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Coimisiún na Scrúduithe Stáit State Examinations Commission
Leaving Certificate 2012
Marking Scheme
Higher Level
Design and Communication Graphics
Coimisiún na Scrúduithe StáitState Examinations Commission
Leaving Certificate 2012
Marking Scheme
Applied Mathematics
Higher Level
General Guidelines
1 Penalties of three types are applied to candidates' work as follows: Slips - numerical slips S(–1) Blunders - mathematical errors B(–3) Misreading - if not serious M(–1) Serious blunder or omission or misreading which oversimplifies: - award the attempt mark only. Attempt marks are awarded as follows: 5 (att 2). 2 The marking scheme shows one correct solution to each question.
In many cases there are other equally valid methods.
Page 2
1. (a) A particle falls from rest from a point P. When it has fallen a distance 19·6 m a second particle is projected vertically downwards from P with initial velocity 39·2 1m s− . The particles collide at a distance d from P. Find the value of d.
( )
( )
( )
212
212
212
212
2 21 12 2
2 2
212
19·6 02 s
0 2
39·2
2 39·2
4 4 8
1
0 24·9 944·1 m
s ut ft
gt t
d g t
d t gt
g t t gt
t t t t
t
d g t
= +
= +=
= + +
= +
+ = +
+ + = +
=
= + += ×=
5 5 5 5 5
25
Page 3
1. (b) A car, starts from rest at A, and accelerates uniformly at 1 2s m − along a straight level road towards B, where AB = 1914 m. When the car reaches its
maximum speed of 32 1s m − , it continues at this speed for the rest of the journey.
At the same time as the car starts from A a bus passes B travelling towards A with a constant speed of 36 1s m − . Twelve seconds later the bus starts to decelerate uniformly at 0·75 2s m − .
(i) The car and the bus meet after t seconds. Find the value of t.
(ii) Find the distance between the car and the bus after 48 seconds.
( ) ( )
( ){ } ( ) ( )( ){ }
s 40 1984061630
1212364323232512
1914
432 01236
bus
512 3212023 1032
2 car (i)
2
243
21
221
2
22
=+−=
−−+−++−+=
+=
=+×=
+=
==+=+=
+=+=
ttt
ttt
ss
s
ftuts
stst
fsuvftuv
bc
( ) ( ) ( )( )
212
212
3 314 2 4
(ii) car 32 8 0
256
bus
36 40 12 15 8 64
s ut ft
v u ft s ut ft
= += × +=
= + = +
= − − = + − 15 96
distance 256 96 352 m
v s= =
= + =
5, 5
5
5
5
25
Page 4
2. (a) Rain is falling with a speed of 25 1s m − at an angle of °20 to the vertical. A car is travelling along a horizontal road into the rain. The windscreen of the car makes an angle of °32 with the vertical. The car is travelling at 20 1s m −
. Find the angle at which the rain appears to strike the windscreen.
( )
1
25sin20 25cos 20
20 0
25sin20 20 25cos 20
28·55 23·49
28·55tan 3223·49
82·55 .
r
c
rc r c
V i j
V i j
V V Vi j
i j
α −
= −
= − +
= −
= + −
= −
= +
= °
5
5 5 5 20
25 1s m −
Page 5
2 (b) At noon ship A is 50 km north of ship B. Ship A is travelling southwest at 1h km 224 − . Ship B is travelling due west at 1h km 71 − . (i) Find the magnitude and direction of the velocity of B relative to A.
A and B can exchange signals when they are not more than 20 km apart. (ii) At what time can they begin to exchange signals? (iii) How long can they continue to exchange signals?
( ) ( )
==
+=
−−−+−=
−=
−
247tan 25
24 7 24 24 0 71
(i)
1θBA
ABBA
V
jijiji
VVV
2 2
7 (ii) 50sin 50 1425
20 14 2 51 1·2829
50cos 14·28292450 14·2829 33·717125
1·348725
time 13 : 21
(iii) 2 28·5658
1·14225
BC
CE
AD
CD
BC
BCt
CE CD
CEt
θ
θ
= = × =
= − = =
= −
= × − =
= =
=
= × =
= = 6
time 1 h 9 min=
5
5
5 5 5
5
θ B
A
50 E
D
C
30
Page 6
3. (a) A particle is projected with a speed of 98 1m s− at an angle α to the horizontal. The range of the particle is 940·8 m. Find
(i) the two values of α
(ii) the difference between the two times of flight.
