M32
Coimisiún na Scrúduithe Stáit
State Examinations Commission
Scéim Mharcála Scrúduithe Ardteistiméireachta, 2006
Matamaitic Fheidhmeach Ardleibhéal
Marking Scheme Leaving Certificate Examination, 2006
Applied Mathematics Higher Level
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Mathematical Problems Involving Vectors, Trigonometry, and Calculus, Exams of Applied Mathematics
A series of mathematical problems involving vectors, trigonometry, and calculus. The problems include finding distances, velocities, accelerations, and angles. Some problems involve trains, aeroplanes, and spheres colliding, while others involve moments of inertia and buoyancy of objects. The document also includes some alternative solutions for certain problems.
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M
Coimisiún na Scrúduithe Stáit
State Examinations Commission
Scéim Mharcála Scrúduithe Ardteistiméireachta, 2006
Matamaitic Fheidhmeach Ardleibhéal
Marking Scheme Leaving Certificate Examination, 2006
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 Mark all answers, including excess answers and repeated answers whether cancelled
or not, and award the marks for the best answers.
3 Mark scripts in red unless candidate uses red. If a candidate uses red, mark the script
in blue or black.
4 Number the grid on each script 1 to 10 in numerical order, not the order of answering.
5 Scrutinise all pages of the answer book.
6 The marking scheme shows one correct solution to each question. In many cases there
are other equally valid methods.
1 ( b ) Two trains P and Q, each of length 79.5 m, moving in opposite directions
along parallel lines, meet at o , when their speeds are 15 m/s and 10 m/s
respectively.
The acceleration of P is 0.3 m/s
2 and the acceleration of Q is 0.2 m/s
2 .
It takes the trains t seconds to pass each other.
(i) Find the distance travelled by each train in t seconds.
(ii) Hence, or otherwise, calculate the value of t.
(iii) How long does it take for 5
of the length of train Q to pass the
point o?
2
2 2
1 Q
2
2 2
1 P
t t
S ut at
t t
i S ut at
- 0848 or 3. 1 s.
6 s
2
2
2
2
2 2
P
t
t t
t t
S t t
iii
t
t t
t t
t t t t
ii S S
Q
Q
2. (a) Two aeroplanes A and B, moving horizontally, are travelling at 200 km/h
relative to the ground. There is a wind blowing from the east at 60 km/h.
The actual directions of flight of A and B are north-west and north-east
respectively.
Find (i) the speed of aeroplane A in still air
(ii) the magnitude and direction of the velocity of A relative to B.
V 1
VA = 200
VB = 200
direction: west
magnitude: 200 2 or 282. 84 km/h
0 or 200 2 0 2
- 2 km/h
() 60 200 260 200 cos
1
2 2 2 1
i j i j
i j i j
ii V V V
v
i v
AB A B
r r r r
r r r r
r r r
= + −^5
3. (a) A particle is projected from a point o
with velocity i j
r r 9.8 + 29.4 m/s where i j
r r and
are unit perpendicular vectors in the
horizontal and vertical directions,
respectively.
(i) Express the velocity and
displacement of the particle
after t seconds in terms of i j
r r and.
o
(ii) Find, in terms of t , the direction in which the particle is moving
after t seconds.
(iii) Find the two times when the direction of the particle is at right angles to
the line joining the particle to o.
( ) ( )
( ) ( )
( )( )
4 s and 5 s
productofslopes 1
linesareperpendicular
or 3
- 8
tan
( ) directionof:linejoiningtheparticleto
or 3
- 8
tan
( ) direction:
or 9. 8 29. 4
cos sin
or 9. 8. 29. 4.
() cos. sin.
2
2
1
2
1
2 2
1
2 2
1
2 2
1
t t
t t
t t
t t
t t
t gt
r
r
iii o
t
gt
V
V
ii
i gt j
v u i u gt j
t i t gt j
i r u t i u t gt j
i
j
i
j
r
r
r
r
r r
r r r
r r
r r^ r
3 (b) A particle is projected up an inclined plane with initial velocity u m/s.
The line of projection makes an angle 30 °with the plane and the
plane is inclined at 30 °to the horizontal.
The plane of projection is vertical and contains the line of greatest slope.
Find, in terms of u , the range of the particle on the inclined plane.
sin 30 3
cos 30
Range cos 30 sin 30
or cos 30
2 sin 30
sin 30 cos 30 0
2
2 2
2
2
1
2 2
1
2 2
1
g
u
g
u
g
u
g
u g .. g
u u.
u .t g .t
g
u
g
u t
u .t g .t
r (^) j
4 (b) A smooth wedge of mass 3 m
and slope α rests on a smooth
horizontal surface.
A particle of mass m is placed
on the smooth inclined face of the wedge
and is released from rest.
A horizontal force F is applied to the wedge to keep it from moving.
(i) Show, on separate diagrams, the forces acting on the wedge and on the
particle.
(ii) Prove that the reaction between the wedge and the horizontal surface is
( α)
2 mg 3 +cos.
