Model Solutions

7/18/2019 Model Solutions

http://slidepdf.com/reader/full/model-solutions 1/15

Part A:

Mathematics for Physics

[SO

Vfarks]

7. If p: JJ urld g

:

J, evafuate

2.

Find

the set of

real numbers .\ for

which the

quadratic

equation

12+(\-3)r*) :0

has distinct, real

roots for

r.

, "#t

a{

f , :

3-}

I

: J.LT,C

#,^R

V

Z

,*{,+ (l-l )'>

[ - tJ:

4x

t ' -

C

+

q

=

+I

Xe-

tOA+

'O

6*')(.r-t)

o

>qA

: .

I

(

|

o-

so

z'\ \'

q'

+)

-4o

FI,

frw 6-t)'

)>I



3

(i)

Draw sketchesof

the functions

sin r and

sin2 over the

range

-2tr

<.

r

12r.

(Label

the axes).

(ii) Explain why,

' .5tanL = g

h

for the range

0 < r

<.rf2,

sinr

is smaller han tanr.

Cu"-l'= .(r

01

t- t

b

?Syn"$''iJuiae,n

(iii)

Using

the equality cos2

:

+(t

*

cos

20)

or otherwise, expresscosa in

terms of cos

20 and cos40.

t3l

u)t

g:

@r'8

'

=

t(

r

+

uzs)z

t(,. L,,>s?0ur;20)

?-ur','18

Lz(

+

u)sUn))

+

7*s2A

,

Lr.

t4g)

=

t( , -

;

L(r.

-+(tnQ'>LO L u'a})



tF l

lo l

: '

(dt

Wr;r.*lq

+

c'= <il+6t

=

3?+

S+z)(

=

I

+36

=

46

-

t f*2.

=

lO+4

=Zo

Ir'

1

c

eb=

=+

1-4

lL=

{

{=

IV

/ ' - -

|

Arur.

lzo

E30

F

Z

I

z

L,

,z

{"-6

l 'q*

- l t

( i i ) log*2:2,

(i i i ) logr2: r.

I r l

l r l

I1l

[1]

4.

Show

hat

(3,

4),

(-4,0)

and

(0,-2)

are

the

verticesof a right-angled

triangle, a,nd

ind its area.

-11t.{

(

tl'

5.

Find the value of r

for

which

(i) logrr :2,,

7

L:

Z-



Evaluate

(2.002)o

o 4 decimal

places.

( l aoi lu- 2 ' ( t oct)d

Y '

=

dq(l+o'aot '

+t4

G")"=

--()

xtry

+

14)

l+ nx + n(a-")?,..

2t

, .

\

' /

t r t .

6tt

o

-

:l

4

t6

b-+

w

)u

rl

16

o

=

64.

gt+96

=

6+

gsSo

&

+

J,*;,i

pfu'.

7. A ball is dropped vertically from a height h onto a flat surface. After the

nth bormce t returns to aheight

hl$\.Find

the

totai distance traveiled

by the ball.

t)tsto*rr

t

^^r1&

a

=

[4]

h

5'{\

f

er)

h+

:h(

z

\ / t rn t

1LG\ I

n=o-

, /

4

,/- \

3)

t-

l.^

I

5/

-h(

=Zh

w

n>

L(_

n

*- l

l+ 7-

f

5

cl,r^

l \

=o

=A

l - r

\-\^-

*Te

Y3'rrus

I

r )



8.

( i )

Sketchhe curve

:2lr l

+1

1

3

i

3

- l

?_+

[2)

Ar=

Z

Az

=

f,*"*'"

L,i.lh

-

&n'

A,

/+=

A,+A.n,4.

91

L- l

1(r{1.

L

-t

I

(ii)

Find the area between he

curve

A

:

2lr

|

+1,

the z-axis, and

the lines

tr : landr--1.

Ate^

--

I

( l {L

L_ l

:-



9.

Two identical dice are thrown,

one after the

other.

What

are the

probabilities

that

(i) the total of the numbers shown s 6, [2]

C,:r,.

*afr,.= tr^*

v,'4

^,

6

c^R.

,

[),(5,')

,

(Z

4).'

