model solutions updated pdf 96471

7/15/2019 Model Solutions Updated PDF 96471

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Part A: Mathematics for Physics [SOVfarks]

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 = gh

for the range0 < r <.rf2, sinr is smaller han tanr.

Cu"-l'= .(r 01t- t

b ?Syn"$''iJuiae,n

(iii) Using the equality cos2 :+(t * cos20) or otherwise, expresscosa in

terms of cos20 and cos40. t3l

u)t g:@r'8' = t( r+ uzs)z

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

?-ur','18Lz( + 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

E30F

Z

Iz

L,

,z {"-6

l 'q* - l t

( i i ) log*2:2,

(i i i ) logr2: r.

I r ll 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)dY '= dq(l+o'aot ' +t4G")"=--()

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 thenth bormce t returns to aheight hl$\.Find the totai distance traveiled

by the ball.

t)tsto*rrt ^^r1& a =

[4]

h5'{\

fer)

h+

:h(

z \ / t rn t1LG\ I

n=o- , /

4,/- \

3)

t- l.^ I5/

-h(

=Zh

w

n>L(_

n *- l

l+ 7-

f

5 cl,r^

l \=o

=Al - r

\-\^-

*TeY3'rrus

I r )



8. ( i ) Sketchhe curve :2lr l +1

13

i

3

- l

?_+

[2)

Ar=Z

Az= ! f,*"*'"L,i.lh -&n' A,

/+=A,+A.n,4.

91L- 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 {LL_ l

:-



9. Two identical dice are thrown, one after the other. \\trat are the

probabilities that

(i) the totai of the numbersshown is 6, tzle,,ru\;ah^=er* tu44' , e.,<. , ( ),(5,

') , (z 4)., +,2) (?,2 ,{:'?)i{. q u*i";J-'r o* ,\,N eul "1.dxL '36

. ?= Wse= i/a

7^,1[4]

hAr,rrL4r>{lt

z6t,

i

II

,,--"i

t itt**-a--"

"')-..-">*-.,-

4'-*- "i

t - il

:

-.- ' - i - - . - .l

6i i

c\

sr&{r?-

c^.r)&c{t

fr, &{-zA+34

q=

lz=--'

,{ = 4f -aa\

-Zj

: 3ar-3a. - Zarl-Lx' QZr'

': -'" '"-l--**4:-::f-

';. '1.'.{. .i "

t ::i

P: Ls6

10. A geometric progressionand 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 geometricprogression. 01

t- l

nI

1L

--

\

c^'{ l[ - :3t fq==3i1=l . t 4 s=1t2, ' I?l

c\oA f , "_4V- "4, ; . -z/ ,+ f_=l

. ,A=

0.+4,1(ii) Show that the fifth term of the arithmetic progressibn s zero.

Jt f^: l lz, ** A= -^/+

- 3r+ l=o-a/+

o -. y_=l l2fo l

IJ J :

hs qt4"*^ + AP"

. lIt -l

c\ + Yc^

/ ,= (L ++ -d\

1==O

4-

---



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 = O6t-.

a

a'-

'^-L

J

drt"h. o,^rt g*kt

tf-1)=

3(6)/ /

{tv'+X = CJ l)

sJ

{a^L ;> -l (rJ

ot'r6*,

-!o +2{+A= - LJ

4ry-21 +36+l = tc

nq:6o+ | = 66.

>

zq+I

+ | =-8+24+l

: .%\1.+

1- f crr\-- -/

(_

{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 2cm from the detector, the count rate risesto 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 rt 'J

13. An aeroplanehas 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 downturn 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?

111L- t

11lL^ t

Gl I 2 yourheightlY 314yourheight

I- - ]_.

B 7f 4 yow height

D your full height

i\



16. A metal disc with a hole in it is heatedand expands

hole has

CA) iocreasedY .1""r"r.".1C stayed the same

D it depends on where the hole is

bv I%. The sizeof 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 seriescombination

D the same current in the two situations t1l

20. An astronaut on the Moon

earth experiencesa

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 lt - - l

towards a plane mirror, at what speed doesyour 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, theposition 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 positionof this image.Mrrtr X

Ir lL- l

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

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 massof the three particles is the sarneas 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 eachof the three particles. [5]

0

@

o

Clu,rorq-0

i l=o3: +h + q

-s + <:W-2a

Ma-

l{s* fl^ * Me- () wl<.

Ms+ lvle= 3'vt"

: . f i ' lh=3m.

Ms= Z'na,

5+e ET

-2s -- 3%

g; 3vz

h___34(

1t = -ZA, "t-sr

=_cl /4"

z

lr -- 2s



24. A black box has three electrical connections abelled A, B, and C, arranged

in a triangle as shownbelow.

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 serieswith 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

WcuJAA,k ^ u/,r-*{- + 3^ A a?f'-*' 1]

(a)

(b )



Long question (20 marks)

25.Consider

an elevatorcar (lift) with a massof 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 30s. Calculate the total

energy needed and the power of motor required. t4)

E = nnah= hn + ar) xlcx9 = lasaoqo '= ll-tar>e" . - . \ '

\ ' -/ :

flcr-str- F --C

t i?crr) . l3oox3 E

3A

3AooW

(b) The elevator system can be made more efficient by incorporating a

counterweight, with a mass of 2000kg 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..*.;Jil"a.tg

d\M155MV

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 = 3rtwa"6 {*r( t eNt@t,-v

i"=

to

Ght KE. =

t4lr / 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*?,rsooe;

61 q T='-

3',<O *C14--

3o

= 3 yy15-l

20

4rY

-qd T '/-z

(d) The elevator takes 10s to accelerateand 10s to decelerateusing 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=

* 9a

"33

s?= af 5 -L^b) ' =: 'z [3/

- S1?r-- 5,+ \t \= 1-*'

q

ry =: -- l : -o+cM7x3o ?xro -??

t 4-

= CR.F = ))1-

? 4LLr v

V*ox s*t

.-, ,0\ -- 5tJ,,t-',

J-

l/v



(e) A man has a mass of 68kg. 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 thelift is stationary, (ii) the change n this reading when the lift is

accelerating upwards, and (iii) the reading on the scaleswhen the lift

is moving upwards at the steady maximum speed calculated above. [6]

,44: d8qilI

a = lAAt-'J

A1

i i6)

6Eo + 6? x o'a16

6Nxo 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.

trL, ' .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

model solutions updated pdf 96471

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