(iii) If the speed of the particle after 1 s is 4.9 m/s find the value of α.
( i )
( )
sin
- 9 0 sin 1
sin
( ) sin
3 cos
3 gcos cos
3 cos
( ) gcos
2
2 1
1
g
v u at
g p
iii mg mp
mg
mg m
R mg R α
ii R m
mg
α
R 1
3 mg
R 2
R 1
F
m
F
3 m α
5. (a) A smooth sphere P, of mass 3 kg, moving with speed 6 m/s, collides directly
with a smooth sphere Q, of mass 5 kg, which is moving in the same direction
with speed 2 m/s. The coefficient of restitution for the collision is e.
(i) Find, in terms of e , the speed of each sphere after the collision.
(ii) If the loss of kinetic energy due to the collision is ( )
2 k 1 − e , find the
value of k.
LossinKE 64
KEafter
( ) KEbefore
NEL 6 2
() PCM 36 52 3 5
2
2 2
2 2
2 1
(^22)
2
1
1 2
1 2
k
( -e )
e e
v (v )
ii
e v
e v
v v e
i ( ) v v^5
6. (a) A particle moves with simple harmonic motion of period 3 π. At time t = 0,
the particle passes through the centre of the oscillation. It passes through a
point distant 4 m from the centre of motion with a speed of 5 m/s away from
the centre.
Find, correct to two decimal places,
(i) the maximum acceleration of the particle
(ii) the time which elapses before it next passes through this point.
( ) ( )
( )
( )
( )
( )
- 7349 3. 24 s. 4
requiredtime 2 Period
4 8. 5 sin
sin.
or
- 24 seconds
4 8. 5 cos
( ) cos.
- 5 3.78m/s
maxaccel
() Period
4
1
3
2
3
1
2
1
(^22) 3
2
2
2 3
2
2 2
⎟^ =
t
t
t
x a t
t
t
ii x a t
a
a
a
v a x
i
6 (b) A hollow cone with its vertex downwards and its
axis vertical, revolves about its axis with a constant
angular velocity of 4 π rad/s.
A particle of mass m is placed on the inside rough surface
of the cone. The particle remains at rest relative to the cone.
The coefficient of friction between the particle and
the cone is 4
.
0
l
The semi-vertical angle of the cone is 30 °and the
particle is a distance l m from the vertex of the cone.
Find the maximum value of l, correct to two places of decimals.
μR
R
mg
( )
( )
- 43 m.
sin 30
cos 30 sin 30
sin30 g Rcos
2
2
2
r
r
r
r g
R
R R
R R mr
R R mr
g
R
m
R m
l
l
7 (b) One end of a uniform ladder rests on a rough horizontal floor and the other
end rests against a rough vertical wall.
The coefficient of friction at each contact is μ.
The ladder makes an acute angle α with the wall.
If the ladder is on the point of slipping find α in terms of λ , the angle of
friction, where μ = tan λ.
W
( ) ( ) ( )
tan 2 tan
tan 1 tan
2 tan
tan 1
tan 1
tan 1
tan 1
tan tan
R sin cos W sin
Takemomentsabout forsystem:
vert
horiz
2
2
2
2
2
2
1 2 2
2
1 2 2
2
1 2 2
(^22)
1 2
1 2
W
W W
μ
μR R W
μ R α
a
W
R
R R W
R R
l l l
μR 1 a
α
R 1
μR 2 R 2
8. (a) Prove that the moment of inertia of a uniform rod of mass m and length 2 l
about an axis through its centre perpendicular to the rod is
2
3
m l.
2 3
1
3 3
2
3
2
2
m
M
x M
momentofinertiaoftherod M x dx
momentofinertiaoftheelement M dx x
massofelement M dx
Let M massperunitlength
l
l
l
l
l
l
−
∫
9. (a) 16 of acid of relative density 1.8 is mixed with 7 of water to
3 cm
3 cm
form 22.35 of dilute acid.
3 cm
Find
(i) the contraction in volume which has taken place.
(ii) the relative density of the dilute acid.
(ii) Massofdiluteacid Massofwater Massofacid
- 65 cm
(i) Contractioninvolume 16 7 2235
6
6
6
3
s
s
s
_-
-_
9 b) A hollow spherical shell of external
radius a and internal radius b floats
in water.
The relative density of the material
of the shell is s.
(i) Find the maximum buoyancy force that the shell could experience
while floating in water.
(ii) Hence, or otherwise, prove that if the shell is not totally immersed in
the water then
3 3
3
a b
a s −
(iii) If 19
s = , prove that the thickness of the shell must be less than 3
a .
( ) ( )
( )
( )
( )
thickness
thickness
(iii)
1 000s
MaximumBuoyancy
() MaximumBuoyancy
3 3
3
3 3
3
3 3 3
3 3 3
3
3
a
a a b a
a b
a b
a
a b
a s
sa b a
a g a b g
W
ii B W
a g
a g
i ρVg