+'Z),

3'12

i

€.

q

u;Lr, J',s

o"t

l,

N

Ur^l

,l-

ixL

'36

,

I

:

( ilb e

second

number is

greater

than

the first?

t4l

, l

2

-3

q:{

Q-

* r

'4.,

}

I n

-

l i

'= '2).+

<

*-

i

' . . , , ,1-"G'*i*--.

*

=

f*

4+

3<Zt

t

: I ,q

i

', ' '

i,l

1-

'

;

Gk(

*^W

4

,o^l;'*J"'ns

z("

qi

, , . i ( ; r2"

e

p

=

tq

,q

-- i l . .

, .

-

;

r

-

|

'

=

-

1

i

l

i

bt

I i i

<- ' -

t'

10. A

geometric progression

and

an arithmetic

progression

have

the same

first

term. The

second and third terms

of the

geometric

progression

(which

are

distinct) are equal to the third

and

fourth

terms

of the arithmetic

progression respectively.

(i) Find the common ratio

of the

geometric

progression.

n

gP

RP

&r=q+LA

|

6\

&.

&{-z=

A+jA

-

=

2

cr

c\+,1

vvr

:.

3ar-3o.

=

Za"r1-L,x

I2r'-

;i;

f-fl,

^

t.":9"

= z::

L,t

+,7',':t?,

Jf f-: llz, ** A= -^l+

i^r

'qt."

*^

+

AP

"

.N

c\

+

+,i

/ ,

:

(^

++

-O

--''---

9t

t- J

[3]

3r+

l=.J

-a/+

: .

;= l l2

-(;\

L

I



11. What are

the largest and smallest values of

y

:

13

-

72r

*

1 for values

of r

in

the

range

-3

to

+5?

Sat

+

fuL

aR

tv\NY

""^^

;

*t

1^

M

=

3z'-

lL

>1)

, )*

'7-t=

4

f F l

Io j

t

)L:2/

-

-z

l- i

d'1

-

6x

Ti,

-

t''r /

d^1

=

O

6t-.

a

a'-

'^-

L

J

drt"h. o,^rt g*kt

tf-1)=

3(6)

/ /

{tv'+X

=

CJ

l)

sJ

{a^L

;>

-l (

rJ

ot'r

6*,

-

o

+2{+

A=

-

L

J

4ry

-21

+36+l

=

tc

nq:6o+

|

=

66.

>

zq+I

+

|

=-8+24+l

:

.%\1

.+

1-

f

c

rr\--

-/

(_

{lfibA\/tttt+,

8

-24+t

-td

:===-

fu*t;,tt*'



Part B:

Physics

[fO

Marks]

Multiple choice (10 marks).

Please circle one

answer to each

question

only.

12. A

radioactive source

s

placed

6cm

from a radiation detector sensitive o all

forms

of

ionizing radiation, and records 74 counts/minute.

When

a 1 cm

thick aluminium

piate

is

placed

in the

gap

then the

count

rate falls to

45 counts/minute.

If the source is removed

entireiy then the count

rate

remains the same. If the aluminium

plate

is removed

and the source

is

placed

2

cm

from the detector, the

count

rate rises

to 5000counts/minute.

\\4rat

forms of radiation

does

the source emit?

Cf)

.fpfta

only B beta

only

Y alpha and beta

D

alpha and

gamma

l r

t 'J

13. An aeroplane

has two engines,

one under each wing.

Suppose

he left

engine stops working.

If

the aeroplane'scontrols are not

changed

hen

the

aeroplane

will

dJ=

+

B

I

slow down and

fall but not

turn

turn

right

and

fall but not slow

down

turn left, fall and slow down

turn

right, fail and slow down

14. A sky-diver

jumps

out of an aeroplane. Which

of the

following

statements

is true after stre eaches

erminal

velocity?

(

A

)

The force of air

resistance

s the same size as her

weight.

Y fft" force of air

resistance s larger

than her weight.

C

The force of air

resistance

s smalier than her

weight.

D

She

begins to slow down

I1l

15.

What

is the

minimum length

of a

plane

mirror

in order for

you

to see a fuil

view of

yourself?

111

L- t

11l

L^ t

Gl

I

2

your

height

lY

314

your

height

I

- - ]_.

B 7f 4

yow

height

D

your

full height

i\



16. A metal disc with

a hole in it is heated

and expands

hole has

CA) iocreased

Y .1""r"r.".1

C stayed the same

D it depends on

where

the hole is

bv I%. The size

of

the

l l l

17.

