Coimisiún na Scrúduithe Stáit State Examinations Commission
LEAVING CERTIFICATE 2010
MARKING SCHEME
APPLIED MATHEMATICS
HIGHER LEVEL
Page 1
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 car is travelling at a uniform speed of 14 1s m when the driver notices a traffic light turning red 98 m ahead. Find the minimum constant deceleration required to stop the car at the traffic light,
(i) if the driver immediately applies the brake
(ii) if the driver hesitates for 1 second before applying the brake.
2
2
2
22
221
2
2
22
s m 17.1or 6
7168
196
168140
14982140
2
14
0114
(ii)
s m 1
196196
982140
2 (i)
f
f
f
fs uv
s
s
ftuts
f
f
f
fs uv
5 5
5 5 20
Page 3
1. (b) A particle passes P with speed 20 1s m and moves in a straight line to Q with uniform acceleration.
In the first second of its motion after passing P it travels 25 m.
In the last 3 seconds of its motion before reaching Q it travels 20
13 of PQ .
Find the distance from P to Q.
m 300
65620
2
062018065
25305160605
25305
160605
45420
25305
15120
10
5
112025
2
2
22207
2207
2
2
221
2
2
207
221
21
2
21
221
PQ
t
tt
tttt
ttPQ
tt
ttPQ
ftutsPQ
tt
ttPQ
ftutsPY
f
f
f
ftutsPX
5
5
5
5
5
5 30
P Q YX
t=1 s=25
t=3 s= PQ20
13
Page 4
2. (a) Two particles, A and B, start initially from points with position vectors
ji
14 6 and ji
2 3 respectively. The velocities of A and B are constant
and equal to ji
3 4 and ji
7 5 respectively.
(i) Find the velocity of B relative to A.
(ii) Show that the particles collide.
South 75.58East directionor
4 slope
4 3or 12 3
2 3
14 6 (ii)
South 75.58East directionor
4 slopeor s m 17 magnitude
4
7 5
3 4 (i)
1
jiji
RRR
jiR
jiR
ji
VVV
jiV
jiV
BAAB
B
A
ABBA
B
A
The particles collide
5
5 5
5 20
Page 5
2 (b) When a motor-cyclist travels along a straight road from South to North at a constant speed of 12.5 1s m the wind appears to her to come from a direction North 45 East.
When she returns along the same road at the same constant speed, the wind appears to come from a direction South 45 East.
Find the magnitude and direction of the velocity of the wind.
Westdirection
s m 2.51 magnitude
0 5.12
5.12
5.125.12 and
5.12
5.12 0
5.12
5.12 0
1
jiV
yx
yxyx
VV
jyiy
VVV
jyiyV
jiV
jxix
VVV
jxixV
jiV
W
WW
MWMW
WM
M
MWMW
WM
M
5
5
5 5
5
5 30
Page 6
3. (a) In a room of height 6 m, a ball is projected from a point P. P is 1.1 m above the floor.
The velocity of projection is 28.9 1s m at an angle of 45 to the horizontal.
The ball strikes the ceiling at Q without first striking a wall. Find the length of the straight line PQ.
m 96.10or 59.4
9.48.9
8.9
.45cos28.9
1
012
09.48.99.4
9.4.45sin28.9
22
2
2
221
PQ
tr
t
tt
tt
gtt
i
5
5
5
5 20
P
Q
1.1 m
6 m 28.9
Page 7
3 (b) A particle is projected up an inclined plane with initial speed 80 1s m . The line of projection makes an angle of 30 with the inclined plane and the
plane is inclined at an angle to the horizontal. The plane of projection is vertical and contains the line of greatest slope.
The particle strikes the plane at an angle of3
2tan 1 .
Find (i) the value of
(ii) the speed with which the particle strikes the plane.
1
22
221
s m 52.9or 720
40320 speed
40
320 (ii)
4.23 4
3tan
tan80340
40
3
2
tan
40
cos
80cos30sin80
tan80340
cos
80sin30cos80
cos
80
cos30sin80 0
0 (i)
v
v
v
v
gg v
gg v
g t
.tg.t
r
j
i
i
j
j
i
j
5 5
5
5
5 5 30
Page 8
4. (a) Two particles of masses 0.24 kg and 0.25 kg are
connected by a light inextensible string passing over a small, smooth, fixed pulley.
The system is released from rest.
Find (i) the tension in the string
(ii) the speed of the two masses when the 0.25 kg mass has descended 1.6 m.
1
22
sm 0.8
64.0
6.12.020
2 (ii)
N 4.2
2.0
49.001.0
24.024.0
25.0.250 (i)
v
v
fs u v
T
f
f g
f g T
f T g
5
5
5 15
0.25 0.24
Page 9
4 (b) A smooth wedge of mass 4m and slope 45º
rests on a smooth horizontal surface. Particles of mass 2m and m are placed on the smooth inclined face of the wedge. The system is released from rest. (i) Show, on separate diagrams, the forces acting on the wedge
and on the particles. (ii) Find the acceleration of the wedge.