2
(i) 98cos . 940·89·6
cos
98sin . 4·9 09·698sin 4·9 0
cos
sin 2 0·962 73·74 , 106·26
36·87 , 53·13
t
t
t t
α
α
α
αα
αα
α
=
=
− =
− =
==
= ° °
1
2
2 1
9.6(ii) cos3·8712·00
9·6 cos53·1316·00
4 s.
t
t
t t
=
=
=
=
− =
5 5
5 5 20
Page 7
Q α
20 3 (b) A particle is projected down an inclined plane with initial velocity 20 1s m −
. The line of projection makes an angle of 90° with the inclined plane and the plane is inclined at α to the horizontal. The plane of projection is vertical and contains the line of greatest slope. The particle strikes the plane at Q. The range of the
particle on the inclined plane is 1600 3 .
g Find
(i) the value of α
(ii) the magnitude of the rebound velocity at Q if the coefficient of
restitution is 1 .2
( ) ( ) s m 70
10340
10 340 (ii)
20802120cos20
3408023sin0
60 cos 32in
31600 cos
1600sin
31600 sin0
31600
cos40 0 cos20
0 (i)
1
22
2
2221
221
221
−=
+=
==
−=××−=×−=
=××=×+=
°==
=×
=×+×
=
=
=×−
=
v
v
v v
ggtg v
ggtg v
s
g
gg
g tgt
gr
gt
tgt
r
j
i
j
i
i
j
α
α
ααα
αα
α
α
α
5 5 5
5 5
5 30
Page 8
4. (a) Two particles A and B each of mass m are connected by a light inextensible string passing over a light, smooth, fixed pulley.
Particle A rests on a rough plane inclined
at α to the horizontal, where 5tan .12
α =
Particle B hangs vertically 1 m above the ground. The coefficient of friction between
A and the inclined plane is 1 .2
The system is released from rest.
(i) Find the speed with which B strikes the ground.
(ii) How far will A travel after B strikes the ground?
12
2 2
2 2
(i) sin cos
5 6 213 13
13
2
0 2 113
213
(ii) 220
mg T mfT mg mg mf
mg mgmg mf
gf
v u fsg
gv
v u fsg
α α− =
− − =
− − =
=
= +
= + × ×
=
= +
= 112 13 131 m.11
g s
s
− + × ×
=
5
5 5
5 5
25
A 1 m
B
α
mg μR
T T
mgcosα
R
mgsinα
Page 9
4 (b) Two particles of mass m kg and 2m kg lie
at rest on horizontal rough tables. The coefficient of friction between each
particle and the table it lies on is 2 .3
μ μ <
The particles are connected by a light inextensible string which passes under a smooth movable pulley of mass 4m kg.
The system is released from rest.
(i) Find, in terms of m and ,μ the tension in the string.
(ii) If the acceleration of the m kg mass is f, find the acceleration of the 2m kg mass in terms of f.
( )( )
( )
( )
( )( )
12
(i)
2 2
4 2 42 2
4 2 2 2 24 1
5
(ii)
4 1
545 5
2
T mg mf
T mg mp
mg T m f pmf mp
mg T T mg T mgmg
T
mf T mg
mgmg
g gf
mp
μμ
μ μμ
μμ
μ
μ
− =
− =
− = × += +
− = − + −+
=
= −
+= −
= −
= ( )( )
2
4 12 2
52 35 5
3 2
T mg
mgmp mg
g gp
p f g
μμ
μ
μ
−
+= −
= −
= −
5
5
5
5 5
25
m
4m
2m
Page 10
5. (a) Three smooth spheres, A, B and C, of mass 3m, 2m and m lie at rest on a smooth horizontal table with their centres in a straight line. Sphere A is projected towards B with speed 5 1m s .− Sphere A collides directly with B and then B collides directly with C. The coefficient of restitution between the spheres is e.
Show that if 2
53 −>e there will be no further collisions.
( )
( )
( )
1 2
1 2
1
2
2 3 4
PCM 3 5 2 (0) 3 2
NEL 5 0
3 2 3 3
PCM 2 (0) 2
NEL
m m mv mv
v v e
v e v e
m v m mv mv
+ = +
− = − −
= −= +
+ = +
( )
( ) ( )( )
( ) ( )( )
( )( )
3 4 2
23
24
1 3 4
1 3
2
2
1 3
0
2 1 232 1 2 1 13
if 3 2 1 2
3 2 2
3 5 3 53 1 0 2 2
3 5 if 2
v v e v
v v e e e
v v e e e
v v v
v ve e e
- e e e
e e e
v v e
− = − −
= − = + −
= + = + +
< <
<− < + −
< + −
− +− + < < <
− < >
25
5
5
5 5 5
Page 11
5 (b) A smooth sphere P collides with an identical smooth sphere Q which is at rest. The velocity of P before impact makes an angle α with the line of centres at impact,
where 0 90 .α° ≤ < ° The velocity of P is deflected through an angle θ by the collision.
