Part A: Mathematics for Physics [SOVfarks]
JJ urld g : J, evafuate
7. If p:
-4o FI,*
2. Find the set of real numbers .\ for which the quadratic equation
12 + (\ -3 )r* ): 0
has distinct, real roots for r.
," # t
a{
3-}
f,:
I
Z
:
J.LT,C
#,^R V
,*{, + (l-l )' > + )
[-tJ: 4x
t ' - Cx + q = + I
Xe- tOA+I'O
=o
6*')(.r-t)
frw
6-t)' >qA
:.
I ( |
o-
so
)>I
z'\t
\' q'
3 (i) Draw sketchesof the functions sin r and sin2r over the range
-2tr <. r 12r. (Label the axes).
(ii) Explain why, for the range 0 < r <.rf2, sinr is smaller than tanr.
'
Cu"-l'= .(r
.5tan)L = g
01
t-t
h
b ?Syn"$''iJuiae,n
(iii) Using the equality cos2t : +(t * cos 20) or otherwise, expresscosag in
terms of cos20 and cos40.
t3l
u)t g: @r'8)' = t( r+ uzs)z
t(,.
+ ur;20)
L,,>s?0
=t( ,-?-ur','18+ Lz(l+ u)sUn))
; L(r.+ 7*s2A , Lr.*t4g)
- +(tn Q'>L O L u 'a })
4. Show that (3, 4), (-4,0) and (0,-2) are the verticesof a right-angled
triangle, a,ndfind its area.
:'
(dt
Wr;r.*lq
+
(
-11t.{
tl' =
q
Ir' -
tF l
lol
c'= <il+6t
3?+ S+z)(= I +36 = 46
tf*2. = lO+4 =Zo
1
c
I
Arur.= ,z eb=
5. Find the value of r for which
(i) logrr :2,,
7
Z-
L:
z {"-6
L,l'q*
lzo
E30
F
Z
-l t
Ir l
lrl
=+
( ii) log*2:2,
I1l
1- 4
l L=
{
{= IV
(iii) logr 2: r.
[1]
/'--
|
Evaluate (2.002)oto 4 decimal places.
(l
Y
'
=
G")"=
o c t)d
2 ' (t
aoilu-
14)
l+ nx + n(a-")r?,..
--()
+ t 4 xtry +
dq( l+ o' aot '
,. '/
2t
\
t rt .
6tt
o
b:l
- 4
& + J,*;,i pfu'.
= 6+ gsSo
t6
b-+ w
)u
64. gt+96
=
rl
16 o
7. A ball is dropped vertically from a height h onto a flat surface. After the
nth bormce it returns to aheight hl$\.Find
the totai distance traveiled
by the ball.
[4]
w
t)tsto*rrttz^^r1& a =
h+
n>
L(_
h
5'{\
n *- l
:h (
l+
f
7\-\^-
f er)
5 cl,r^ =A
l\ =o
Y3'rrus
*Te
l-r
z \/trn
1r L n=oG\
4
,/-
-h (
3)
=Zh
I
,/
\
t- l.^
I
5/
I t r)
t
8 . ( i) S ke tchth e cu rveA :2 l rl
L
-t
I
91
1 (r{ 1 .
+1
L-l
1
3
i
3
(ii) Find the area between the curve A : 2lr | +1, the z-axis, and the lines
tr :landr--1.
[2)
Ar =Z
Az = ! f,*"*'"L,i.lh &n' A,
/+= A,+A.n,4.
-l
Ate^ --
?_+
I
(l
{L
L_l
:-
9. Two identical dice are thrown, one after the other. \\trat are the
probabilities that
(i) the totai of the numbers shown is 6,
tzl
H't, e.,<.0, ( ), (5,') , (z,4).,(+,2), (?,2 ,{:' ?)
e,,ru\;ah^=er* tu44
i{. q u*i";J-'r o* ,\, N eul "1.dxL '36
. ?=tW/se
= i /a
,,--"i
I
I
7^,1
ti
tt
[4]
hAr,rrL4r>{lt
**-a--"
ir'l
t,
tst
:.. "
')
l-)a -..-">*-.,
*-*-
4 '"i
t-i
Fl
z6
': -'"'"-l--**4:-::f
';.