A baby's bath should be at a temperature

of 37'C. The bath already

contains

10kg of water at 15'C. Approximately

how much hot water, with a

temperature

of

50'C,

should be added

to achieve he desired emperature?

B

15kg

D 20kg

18. A stone s thrown straight upwards. At the top

of

its

path

its velocity is

momentarlly

zero.

\rhat

is

the magnitude of its

acceleration at this

point?

B

5ms-2

D 20ms*2

19. A battery is connected

first

across one

bulb and then across two bulbs in

series. If all the bulbs are

identical

then the

battery delivers

(

A) less current to the seriescombination

rf lnore current to the seriescombination

C a lower

potential

difference across the series

combination

D

the

same current in the two

situations

t1l

20. An astronaut on the

Moon

earth experiences

a

A solar eclipse

C both

A

10ke

@ rzr.s

A

0ms-2

Q)

ro-.-'

If

you

run at

5

ms-i

approach

you?

A 5

ms-1

/^ \

IC)

10ms-'

[1]

[1]

[1]

21.

B 2.5ms*1

D

15ms-]

I

*nrecripse

observesa solar

eciipse; at the same time the

l i l

t - - l

towards a

plane

mirror,

at what speed does

your

image



Written answers

(20

marks)

22. The diagram

below

shows two mirrors X and Y, and a solid object with

white spots at

P and

Q.

On this diagram

answer the following

questions.

(a)

An

observer

at

A

seesan image of P reflected in

mirror

Y. Mark

R,

the

position

of this

image,

and draw a ray from P

to the observer at A.

[2]

(b)

In which mirror

would

an observer at A see an image

of

spot

Q?

Ma.rk

S,

the

position

of this

image.

Mrrtr

X

Ir l

L- l

(c) An observer at B can see an image of P resulting from reflections at

both mirrors. Draw

a ray

of

light

from P to B

which enables

his

image

to

be seen.

l2l

B



23. Suppose

hat three

new

particles

are discovered,called the slepton,

the

hozon and

the elephoton.

These

cannot

be investigated on their

own, but

the properties of certain combinations have been measured. Each

particle

has its own

antiparticle,

with the same

mass

but opposite

charge. Given

that the

following observations

are

made

o

The combination

of two sleptons and

a hozon has no overail

charge

o

The

combination

of three sleptons,

ahozon and an eiephoton

has

a

chargeof

+1q

o

The total mass

of the three

particles

is

the sarne

as that of six

eiephotons

o

The combination

of an anti-slepton

and an elephoton has the same

mass as three elephotons and a total charge of -2 q

work out the

charge

(in

units

of

q)

and the mass

(in

units of

the elephoton

mass)

of each

of the three

particles.

[5]

0

@

o

Clu,rorq-

0

i l=o

3: +h + q

-s

+

<

:W

-2a

Ma-

l{s*

fl^

*

Me-

()

wl<.

Ms+

lvle= 3'vt"

: .

f i ' lh=

3m.

Ms

=

Z'na,

5+e

E

T

-2s --

3%

g;

3v

z

h__

_34(

1t

=

-ZA,

"t-

s

r

=_cl /

4"

z

lr

--

2s



24. A black box

has three

electrical connections abelled A, B, and C, arranged

in a triangle

as shown

below.

The box contains

three components: a

resistor,

a smail capacitor

and a

diode.

You know that

one

component

is connected between each

pair

of

terminals, but

you

cannot see exactly how they are arranged.

You make

the

following observations

with a 9 V battery connected

in

series

with an

ammeter

o

Mlhen

the battery

is connected with

*

to A and

-

to B a current of

3mA

flows

fgS^klr

o

When

the battery

is connected with

f

to B and

-

to

C

a

very

large

current

flows

o

When

the battery

is connected with

*

to

C

and

-

to A no current

is

measured

o

When

the battery

is

connected with

-

to

B

and

*

to C

no

current

is

measured

On

the diagram

sketch the

arrangement

of the

three components

in

the box

indicating the terminals clearly.

l4l

Caiculate

the

resistance of the resistor.

'.21

K-- Y:9

=

Sxto- ' '

3K.5L'

T 3xc;-->

(c)

What

would happen

if

the battery

were

collnected with

-

to A and

*

h

to B?