The coefficient of restitution between the spheres is 1 .3
Show that 2
2 tantan .1 3 tan
αθα
=+
( ) ( )
( )
( )
tan31tan2tan
tantan3tan3tantan
tan3
cossin3
tantan1tantan
sintan
3
cos
0cos31 NEL
0cos PCM
2
2
1
1
21
21
ααθ
θααθα
α
αα
θαθα
αθα
α
α
α
+=
−=+
=
=−
+
=+
=
−−=−
+=+
uuv
u
uv
u v v
mv mv m um
5
5 5 5 5
25
Q P θ
Page 12
6. (a) A particle of mass 0·5 kg is suspended from a fixed point P by a spring which executes simple harmonic motion with amplitude 0·2 m. The period of the motion is 2 seconds.
Find (i) the maximum acceleration of the particle
(ii) the greatest force exerted by the spring correct to one place of
decimals.
2
2
2
2
2
2( ) 2
0·2
5
( )
0·55
2 105·9 N
i
a A
ii F m a
T mg
gT
πω
ω π
ωππ
π
π
=
=
== ×
=
= ×
− = ×
= +
=
5 5 5 5
20
Page 13
6 (b) A particle of mass m kg lies on the top of a smooth fixed sphere of radius 30 cm.
The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B.
(i) Find the speed of the particle at B.
(ii) The horizontal distance, in metres, of the particle from the centre of the
sphere t seconds after it has left the surface of the sphere is 5 .10
kt+
Find the value of k correct to two places of decimals.
( )
( )
( )( )
212
2
2
2
2
1
(i) 0·3 0·3cos
0·6 1 cos
cos0·3
0 0·3 cos
0·3 cos 0·6 1 cos
cos 2 1 cos2cos3
0·3 cos 0·21·4 m s
(ii) 0·3sin 1·4cos
5 20·3 1·43 3
mv mg
v g
mvmg R
R v g
g g
v g gv
x t
α
α
α
α
α αα α
α
α
α α
−
= −
= −
− =
= =
= −
= −
=
= = =
= + ×
= × + ×
5 2·810 3
14 0·9315
t
t
k
×
= + ×
= =
30
5 5 5 5
5 5
B
Page 14
7. (a) A uniform wire ABC is bent at right angles at B. When it is suspended from B the parts AB and BC make angles of 30 and 60 respectively with the vertical. If the mass per unit length of the wire is m and BChAB = find the value of h.
12
12
2
sin 30
sin 60
1 32 2
3
1·316
mgh BC h BC
mg BC BC
h
h
× =
×
× =
=
=
20
5, 5 5
5
60 30
A
B C
Page 15
7. (b) Two rough rings of equal weight W are
a distance d apart on a horizontal rod. A light smooth inelastic string of length 2 connects the rings. Another ring of weight 2W slides on the string. The coefficient of friction between the rough rings and the rod is .μ
Show that the system remains at rest if 2
4 .1 4
d μμ
<+
( )
( )
1
22
1 1
22
22
2 2 2 2 2
2 2 2 2
2
2
2 cos 2cos
cos2
2 2 4
2 22 2 4
44
16 4
1 4 16
41 44
1 4
T WT W
R W TW
d d dR W R
d d dW W W
dd
d d
d
d
d
θθ
θ
μ
μ
μ
μ μμ μ
μμ
μμ
==
= +=
= + −
= + −
= −
= −
+ =
=+
<+
5 5
5, 5 5 5
30
d
μR1
W
d
μR2
2W
W
R1 R2
θ
θθ
T
T
T T
Page 16
8. (a) Prove that the moment of inertia of a uniform circular disc, of mass m and
radius r, about an axis through its centre perpendicular to its plane is 21 .2
m r
{ }
{ }
221
4
0
4
0
3
2
2 M
4 M 2
d M 2 disc theof inertia ofmoment
d 2M element theof inertia ofmoment
d 2M element of mass
areaunit per mass MLet
rm
r
x
xx
xxx
xx
r
r
=
=
=
=
=
=
=
π
π
π
π
π
5 5 5 5
20
Page 17
8. (b) A string is wrapped around a smooth pulley wheel of radius r. A particle of mass m is attached to the string.