(i)
2sm 67.2or 11
3
833
42
1
445sin45sin 4
2
1
45sin45cos
2
45sin245cos2 2 )(
gf
mfmfmg
mfmfmgmfmg
mf RSm
mfmgS
mfS mg m
mfmgR
mf R mgm ii
5
5
5 5
15 35
45º
m
2m
4m
45º
S
R
4mg
T mg 2mg
SR
Page 10
5. (a) A sphere, of mass m and speed u, impinges directly on a stationary sphere of mass 3m. The coefficient of restitution between the spheres is e.
(i) Find, in terms of u and e, the speed of each sphere after the collision.
(ii) If 4
1e , find the percentage loss in kinetic energy due to the collision.
% 3.70100128
45
KEin loss Percentage
128
45
128
19 KEin Loss
128
19or
512
76
16
53
16
3after K.E.
before K.E.
16
5 and
16
4
1 )(
4
1
4
31
0 NEL
303 PCM )(
221
2
222
21
22
2
21
2
21
222
1212
1
221
21
2
1
21
21
mu
mu
mumumu
mumu
um
um
vmmv
mu
u v
u v
e ii
eu v
eu v
u e v v
mv mv) m( umi
25
5
5
5
5 5
Page 11
(b) A smooth sphere, of mass m, moving with
velocity ji
2 6 collides with a smooth sphere, of mass km, moving with velocity ji
4 2 on a smooth
horizontal table.
After the collision the spheres move in parallel directions. The coefficient of restitution between the spheres is e.
(i) Find e in terms of k.
(ii) Prove that 3
1k .
3
1
423
14k2
k3
1 (ii)
4k2
k3
42623
1
4262
1
246
2
42
equal are slopes directions parallel
1
246
1
426
26 NEL
26 PCM (i)
12
21
2
1
21
21
k
kk
e
e
ekkkek
ekk
k
ke
vv
vv
k
kev
k
ekkv
e v v
kmv mv km m
5
5
5
5
5 25
km m
4
2
2
6
Page 12
6. (a) A particle of mass m kg lies on the top of a
smooth sphere of radius 2 m. The sphere is fixed on a horizontal table at P.
The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B and strikes the table at Q.
Find (i) the speed of the particle at B
(ii) the speed of the particle on striking the table at Q.
11
212
121
2212
121
1
32
21
221
2
2
s m 8
3
42
3
4
cos22
at energy Total at energy Total )(
s m 3
4
cos
cos22cos2
cos22
cos2 0
2
cos )(
gv
mgg
mmv
mgmvmv
B Qii
gv
mggm
mgmv
gvR
mvR mgi
5
5
5
5 5 25
P Q
B
P Q
α
R
mg
Page 13
6 (b) A particle moves with simple harmonic motion of amplitude 0.75 m. The period of the motion is 4 s. Find (i) the maximum speed of the particle
(ii) the time taken by the particle to move from the position of
maximum speed to a position at which its speed is half its maximum value.
s. 3
2 s.
3
2
3
11time
2sin
4
3
8
33
3
1
2cos
4
3
8
33
cos
8
33
4
3
216
3
16
3 )(
s m 8
3
4
3
2
2
42
4 Period )(
2222
2222
max21
1
max
t
tt
t
tax
x
x
xav
vii
av
i
25
5 5
5
5 5
Page 14
7. (a) One end of a uniform ladder, of weight W, rests against a smooth vertical wall, and the other end rests on rough horizontal ground. The coefficient of friction between the ladder and the ground is μ. The ladder makes an angle with the horizontal and is in a vertical plane which is perpendicular to the wall.
Show that a person of weight 3W can safely climb to the top of the ladder if
tan8
7 .
tan8
72
7tan4
2
7tan
cos3cossin
: about moments
4
4 vertical
horizontal
2
21
2
2
1
12
WW
WR
WWR
c
WR
WR
RR
25
5
5
5 5
5
∟c
W
3W R1
R2
μ R1
Page 15
7. (b) Two uniform smooth spheres each of weight W
and radius 0.5 m, rest inside a hollow cylinder of diameter 1.6 m.
The cylinder is fixed with its base horizontal.
(i) Show on separate diagrams the forces
acting on each sphere.
(ii) Find, in terms of W, the reaction between the two spheres.
(iii) Find, in terms of W, the reaction between the
lower sphere and the base of the cylinder.
in cos 54
53 s
W
WW
WR
WR
WR
WR
2R
R
sinR B Sphere (iii)
4
5
5
4
sin A Sphere (ii)
3
3
23
2
2
2
5 5 5
5
5
1.6
0.5
0.5
0.6 θ
1
A
B
R2 R4
W
W
R1 R2
R3
25
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 2
2
1rm .