:
-.-' -i--.- .l
6ii
{.
. i"
P: L
-
'1.'.
i
q
=
lz
s6
t:
=--'
:i
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.
01
(i) Find the common ratio of the geometric progression.
t- l
n
I
c\
1
sr
L
fr,
c^' {l
+ c^.r) c\ oA
--
\
&{r?-
&c{t
0.+4,1
= A+34
&{-z
,{ = 4f -a
a\
:
-Zj
3ar-3a. - Zarl-Lx' QZr' - 3r+ l=o
' I ?l -a/+
, A=
4 s = 1t .2,
o -. y_=l l 2
f _= l
+
progressibn is zero.
, "_4
tfq==
"4,l.; .t- z / ,
f[-:3
V- 3i1=
(ii) Show that the fifth term of the arithmetic
Jt f^: llz, **
fol
IJJ
A= -^/+
.l
It
hs qt4"r*^ + AP"
-l
c\ + Yc^
= (L ++
==O
4-
---
/,
- d\
1
:
11. What are the largest and smallest values of y : 13 - 72r * 1 for values of r
in the range -3 to +5?
Ioj
f F l
Sat +
*t
tv\NY
1""^^ ;
fuL aR
1^
lL >1)
M = 3z',)*
'7-t= 4
l-
f
c
(_
-/
6x
Ti,
t
rr\--
i
d'1 -t''r
d^1
)L:2/
1-
=
6t-.
>
/
O
a
'J^-
zq+I
a' L
{lfibA\/tttt+,
8 -24+t
-t d
:===-
ot'r
!- -z
fu*t;,tt*'
6*,
o
L +2{+ + | =-8+24+ l
-!
(A= J
: . % \1
.+
drt"h. o,^rt g*kt
4ry
- 21 + 3 6 + l
tf-1)=
= nq:6 o +
3(6)
{tv'+X
=
{a^L
rJ
;>
//
CJtl)
sJ
-l(
= tc
| = 66.
Part B: Physics [fO Marks]
Multiple choice (10 marks).
Please circle one answer to each question only.
12. A radioactive sourceis placed 6cm from a radiation detector sensitiveto 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)
Y
.fpfta only
alpha and beta
B
D
beta only
alpha and gamma
lr l
t 'J
13. An aeroplanehas two engines,one under each wing. Supposethe left
engine stops working. If the aeroplane'scontrols are not changedthen the
aeroplane will
A
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
dJ=
+
111
L-t
14. A sky-diver jumps out of an aeroplane. Which of the following statements
is true after strereachesterminal velocity?
( A ) The force of air resistanceis the same size as her weight.
Y
fft" force of air resistance is larger than her weight.
C The force of air resistance is 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?
Gl
lY
I l2 yourheight
314yourheight
I
--]_.
i\
B
D
7f 4 yow height
your full height
11l
L^t
16. A metal disc with a hole in it is heated and expands bv I%. The size of the
hole has
CA) iocreased
Y
C
D
.1""r"r.".1
stayed the same
it depends on where the hole is
lll
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 achievethe desired temperature?
A
@
1 0 ke
B
D
rzr.s
15kg
20kg
[1]
18. A stone is thrown straight upwards. At the top of its path its velocity is
momentarlly zero. !\rhat is the magnitude of its acceleration at this point?
A
B
D
0ms-2
Q) ro-.-'
5 ms -2
20ms*2
[1]
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 observesa solar eciipse; at the same time the
earth experiencesa
A
C
solar eclipse
both
I
*nrecripse
li l
t--l
2 1 . If you run at 5 ms-i towards a plane mirror, at what speed does your image
approach you?
A
/^\
5 ms-1
I C ) 1 0 ms-'
B
D
2 . 5 ms * 1
1 5 ms -]
[1]
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.
B
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 positionof this image.
