Revision of Chapters 15–16

C

H

A

P

T

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R

Revision

17

Revision of

Chapters 15–16

17.1

Multiple-choice questions

1 If loga 8=3 thenais equal to

A 1 B 2 C 3 D 4 E 0

2 5n−1_×5n+1_{is equal to}

A 5n2

B 52n _{C 10}2n _{D 25}2n _{E 25}n2₋₁

3 If 2x = 1

64 thenxis equal to

A 6 B −6 C 5 D −5 E 1

6

4 125a_×5b_{is equal to}

A 625a+b _{B 625}ab _{C 125}a+3b _{D 5}a+3b _{E 5}3a+b

5 The solution of the equation 4x₌₁₀₋₄x+1_is

A x=4 B x=2 C x= 1

4 D x = 1

2 E x=1

6 7

n+2₋₃₅₍₇n−1₎

44(7n+2₎ is equal to

A 1

49 B 1

44 C 1

28 D

1

7 E 7

7 Iff(x)=2+3x_{thenf(2x)−f(x) is equal to}

A 3x B 32x C 2+3x D 3x(3x−1) E 3x(3x+1)

8 If 72x_×492x−1_{=1 thenxis equal to}

A −1 B −1

3 C 1

3 D 1 E 3

Revision

9 The graphs ofy=2xandy=

1 2

x

have

A the samex-axis intercept B the samey-axis intercept

C no point in common D two points in common

E three points in common

10 If f(x)=(2x)0 _{+ x}−2_{3,thenf(8) is equal to}

A 5

4 B

65

4 C 5 D 20 E None of these

11 loga2_+logb2_{−2 logabis equal to}

A 0 B 1 C a D b E a2_b2

12 An angle is measured as 2xradians. The measure of the angle in degrees is

A _x 90 ◦ B 90x ◦ C 180x ◦ D 360x ◦ E _x 360 ◦

13 The figure shows the graph ofy=sin 2x+1. The coordinates ofQare

A

4,2

B

12,2

C (,1)

D

4,1

E

2,1

y _Q

y = sin 2x + 1 1

0

2

π π θ

14 The smallest values of 1−3 cosis

A −5 B −4 C −3 D −2 E −1

15 A rollercoaster is constructed in such a way that any car isymetres above the ground when it isxmetres from the starting point, wherey=16 + 15 sin

60x

.

The height of the car, in metres, whenx=10 is

A 31 B 1 C 16 D 23.5 E 16−5√2

16 sin ( + ) + cos ( + ) is equal to

A sin + cos B −sin + cos C sin−cos

D −sin−cos E −sincos

17 For∈[0,],which of the following equations has exactly two solutions?

A sinx=0 B cosx=0 C sinx=1

D sinx= −1 E cosx=1

18 Which of the following is the graph ofy=sin

2 for one cycle?

Revision

19 The minimum value of 2−3 sinis

A 2 B 1 C 0 D −1 E 3

20 1 0 90 180 270 360 –1 y x

The above is the graph of

A y=cos (x−30)◦ B y= 1

2cos (x + 30)

◦ _{C y=} 1

2cos (x−30)

◦

D y=cos (x+30)◦ E y= 1

2cosx

21 The functionf:R→R,f(x)= −2 cos 3xhas

A amplitude 2 and period B amplitude –2 and period 2 3

C amplitude 2 and period 6 D amplitude 3 and period

E amplitude 2 and period 2 3

22 IfCd_{=3 thenC}4d_{−5 equals}

A 76 B 7 C 22 D 86 E 35

23 The value of log256−log27+log22 is

A log251 B 1 C 2 D 3 E 4

24 If logb a=cand logxb=c, then logaxequals

A a B c−2 _{C b}2 _{D b E abc}2

25 If cos−sin= 1

4,then sincosequals

A 1 16 B 15 16 C 1 32 D 15 32 E 1 2

26 The coordinates of a point of intersection of the graphsy = 1

2sin(2x) andy= 1 2are A 1 2, 3 B 4, 1 2 C 2, 1 2 D 1 2, 6 E

2,1 2

Revision

17.2

Extended-response questions

1 The height of the tide,hmetres, at a harbour at any time during a 24-hour period is given by the equationh(t)=10+4 sin(15t)◦, wheretis measured in hours.

a Sketch the graph ofhagainsttfor 0≤t≤24.

b Find the times at whichh=13 during the 24-hour period.

c A boat can leave the harbour when the height of the tide is at 11 m or more. State the times during the 24 hours when the boat can leave the harbour.

