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 2013
Marking Scheme
Applied Mathematics
Higher Level
Note to teachers and students on the use of published marking schemes
Marking schemes published by the State Examinations Commission are not intended to be standalone documents. They are an essential resource for examiners who receive training in the correct interpretation and application of the scheme. This training involves, among other things, marking samples of student work and discussing the marks awarded, so as to clarify the correct application of the scheme. The work of examiners is subsequently monitored by Advising Examiners to ensure consistent and accurate application of the marking scheme. This process is overseen by the Chief Examiner, usually assisted by a Chief Advising Examiner. The Chief Examiner is the final authority regarding whether or not the marking scheme has been correctly applied to any piece of candidate work. Marking schemes are working documents. While a draft marking scheme is prepared in advance of the examination, the scheme is not finalised until examiners have applied it to candidates’ work and the feedback from all examiners has been collated and considered in light of the full range of responses of candidates, the overall level of difficulty of the examination and the need to maintain consistency in standards from year to year. This published document contains the finalised scheme, as it was applied to all candidates’ work.
In the case of marking schemes that include model solutions or answers, it should be noted that these are not intended to be exhaustive. Variations and alternatives may also be acceptable. Examiners must consider all answers on their merits, and will have consulted with their Advising Examiners when in doubt.
Future Marking Schemes
Assumptions about future marking schemes on the basis of past schemes should be avoided. While the underlying assessment principles remain the same, the details of the marking of a particular type of question may change in the context of the contribution of that question to the overall examination in a given year. The Chief Examiner in any given year has the responsibility to determine how best to ensure the fair and accurate assessment of candidates’ work and to ensure consistency in the standard of the assessment from year to year. Accordingly, aspects of the structure, detail and application of the marking scheme for a particular examination are subject to change from one year to the next without notice.
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 1
Page 2
1. (a) A ball is thrown vertically upwards with a speed of 44·1 .s m 1
Calculate the time interval between the instants that the ball is 39·2 m above the point of projection.
s 718
8 ,1
081
089
8·91·449·23
1
2
221
221
t
tt
tt
tt
tt
atuts
5 5,5 5
20
Page 3
1. (b) A lift ascends from rest with constant acceleration f until it reaches a speed v. It continues at this speed for 1t seconds and then decelerates uniformly to rest with deceleration f.
The total distance ascended is d, and the total time taken is t seconds.
(i) Draw a speed-time graph for the motion of the lift.
(ii) Show that 121 ttfv .
(iii) Show that fdtt 42
1 .
(i)
fdtt
ttfd
tt fttd
vttt
vttdvttvtvttd
tt fvtt
vf
4
4
or (iii)
(ii)
21
21
2
121
121
1121
21
121
141
1141
121
121
5 5
5 5
5 5
30
121 tt
121 tt
1t
v
Page 4
2. (a) Two cars, A and B, travel along two straight roads which intersect at an angle . Car A is moving towards the intersection at a uniform speed of 9 1s m . Car B is moving towards the intersection at a uniform speed of 15 1s m .
At a certain instant each car is 90 m from the intersection and approaching the intersection. (i) Find the distance between the cars when B is at the intersection. (ii) If the shortest distance between the cars is 36 m, find the value of .
.13·53.159cos
0cos159
sin15 cos159
sin15 5cos1
9 (ii)
m 361590990 (i)
1
ABV
jiVVV
jiV
iV
AB
AB
BAAB
B
A
5 5 5 5 5 5
25
B
Blank Page
A
Page 5
2 (b) An aircraft P, flying at 600 ,h km 1 sets out
to intercept a second aircraft Q, which is a distance away in a direction west 30 south, and flying due east at 600 .h km 1
Find the direction in which P should fly in order to intercept Q.
W30 Sor S 60 W 6021cos
2cos2cos401cos2coscos131cos2cosin3
1cossin3
600cos600
sin60030tan
sin600 600cos600
600
sin600 cos600
2
22
22
s
ji
VVV
iV
jiV
QPPQ
Q
P
5 5 5 5 5 25
P 30
600
Q
600
Page 6
3. (a) A particle is projected from a point on horizontal ground.
The speed of projection is u 1s m at an angle to the horizontal. The range of the particle is R and the maximum height reached by the particle is
.34
R
(i) Show that g
uR cossin2 2
.
(ii) Find the value of .
gu
guu
tuR
gut
gttu
cossin2
sin2cos
.cos
sin20.sin (i)
2
221
303
1tan
sinsin32
cos
sin2
sinsin34cossin2
.sin34
sin (ii)
21
22
212
11
1
gug
guu
gu
gttuRg
ut
5 5
5 5 5 5
25
Page 7
3 (b) A plane is inclined at an angle 21tan 1 to the horizontal.