221
4
0
4
0
3
2
4 M 2
4 M 2
M 2 disc theof inertia ofmoment
2M element theof inertia ofmoment
2M element of mass
areaunit per mass MLet
rm
r
x
dxx
xxdx
xdx
r
r
5
5 5
5 20
Page 17
8. (b) An annulus is created when a central hole of radius b is removed from a uniform circular disc of radius a. The mass of the annulus (shaded area) is M.
(i) Show that the moment of inertia of the annulus about an axis through
its centre and perpendicular to its plane is
2
22 baM .
(ii) The annulus rolls, from rest, down an incline of 30˚. Find its angular
velocity, in terms of g, a and b, when it has rolled a distance 2
a.
22
2
212
22
2
212
21
2212
21
22
44
22
4
1
31
3
422
1
30sin2
PEin Loss KEin Gain (ii)
2
4 2
4 2
2 annulus of inertia ofmoment (i)
ba
ga
aMgaM
baM
aMgaMI
MghMvI
baM
ba
ba
M
xM
dxxM
a
b
a
b
30
5
5
5
5 5
5
a
b
Page 18
9. (a) State the Principle of Archimedes.
A buoy in the form of a hollow spherical shell of external radius 1 m and internal radius 0.8 m floats in water with 61% of its volume immersed.
Find the density of the material of the shell. Principle of Archimedes
3
33
3
m kg 1250488.0
610
3
4610
3
4488.0
3
4488.0
8.03
41
3
4
3
4610
13
4
100
611000
gg
BW
g
g
VgW
g
g
VgB
5
5
5 5 5 25
1
0.8
Page 19
9 (b) A uniform rod, of length 1.5 m and weight W, is freely hinged at a point P. The rod is free to move about a horizontal axis through P. The other end of the rod is immersed in water. The point P is 0.5 m above the surface of the water. The rod is in equilibrium and is inclined at an angle of 60 to the vertical. Find (i) the relative density of the rod (ii) the reaction at the hinge in terms of W.
5
25
3
5
3
3 )(
9
5
33
53
1 and
60sin4
360sin
4
5
:about moments2
12
160cos5.1
part immersed oflength )(
31
WR
WRW
WRB
W
s
W Bii
s
Ws
Ws
W
s
WB
WB
P
x
x
xi
25
5
5 5
5 5
P
60 0.5 m
Page 20
10. (a) Solve the differential equation
2xyxdx
dyy
given that y = 0 when x = 0.
e y
e y
x y
C
xy
C x y
dxxdyy
y
y
yx
dx
dy
xyxdx
dyy
x
x
1
1
2
11ln
2
1
0
0 ,0
2
11ln
2
1
1
1
2
22
22
22
2
2
2
5
5 5
5 20
Page 21
10 (b) The acceleration of a cyclist freewheeling down a slight hill is
22 s m 0006.012.0 v
where the velocity v is in metres per second. The cyclist starts from rest at the top of the hill. Find (i) the speed of the cyclist after travelling 120 m down the hill
(ii) the time taken by the cyclist to travel the 120 m if his average speed is 2.65 1s m .
s 3.45
12065.2
time
distance speed average )(
s m 18.5
155.10006.012.0
12.0
144.00006.012.0
12.0ln
1200006.012.0
12.0ln
0012.0
1
12012.0ln0012.0
10006.012.0ln
0012.0
1
0006.012.0ln0012.0
1
0006.012.0
0006.012.0 )(
1
144.02
2
2
2
120 0
0
2
120
0 2
0
2
tt
ii
v
ev
v
v
v
x v
dx dv v
v
v dx
dvvi
v
v
5
5 5
5
5
5 30
Page 22
Marcanna Breise as ucht freagairt trí Ghaeilge
Ba chóir marcanna de réir an ghnáthráta a bhronnadh ar iarrthóirí nach ngnóthaíonn níos mó ná 75% d’iomlán na marcanna don pháipéar. Ba chóir freisin an marc bónais sin a shlánú síos. Déantar an cinneadh agus an ríomhaireacht faoin marc bónais i gcás gach páipéir ar leithligh. Is é 5% an gnáthráta agus is é 300 iomlán na marcanna don pháipéar. Mar sin, bain úsáid as an ngnáthráta 5% i gcás iarrthóirí a ghnóthaíonn 225 marc nó níos lú, e.g. 198 marc 5% = 9·9 bónas = 9 marc. Má ghnóthaíonn an t-iarrthóir níos mó ná 225 marc, ríomhtar an bónas de réir na foirmle [300 – bunmharc] 15%, agus an marc bónais sin a shlánú síos. In ionad an ríomhaireacht sin a dhéanamh, is féidir úsáid a bhaint as an tábla thíos.
Bunmharc Marc Bónais
226 11
227 – 233 10
234 – 240 9
241 – 246 8
247 – 253 7
254 – 260 6
261 – 266 5
267 – 273 4
274 – 280 3
281 – 286 2
287 – 293 1
294 – 300 0