2 Medical researchers studying the growth of a strain of bacteria observe that the number of bacteria present afterthours is given by the formulaN(t)=40×21.5t.

a State the number of bacteria present at the start of the experiment.

b State the number of bacteria present after:

i 2 hours ii 4 hours iii 12 hours

c Sketch the graph ofNagainstt.

d How many minutes does it take for the number of bacteria to double?

3 For a ride on a Ferris wheel, the height above the ground of a person,hm, at timet(s) is given by

20 m

2 m

h(t)=11+ 9 cos

30(t−10)

a How long does it take for the Ferris wheel to make one complete revolution?

b Sketch the graph ofhagainsttfor one revolution.

c State the range of the function.

d At what time(s) is the person at a height of 2 metres above the ground?

e At what time(s) is the person at a height of 15.5 metres above the ground?

4 The voltage,V, in a circuit aftertseconds is given byV=120 cos 60t.

a Sketch the graph ofVagainsttfor one cycle.

b Find the first time the voltage is 60.

c Find all times at which the voltage is maximised.

5 The figure shows a waterwheel rotating at 4 revolutions per minute. The distance,d, of a point,P, from the surface of

the water as a function of time,tin seconds, can be modelled by a rule of the form

P

3 m

2 m

d d=a+bsinc(t−h)

a Find:

i the period ii the amplitude iii c b Ifd=0 whent=0 findh.

Revision

6 A forest fire has burnt out 30 hectares by 11.00 am. It then spreads according to the formula

h=30(1.65)t

wheretis the time in hours after 11.00 am.

a Findhwhen:

i t=0 ii t=1 iii t=2

b Findksuch thath(N+1)=kh(N).

c How long does it take for 900 hectares to be burnt?

d Sketch the graph ofhagainstt.

7 A bowl of water is initially at boiling point (i.e. 100◦C). The temperature,◦C, of the watertminutes after beginning to cool is=80(2−t)+20.

a Copy and complete this table:

t 0 1 2 3 4 5

b Draw a graph ofagainstt.

c At what time is the temperature 60◦C?

d Findwhent=3.5.

8 A,BandCare three countries. Each of them now has a population of 70 million. CountryA’s population grows uniformly by 30 million in every

period of 10 years.

CountryB’s population grows uniformly by 50 million in every period of 10 years.

CountryC’s population is multiplied by 1.3 every 10 years.

a Give an equation for the population of each country at timet.

b On the same set of axes carefully draw the graphs of the three equations.

c From your graph find out when the population ofCovertakes:

i the population ofA ii the population ofB

9 An estimate for the population of the Earth,Pin billions, is

P=4(2)

(t−1975)

35

wheretis the year.

a EvaluatePfor:

i t=1975 ii t=1995 iii t=2005

Revision

10 Two tanks simultaneously start to leak. Tank A containsV1(t) litres and tank BV2(t) litres

of water, where

V1(t)=1000e

−t

10 _t ≥₀

V2(t)=1000−40t 0≤t ≤25

andt is the time after the tanks start to leak.

a FindV1(0) andV2(0).

b Sketch the graphs ofy=V1(t) andy=V2(t) for 0≤t≤25.

c How much water is in tank A when tank B is empty?

d Find the times at which the two tanks have equal amounts of water in them.

11 A river gate is used to control the height of water in a river.

On one side of the gate the height of the water is subject to tides. The depth of the water in metres on this side is given by

h1(t)=18 + 10 sin

6t

wheretis the time in hours past midnight.

On the other side of the gate the water rises according to the rule

h2(t)=8+6t

wheretis the time in hours past midnight andh2(t) is the height of the water at timet.

a Sketch the graphs ofy=h1(t) andy=h2(t) for 0≤t≤6, on the one set of axes.

b Find the time at whichh1(t)=h2(t).

c When the water levels on both sides of the gate are the same, the gate is opened and the height of the water on both sides is given byy=h1(t).

i The gate is closed again whenh1(t) reaches its minimum value. At what time does

this happen?