A particle is projected up the plane with initial speed u 1s m at an angle to the inclined plane. The plane of projection is vertical and contains the line of greatest slope.
Find the value of that will give a maximum range up the inclined plane.
7·31
22tan 2sin2cos2
0cossin22cos2 0
cossin22cos22
5
sin2sin2
5
sinsincos5
sincos
sin5cossin2
0 cossin
0
2
22
221
2
221
221
ddR
gu
ddR
gu
gu
tgtuR
gu
gut
tgt u
rj
5 5 5 5 5 25
Page 8
4. (a) Two particles of masses 6 kg and 7 kg are connected by a light inextensible string passing over a smooth light fixed pulley which is fixed to the ceiling of a lift. The particles are released from rest. Find the tension in the string
(i) when the lift remains at rest (ii) when the lift is rising vertically with constant
acceleration .8g
71·24or 26
1898
666
1049
8
66
8
77 (ii)
63·32or 13
8466
13
66 77 (i)
gT
gfgT
gf
gf g T
gf T g
gT
fgT
gf
f g Tf T g
5
5 5 5 5
25
6 7 6
Page 9
4 (b) A light inextensible string passes over a smooth
fixed pulley, under a movable smooth pulley of mass 3m , and then over a second smooth fixed pulley. A particle of mass 1m is attached to one end of the string and a particle of mass 2m is attached to the other end. The system is released from rest.
Find the tension in the string in terms of . and , 321 mmm
321
213231
321
213231321
321313221321
21
33
21
33
22
11
4114or
44
44
242
22
2
mmm
g
mmmmmmgmmmT
mmmmmmTgmmm
gmmmTmmTmmTmmgmmm
gmTg
mTmTgm
qpmTgmq mg mTp mg mT
5
5
5
5 5
25
1m 3m 2m
Page 10
5. (a) A smooth sphere A, of mass 3m, moving with speed u, collides directly with a smooth sphere B, of mass 5m, which is at rest.
The coefficient of restitution for the collision is e. Find (i) the speed, in terms of u and e, of each sphere after the collision (ii) the value of e if the magnitude of the impulse imparted to each sphere as a
result of the collision is 2mu.
151
11516
18
152
05
183
538
0 NEL
53(0)53 PCM
2
2
1
21
21
e
emu mu
emu mu
mv I
eu v
eu v
u e v vmvmv m um
20 5
5
5
5 5
Page 11
5 (b) A ball is dropped on to a table and it rises after impact to one-quarter of the height of the fall.
(i) Find the value of the coefficient of restitution between the ball and the table.
If sheets of paper are placed on the table the coefficient of restitution decreases by a factor proportional to the thickness of the paper. When the thickness of the paper is 2·5 cm it rises to only one-ninth of the height of the fall. (ii) Find the value of the coefficient of restitution between the ball and this
thickness of paper. (iii) What thickness of paper is required in order that the rebound will be one-
sixteenth of the height of the fall?
cm. 3·33or 3
10
416·0
21
6·0
315·2
215·2
41
16220
2 (iii)
31
9220
2 (ii)
21
4220
2
2
20
2 (i)
2
1
2
22
22
22
1
12
11
22
2
22
2
22
x
xexke
k
keke
e
hgghe
asuv
e
hgghe
asuv
e
hgghe
asuv
ghv
ghv
asuv
5 5 5 5 5 5
30 5
Page 12
6. (a) A rectangular block of wood of mass 20 kg and height 2 m floats in a liquid. The block experiences an upward force of 400d N, where d is the depth, in metres, of the bottom of the block below the surface. Find
(i) value of d when the block is in equilibrium (ii) the period of the motion of the block if it is pushed down 0·3 m from the
equilibrium position and then released.
s. 4·1or 5
5222
52or 20
20
40040020 )(
49·0
20400 )(
T
xmFa
xxdg Fii
d
gd i
5 5 5 5
20 5
2 m
d 2 m
d+x
400 d
20 g
400 (d+x)
20 g
Page 13
6 (b) A vertical rod BA, of length 4l, has one end B fixed to a horizontal surface with the other end A vertically above B. The ends of a light inextensible string, of length 4l, are fixed to A and to a point C, a distance 2l below A on the rod. A small mass m kg is tied to the mid-point of the string. It rotates, with both parts of the string taut, in a horizontal circle with uniform angular velocity . (i) Find the tension in each part of the string in terms of m, l and . (ii) At a given instant both parts of the string are cut.
