that is,dP

dt= k(P − I), where k may be positive or negative.

(a) Prove that P = I + Aekt is a solution of this equation.(b) Initially 10 000 animals are released. A census is taken 7 weeks later and again at

14 weeks, and the population grows to 12 000 and then 18 000. Use these data to findthe values of I, A and k.

(c) Find the population after 21 weeks.

QUESTION 16

[Alternative proof of the modified natural growth theorem] Suppose that a quantity P changes at a rate proportional to the difference between P and some fixed value B,

It is assumed that the population of a newly introduced species on an island will usually grow or decay in proportion to the difference between the current population P and the

4

6

QUESTION 17

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Section ISection IIExam ReportExam ReportExam ReportExam ReportExam Report2013 HSC Mathematics Extension 1Section I 1 2 3 4 5 6 7 8 9 10

Section II Question 11 Question 12Question 13 Question 14

Standard Integrals

2013 HSC Mathematics Extension 1Section I 1 2 3 4 5 6 7 8 9 10


Standard Integrals



Standard Integrals



Standard Integrals


Section II Question 11 Question 12Question 13

Mathematics Extension 1Section I

3 Unit Maths-TRial HSC Test 7 Trial...Pre HSC ReVISION MATHEMATICS EXTENSION 1 (d) The concentration...

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