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The marking scheme for the leaving certificate examination, 2006 in applied mathematics at the ordinary level. It includes guidelines for marking and specific problems with solutions for calculating acceleration, deceleration, distance, average speed, and maximum height of a particle, as well as the velocity and shortest distance between two ships.
Typology: Exams
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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:
Attempt marks are awarded as follows: 5 (att 2), 10 (att 3).
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 9 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.
constant speed of 39 km/h, where 12
90 km
39 km/h
16 km/h
Ship B is travelling due east with a constant speed of 16 km/h.
At 2 pm ship B is positioned 90 km due north of ship A.
(i) Express the velocity of ship A and the velocity of ship B in r r terms of i and j.
(ii) Find the velocity of ship A relative to ship B in terms of and.
r i
r j
(iii) Find the shortest distance between the ships.
( ) ( )
20 i 15 j
36 i 15 j 16 i
(ii) V V V
V 16 i 0 j
36 i 15 j
j 13
i 39 13
(i) V 39 cos i 39sin j
AB A B
B
A
v^ r
v r v
r r r
r v r
v r
v r
r v^ r
(iii)
θ
θ
72 km
90 cos
shortestdistance
3. A particle is projected from a point on a level horizontal plane with initial r velocity 10 + 35 m/s, where
r i j
r i and
r j are unit perpendicular vectors in the
horizontal and vertical directions respectively.
Find (i) the time it takes to reach the maximum height
(ii) the maximum height
(iii) the two times when the particle is at a height of 50 m
(iv) the speed with which the particle strikes the plane.
speed 10 35
velocity 10 35 70
(iv) time 7 seconds
2 s and 5 s
(iii) 35 5 50
(ii) maximumht. 35
(i) 0
2 2
2
2
2
2 2
1
i j
i j
t t
t t
t t
t t
t at
t t
t t
v (^) y v u at
r r
r r
5. A smooth sphere A, of mass 7 kg, 2 m/s 1 m/s
collides directly with another smooth sphere B, of mass 3 kg, on a smooth horizontal table. A B 7 kg 3 kg
A and B are moving in opposite directions with speeds of 2 m/s and 1 m/s respectively.
The coefficient of restitution for the collision is 3
Find (i) the speed of A and the speed of B after the collision
(ii) the loss in kinetic energy due to the collision
(iii) the magnitude of the impulse imparted to A due to the collision.
(iii) Impulse 7 2 7 0.
KElost 1 5. 5 7. 1
KEaftercollision 7 0.8 3 1.
(ii) KEbeforecollision 7 2 3 1
(i) PCM 72 3 1 7 3
2 2
(^21) 2
1
2 2
(^21) 2
1
1 2
1 2 1 2
1 2
1 2
v m s v
v v e u u
v v
( ) v v
10
The centre of gravity of the four particles is at the origin.
Find the value of x and the value of y.
q
s
p r
(b) A triangular lamina with vertices p , q and r has the triangular portion with vertices p , s and r removed.
The co-ordinates of the vertices are p (0,0), q (0,6), r (12,0) and s (3,3).
Find the co-ordinates of the centre of gravity of the remaining lamina.
(b) rea: c.g.
(a)
2
1
2
1
y
y
x
x
pqsr x, y
psr
pqr
a
y
y y
x
x x
8. (a) A particle describes a horizontal circle of radius 2 metres with constant angular
velocity ω radians per second. The particle completes one revolution every 5 seconds.
(i) Show that ω is equal to 5
(ii) Find the speed and acceleration of the particle. Give your answers correct to one place of decimals.
(b) A conical pendulum consists of a particle of mass 4 kg attached by a light inelastic string of length 2 metres to a fixed point p.
The particle describes a horizontal circle of radius r. The centre of the circle is vertically below p. The string makes an angle of 30
0 with the vertical.
p
4 kg
2 m
r m
Find (i) the value of r (ii) the tension in the string (iii) the speed of the particle.
(a)
2
(^22) 2 3.2m/s 25
2.5m/s 5
a r
ii v r
i
(b)
( ) sin 30
( ) cos 30 4
() 2 sin 30 1 m
2
2
v
v
r
mv iii T
ii T g
i r
9. (a) State the Principle of Archimedes.
A solid piece of metal weighs 150 N in air and 131 N in water. Find the volume of the piece of metal.
(b) A solid sphere of radius 5 cm and relative density 8 is completely immersed in oil of relative density 0.9.
The sphere is held at rest by a light inelastic vertical string which is tied to a fixed support.
Oil
Find the tension in the string.
[Density of water = 1000 kg/m 3 ].
(a)
3
(ii) 150 131 19
(i) :Principle ofArchimedes
B Vg
(b)
3
s
W s T
L
π