(d)

What

would

happen if

the

battery were connected with

-

to C and

*

to

A?

fr l

WcuJA

A,k

^

u/,r-*{-

+

3^

A

a?f'-*' 1]

(a)

(b )



Long

question

(20

marks)

25.

Consider

an elevator

car

(lift)

with a mass

of 700k9

which call carry

up

to

600kg

of

passengers.

se

g

:70ms-2.

(a)

An

electric motor

is

used to

raise

the elevator with a full ioad from the

ground

floor to the third floor, 9 m

higher,

in

30

s.

Calculate

the total

energy needed and

the

power

of motor

required.

t4)

E

=

nnah

=

hn

+

ar)

xlc

x9

=

lasa

oqo

'=

ll-tar>e

" . - . \ '

\ ' -

/

:

flcr-str-

F

--

C

t i?crr)

.

l3oox3

E

3A

3Aoo

W

(b)

The elevator system can be made

more

efficient by incorporating a

counterweight, with a mass of 2000

kg

as

shown in

the

diagram

beiow.

What total

mass of

people

is needed n

the car to balance the svstem?

t2l

fl.^lit\.gf6"

4*vJ'*i..*.;J

il"a.tg

d\M155

MV

LT

=

Mr3

f=

M;+Mp

1cc

+

Mp

I6pO=

-7tJa

{-

Mp

MP.

&ul'*

z

Mi,,u

-

?6W(5

lift

fif

Mp - 3ocuE



(c)

When

the system

is

balanced the elevator takes 30 s to

make the

upwards

journey

to the third floor. Calculate the average speed and

the kinetic energy of the system at this speed.

r ,

- - t

t I l -

:

q

=

3

rtwa"6

{*r(

t

eNt@t,-v

i"

=

to

Ght

KE.

=

t4l

r / rs- '

k-)

Ca1rfq;g4

il,l,dw:r

t4

*1

V

c''o\

:e

lo'a^ otl't1r"1f

@

| . . ,7-

KE=

/-M\V

/

u

KEr^*r*'

Lr,

t'soo

"

€r)'

=

lKEo.,r,*uDd,,{r-r*?,rsoo

e;

61

q

T

='-

3',<O

*C

14--

3o

=

3

yy15-l

20

4rY

-qd T

'/-

z

(d)

The elevator

takes 10s to accelerateand 10s

to decelerate

using the

same magnitude

of constant force in each case.

Calculate

the

greatest

speed which the lift attains.

t4)

L=

.3o:.

O\f

3

V*ux=

a|

,1*

-->

c-

r r t?

q4G

-)-

(^t* lbc\k-

S,

t

^k)'

'zFl

i_

S1

=

* 9

a

"33

s?=

af

5

-L^b) '

=: 'z

[3/

-

S1?r--

5,+

\t

\

=

1-*'

q

ry

=:

-- l : -o+cM

7x3o

?xro

-??

t 4-

=

CR.F =

))

1-

?

4L

Lr v

V*ox

s*t

.-,

,

0\

--

5tJ,,t-',

J-

l/

v



(e)

A man

has

a

mass of

68

kg.

While standing

in

the balanced

lift,

he

measureshis weight

(in

N) using spring scales as the lift moves up as

described n part (d). Calculate (i) the reading on the scaleswhen the

lift is

stationary,

(ii)

the change n this reading when

the

lift is

accelerating

upwards, and

(iii)

the reading on the

scales

when the lift

is moving upwards at the steady maximum speed calculated above.

[6]

,44:

d8

q

il

I

a

=

lAAt-'

J

A1

i i6)

6Eo

+

6?

x

o'a16

6Nx

o Dh-f

=

3

.oe,

w=

689

-ob

NJ

44t^

ust'"*l-

y

3AJ

,*

S{AJ,^

wo'rlu;^^'^

tF

'tl

,

@

et\"h"

ucolehh'r

tnJ

,-

6JO

/lJ

/

"r*U*f"-

I

cJil*ke;

,

\

prd-4^-a^F

<h.

tr

L, '

.1

b\

t , . .--

(+'r"

----

L1

ZC

3qO

366

0

rri

)

-

oT-

,1,. tu,

f f i r*)

lqJ

=

n4\

\H,i 68rie'

W=

nrr(t*^)

+q{69'

\06o

)a442

04

Documents

Model Solutions