Find the time (in terms of l ) which elapses before the mass strikes the horizontal surface.
gt
gt
atut s
gmT
gmT
mTT
m
mrTT
mgTT
mgTT
6
03
(ii)
2
3
60sin60sin
2
60cos60cos
60 (i)
221
221
22
21
221
2
221
21
21
30 5
5 5 5
5
5 5 5
4l
A
B
C
4l
A
B
C
α T1
T2 mg
Page 14
7. (a) Two forces 5 N and 12 N are inclined at an angle as shown in the diagram.
They are balanced by a force of 15 N.
Find the acute angle .
18·62
82·117
4667·012056cos
cos12025144225
cos512251215 222
20 5
5, 5 5 5 5
15 N 12 N
5 N
15
12
5
Page 15
7. (b) Two uniform rods AB and BC, of length 1 and weight W, are hinged at B and rest in equilibrium on a smooth horizontal plane.
A weight W is attached to AB at a distance b from A as shown in the diagram. A light inextensible string AC of length 2q prevents the rods from slipping.
(i) Find the reaction at A and the reaction at C.
(ii) Show that the tension in the string is 2
1
2 1
q b W
q
.
2
2
2
21
2
1
21
2
23
21
2
23
21
2
121
12
1
122
2
sin (ii)
24
3
22
2
coscos2 (i)
qqbWT
qTqbW
qTWqqbW
TqWqR
bWR
WRR
bWR
bqWqWqWqR
qbWqWqWq R
5 5 5 5 5 5 5
30 5
A
B
C 2q
W b
W W
R1 R2
α
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
mr
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
5 5 5 5
20 5
Page 17
8. (b) A uniform circular lamina, of mass 8m and radius r, can turn freely about a horizontal axis through P perpendicular to the plane of the lamina.
Particles each of mass m are fixed at four points which are on the circumference of the lamina and which are the vertices of square PQRS. The compound body is set in motion.
Find (i) the period of small oscillations of the compound pendulum
(ii) the length of the equivalent simple pendulum.
35
3522 (ii)
352
12202
2
20
22288
1228 (i)
2
2
2222221
rL
gr
gL
gr
mgrmr
MghIT
mr
rmrmrmrmrmI
mgrrmgrmgrmgrmgMgh
30 5
5,5 5,5 5 5 5
P Q
R S
P
Q
R
S
8mg mg
mg
mg
mg
Page 18
9. (a) 1V cm3 of liquid A of relative density 0·8 is mixed with 2V cm3 of liquid B of relative density 0·9 to form a mixture of relative density 0·88.
The mass of the mixture is 0·44 kg.
Find the value of 1V and the value of .2V
332
331
11
21
12
12
2121
cm 400or m 0004·0
cm 100or m 0001·0
44·04880
44·0880
4
8020
880900800
V
V
VV
VV
VV
VV
VVVV
mmm MBA
5 5 5 5 5
25 5
Page 19
9 (b) Liquid C of relative density 0·8 rests on liquid D of relative density 1·2 without mixing. A solid object of density floats with part of its volume in liquid D and
the remainder in liquid C.
The fraction of the volume of the object immersed in liquid D is a
a2 .
Find the value of .a
400
400800
8001200800
1800
12001
1
1
1200800
1200008
2
121
2
21
2121
a
VVVV
V
VV
gVgVgVV
BBW DC
25 5
5 5, 5 5 5 5
Page 20
10. (a) If
072 dxdyx
and 1y when 7,x find the value of y when 14x .
(b) A particle starts from rest at O at time .0t It travels along a straight line with acceleration 224 16 m st , where t is the time measured from the instant when the particle is at O. Find
(i) its velocity and its distance from O at time 3t
(ii) the value of t when the speed of the particle is 80 .s m 1
(c) Water flows from a tank at a rate proportional to the volume of water remaining in
the tank. The tank is initially full and after one hour it is half full.
After how many more minutes will it be one-fifth full?
5·1
11417
71
14171
1 7
7
7
7 (a)
14
7
1
14
7
2
1
2
2
y
y
y
xy
dxx dy
dxxdy
dxdyx
y
y
5 5 5
5
10 5
Page 21
t
tt
tt
tt
ttvb
s
tts
dtt ds
v
ttv
dtt dv
t dtdvb
s
v
s3
10
02103
02043
161280
1612 (ii) )(
m 36
84
16t)-2(1
s m 60
1612
16)-24(
1624 (i) )(
2
2
2
30
23
3
0
2
0
1
30
2
3
0
0
min 3·791
h 322·22ln5ln
ln5
ln
693·0or 2ln
ln2
ln
1
)(
1
1
0
V
21
tt
t
k tVV
k
kVV
dtk dV V
kV dtdVc
V
5 5 5 5
20 5
5 5 5 5 5 5
20 5
Page 22
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