Irl
Mrrtr X
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 this
image to be seen.
l2l
23. Supposethat 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]
Ma-
Clu,rorq0
0
il=o
@
3: +h + q
o
lr -- 2s
-s + <
:W
- 2a
Ms+ lvle= 3'vt"
:. fi'lh=3m.
Ms= Z'na,
i^ t ( l*h" (f
5+e
6{0
l{s* fl^ * Me- () wl<.
E
T
-2s ---3%
g ; 3v
z
h _ _ _ 34(
1t =
-ZA, "t-s
r
= _cl/
4"
z
24. A black box has three electrical connectionslabelled A, B, and C, arranged
in a triangle as shown below.
A
h
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
(a) On the diagram sketch the arrangement of the three components in
the box indicating the terminals clearly.
l4l
'.21
(b) Caiculate the resistance of the resistor.
K-- Y:9
T
= Sxto-'' 3K.5L'
3xc;-->
(c) What would happen if the battery were collnected with - to A and *
to B?
WcuJAA,k ^ u/,r-*{- +
[1]
3^ A a?f'-*'
(d) What would happen if the battery were connected with - to C and *
to A?
frl
Long question (20 marks)
25. Consider an elevator car (lift) with a mass of 700k9 which call carry up to
600kg of passengers.Use 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)
nnah= hn
E = ".-.\'
\'-
'= ll-tar>ef
+iar) / xlc x9 = lasaoqo
:
= F -- t i ? c r r )
flcr-strC
.
l3 o o x3
E
3Aoo W
3A
(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 in the car to balance the svstem?
t2l
fl.^lit\.gf6"
4*vJ'*i..*.;J
il"a.tg
lift
d\M155MV
LT = M r 3
&ul'*
z
f=
M ;+ M p
fif
1cc + Mp
Mi,,u-- ?6W(5
I6pO=
-7tJa
{- Mp
Mp-- 3ocuE
MP.
(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.
t4l
r,
t
--t
rtwa"6 {*r( t
il,l,dw:r t4 *1 V
Ca1rfq;g4
|
=
=
3
to
i"
r / r s -(' k -)
= 3
c''o\ :e lo'a^ otl't1r"1f@
u
lKEo.,r,*uDd,,{rLr*?,rsoo
" e;
yy15-l
20
4rY
Lr, t'soo" €r)' =
KEr^*r*'
. . , 7-
/-M\V
/
KE=
q
I l=:
eNt@t,-v
-- q '/-d T
z
Ght KE. = 61 q T
='-
(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)
v
V*ux= a|
O\f
3
,1* L= .3o:.
-->
r
r
t?
q4G - ) - ( ^t * lbc \ k c-
S,= t ^k)'
''zFl
S1 = *
"33
i_
9 a
s?=
af 5 -L^ b)'
5
=:'z
[ 3/
-
V*ox=
t
CR.F =
.-, ,
5tJ,,t-',
J-
l/
v
\
= 1-*'
q
=: --l:-o+cM
s*t
ry
-??
?xro
7x3o
))
1-
?
0\ --
4-
S1?r-- 5,+ \t
4L
Lrv
3',<Ot*C
14--
3o
(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
describedin part (d). Calculate (i) the reading on the scaleswhen the
lift is stationary, (ii) the change in 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]
lqJ =
il
I
,44: d8q
a = lAAt-'
n4\
J
A1
: 68rie'
\H,i
ii6)
W= nrr(t*^)
/
"r*U*f"- ,
I cJil*ke;
6? x o'a16
6Eo +
\
tr
L,' .1
b\
(+'r"
+q{69' \06o
t,..--
6 N x o Dh-f
----
L1 Z C
3qO
= 3 .oe,
689 -ob NJ
w=
366 0
44t^
ust'"*l-
y 3AJ
-
rri)
,*
S{AJ,^ wo'rlu;^^'^ tF 'tl ,
tnJ ,- 6JO /lJ
)a442
04
@ et\"h" ucolehh'r
oTjffir*)
,1,.tu,
s<h.
prd-4^-a^F