The axis of rotation of the wheel is horizontal, perpendicular to the wheel, and passes through the centre of the wheel.
The moment of inertia of the wheel about the axis is I. The particle is released from rest and moves vertically downwards. (i) Find, in terms of I, m and r, the tension in the string.
(ii) If the acceleration of the particle is ,5g find the mass of the pulley
wheel in terms of m.
( )
mMMrmr
mrImr
gmrI
mgr f
mrImgIT
mrImrmg
mfmgTmfTmg
mrImgr
mrImgf
mgmrIf
mghfhmrfhI
fhfsuv
mghmvrvI
mghmvI
845
5 (ii)
1
22
202
(i)
2212
22
2
2
2
2
2
2
2
2
2
21
221
22
221
2
2
21
2212
21
=
=
+=
=+
=
+=
+−=
−==−
+=
+=
=
+
=+
+=+=
=+
=+ω
30
5 5 5 5 5 5
m
Page 18
9. (a) Stainless steel is an alloy of iron, chromium and nickel. A piece of stainless steel consists of 70% iron, 20% chromium and 10% nickel by volume.
The relative densities of iron, chromium and nickel are 7·8, 7·2 and 8·9
respectively.
Find the relative density of stainless steel.
7800 0·7 7200 0·2 8900 0·1
7790
7·79
I C N SSm m m m
V V V V
s
ρ
ρ
+ + =
× + × + × =
=
=
5 5, 5 5
20
Page 19
9 (b) A uniform rod, of length 2 m and weight W, is freely hinged at a point P. The rod of relative density 0·756 is free to move about a horizontal axis through P. The other end of the rod is immersed in a liquid of relative density 0·9. The point P is 0·4 m above the surface of the liquid. The rod is in equilibrium and is inclined at an angle of θ to the vertical. Find (i) the length of the immersed part of the rod (ii) the value of .θ
2
2
2
0·9(i) 0.756
1sin 2 sin2
0·9 20·756 2
1·68 22
4 3·36 0
1·2 m.
(ii)
x
x
W B
xW B
W xW
xx
x x
x
θ θ
×=
× = × −
× = −
= −
− + =
=
0·4 1 cos 2 1·2 2
60
θ
θ
= =−
= °
30
5 5, 5 5 5 5
P
θ 0·4 m
P
θ 0·4 m B
W x
Page 20
10. (a) Newton’s law of cooling states that ‘the rate of cooling of a body is proportional to the difference between the temperature of a body and the temperature of its surroundings.’ If θ is the difference between the temperature of a body and the temperature of its surroundings then
.d kdtθ θ= −
A body cools from 80 C to 60 C in 10 minutes. The temperature of the surroundings is maintained at 20 C. Find (i) the value of k
(ii) the temperature of the body after a further 15 minutes.
[ ] [ ]
40 10
60 0
40 10
60 0
(i)
1
1
ln
ln 40 ln 60 106010 ln40
1 ln1·5 0·0405
10
(ii)
d kdt
d k dt
d k dt
kt
k
k
k
θ θ
θθ
θθ
θ
= −
= −
= −
= −
− = −
=
= =
[ ] [ ] 20 15
40 0
0·6075
ln
20ln 15 40
2040
20 21·7884
41·8
θ kt
k
e
θθ
θ
θ
θ
−
−
= −
− = −
− =
− =
= °
5
5 5 5
5
25
Page 21
10 (b) A particle of mass m is fired horizontally through a block of resistive gel. The resistance to motion is mkv2 N when v m s–1 is the speed. The particle enters
the gel at a speed of 1000 m s–1 and 1100
seconds later exits the gel at a speed
of 10 m s–1.
(i) Show that k = 99 .10
(ii) Use the fact that 1 ln(9900 1)
9900 1 9900dt t C
t= + +
+ or otherwise to
show that the length of the block of gel is 10 ln100 m.99
[ ]
2
10 0·01
2 1000 0
10 0·1
0 1000
( )
1
1
1 1 0·0110 1000
9910
( )
dvi kvdt
dv k dtv
ktv
k
k
ii
= −
= −
− = − − + = − ×
=
[ ]
2
2
10
1000 0
10
1000 0
9910
1 9910
99ln 10
10 99ln 1000 10
10 ln10099
x
x
dvv kvdx
v
dv dxv
xv
x
x
= −
= −
= −
= −
= −
=
5 5 5 5
5
25
Page 22
Page 23
Page 24