Marking Scheme for Applied Mathematics Ordinary Level Leaving Certificate 2008, Exams of Applied Mathematics

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Coimisiún na Scrúduithe Stáit

State Examinations Commission

LEAVING CERTIFICATE 2008

MARKING SCHEME

APPLIED MATHEMATICS

ORDINARY 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), 10 (att 3).

2. The marking scheme shows one correct solution to each question. In many cases

there are other equally valid methods.

1. Four points a , b , c and d lie on a straight level road.

A car, travelling with uniform retardation, passes point a with a speed of 30 m/s and

passes point b with a speed of 20 m/s.

The distance from a to b is 100 m. The car comes to rest at d.

Find (i) the uniform retardation of the car

(ii) the time taken to travel from a to b

(iii) the distance from b to d

(iv) the speed of the car at c , where c is the midpoint of [ bd ].

10 2 or 14. 1 m/s

(iv) 2

80 m

(iii) 2

4 s.

(ii)

retardation 25 m/s

25 m/s

(i) 2

2

2 2

2 2

2 2

2

2

2 2

2 2

v

v u as

s

s

v u as

t

t

v u at

a.

a

a

v u as

3. A particle is projected from a point on horizontal ground with an initial speed

of 25 m/s at an angle β

0 to the horizontal where tan β = 3

(i) Find the initial velocity of the particle in terms of

→

i and

→

j.

(ii) Calculate the time taken to reach the maximum height.

(iii) Calculate the maximum height of the particle above ground level.

(iv) Find the range.

(v) Find the speed and direction of the particle after 3 seconds of motion.

tan

18. 0 m/s

speed 15 10

V 15 i 10 j

(v) V 15 i 20 - 10 t j

60 m

range 154

(iv) time 4 s

s 20 m s 20 m

(iii) or 2

t 2 s

(ii)

15 i 20 j

(i) V 25 cos i 25 sin j

2 2

(^222)

2 2 2 2

1

r v^ r

r v r

v r

r v^ r

s

s ut at v u as

t

v u at

θ

5 kg 9 kg

4. (a) Two particles of masses 9 kg and 5 kg are

connected by a taut, light, inextensible string

which passes over a smooth light pulley.

The system is released from rest.

Find (i) the common acceleration

of the particles

(ii) the tension in the string.

(b) Masses of 3 kg and 6 kg are

connected by a taut, light,

inextensible string which passes

over a smooth light pulley as

shown in the diagram.

The 3 kg mass lies on a rough

horizontal plane and the

coefficient of friction

between the 3 kg mass and the

plane is μ.

The 6 kg mass lies on a smooth plane which is inclined at 30

0 to the horizontal.

6 kg

30

0

3 kg

When the system is released from rest each mass travels

1 metre in 2 seconds.

Find (i) the common acceleration of the masses

(ii) the tension in the string

(iii) the value of μ.

4 (a) (i)

2 or 2. 86 m/s 14

a

g T a

T g a

(ii)

64. 29 N

T

T

T g a

5. A smooth sphere A, of mass 6 kg,

collides directly with another smooth 4 m/s 2 m/s

A 6 kg B 5 kg

sphere B, of mass 5 kg, on a smooth

horizontal table.

A and B are moving in opposite

directions with speeds of 4 m/s and

2 m/s respectively.

The coefficient of restitution for the collision is 10

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.

18 Ns

(iii) Impulse 6 4 6 1

4 8.6J

KElost 58 9. 4

KEaftercollision 6 1 5 1.

(ii) KEbeforecollision 6 4 5 2

1 m/s and 1. 6 m/s

NEL

(i) PCM 64 5 ( 2) 6 5

2 2

(^21) 2

1

2 2

(^21) 2

1

1 2

1 2 1 2

1 2

1 2

v v

v v e u u

v v

v v 10

6. (a) Particles of weight 5 N, 2 N, 3 N and 8 N are placed at the points

( p , q ) (, 7 , p ) (, − 2 , q ) and ( 1 ,− 6 ), respectively.

The co-ordinates of the centre of gravity of the system are ( 2 , 0 ).

Find (i) the value of p

(ii) the value of q.

d

c

b

a

(b) A quadrilateral lamina has

vertices a , b , c and d.

The co-ordinates of the vertices

are a ( 0 , 0 ), b ( 0 , 6 ), c ( 6 , 9 )and

d ( 12 , 0 ).

Find the co-ordinates of the

centre of gravity of the lamina.

co-ordsofc.g.( 5 , 3 .5)

lamina 72

acd 12 9 54 6, 3

abc 6 6 18 2, 5

(b) rea: c.g.

(a) 2

2

1

2

1

y

y

x

x

x, y

a

q

q q

p

p

8. (a) A particle describes a horizontal circle of radius 2 metres with constant

angular velocity ω radians per second.

Its speed is 5 m/s and its mass is 3 kg.

Find (i) the value of ω

(ii) the centripetal force on the particle.

2 cm

(b) A hemispherical bowl of diameter 10 cm

10 cm is fixed to a horizontal

surface.

A smooth particle of mass 2 kg

describes a horizontal circle of

radius r cm on the smooth inside

surface of the bowl.

The plane of the circular motion

is 2 cm above the horizontal surface.

(i) Find the value of r.

(ii) Show on a diagram all the forces acting on the particle.

(iii) Find the reaction force between the particle and the surface of

the bowl.

(iv) Calculate the angular velocity of the particle.

(a)

( )( )( )

37.5 N

(ii) Force

2. 5 rad/s

(i)

2

2

mr

r v

(b)

( ) cos

( ) sin 2

2

2

2 2

⎟ = ⇒^ =

ω ω

α ω

α

iv R mr

R R

iii R g

ii

i r

R

2g

9. (a) State the Principle of Archimedes.

A solid piece of metal has a weight of 28 N.

When it is completely immersed in water the metal weighs 18 N.

Find (i) the volume of the metal

(ii) the relative density of the metal.

(b) A right circular solid cone has a base of

radius 6 cm and a height of 15 cm. 6 cm

15 cm

The relative density of the cone is 0.

and it is completely immersed in a

tank of liquid of relative density 0.9.

The cone is held at rest by a light

inextensible vertical string which

is attached to the base of the tank.

The upper surface of the cone is

horizontal.

Find the tension in the string.

[Density of water = 1000 kg/m

3 ]

(a)

relativedensity 2.

B weightofwaterdisplaced

PrincipleofArchimedes

V

M

ii

V

Vg V

(i)

(b)

{ ( ( ) ( )) }

1. 7 N

B

2 3

1

2

1

2

1

T

T Vg

T W

T W

W

T W