Leaving Certificate 2009 Applied Mathematics Ordinary Level Marking Scheme, Exams of Applied Mathematics

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Coimisiún na Scrúduithe Stáit

State Examinations Commission

LEAVING CERTIFICATE 2009

MARKING SCHEME

APPLIED MATHEMATICS

ORDINARY LEVEL

1. 3 points p , q and r lie on a straight level road.

Two cars, A and B, are moving towards each other on the road.

Car A passes p with speed 3 m/s and uniform acceleration of 2 m/s 2 and at the same instant car B passes r with speed 5 m/s and uniform acceleration of 4 m/s 2.

A and B pass each other at q seven seconds later.

Find (i) the speed of car A and the speed of car B at q.

(ii) | pq | and | rq |, the distances A and B have moved in these 7 s.

Car A stops accelerating at q and continues on to r at uniform speed.

(iii) Find, correct to one place of decimals, the total time for car A to travel from p to r.

totaltime 14.8 s.

7. 8 s.

(iii)

133 m.

70 m.

(ii)

33 m/s.

17 m/s.

(i)

2 2

1

2

1

2 2

1

2

1

2 2

1

t

t

s ut at

s

s

s ut at

s

s

s ut at

v

v

v u at

v

v

v u at

B

B

A

A

B

B

A

A 10

A

B

p q^ r

2. A ship P is moving north at a constant speed of 20 km/h.

Another ship Q is moving south -west at a constant speed of 10 2 km/h.

At a certain instant, P is positioned 50 km due west of Q.

Find (i) the velocity of P in terms of i and j

(ii) the velocity of Q in terms of i and j

(iii) the velocity of P relative to Q in terms of i and j

(iv) the shortest distance between P and Q in the subsequent motion.




47. 43 km.

50 sin 71. 5651

shortestdistance 50sin

(iv) tan

10 i 30 j

0 i 20 j 10 i 10 j

(iii) V V V

10 i 10 j

(ii) V 10 2 cos 45 i 10 2 sin 45 j

(i) V 0 i 20 j

PQ P Q

Q

P

P Q

20 km/h

50 km

10 2 km/h

V PQ

α P 50 Q

α

3 (b) A straight vertical cliff is 45 m high. A projectile is fired horizontally with an initial speed of x m/s from the top of the cliff. It strikes the level ground at a distance of 30 3 m from the foot of the cliff.

Find the value of x , correct to one decimal place.

30 3 t 0

2 2

1

2

2 2

1

x

x

x

s ut at

t

t

s ut at

x

y

x m/s

30 3 m

45 m

4. (a) Two particles of masses 3 kg and 2 kg are connected by a taut, light, inextensible string which passes over a smooth light pulley at the edge of a smooth horizontal table. The system is released from rest.

(i) Show on separate diagrams the forces acting on each particle.

(ii) Find the common acceleration of the particles.

(iii) Find the tension in the string.

(i)

(ii)

4 m/s^2 5

a

g T a

T a

(iii)

12 N

T

T a

3 kg

2 kg

2 kg

T

2 g

3 kg T

R

3 g

5. A smooth sphere A, of mass 5 kg, collides directly with another smooth sphere B, of mass 2 kg, on a smooth horizontal table.

Before impact A and B are moving in opposite directions with speeds 3 m/s and 5 m/s, respectively.

The coefficient of restitution for the collision is 4

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 B due to the collision.

20 Ns

(iii) Impulse 2 5 2 5

20 J

KElost 4 7.5 27. 5

KEaftercollision 5 1 2 5

(ii) KEbeforecollision 53 2 5

1 m/sand 5 m/s

NEL

(i) PCM 53 2 ( 5) 5 2

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

A B

3 m/s (^) 5 m/s

5 kg 2 kg

6. (a) Particles of weight 4 N, 5 N, 3 N and 2 N are placed at the points (11, 5), ( p , q ), (-4, 1) and (7, p ), respectively.

The co-ordinates of the centre of gravity of the system are (4, q ).

Find (i) the value of p

(ii) the value of q.

(b) A rectangular lamina with vertices a , b , c and d has the triangular portion with vertices a , d and e removed.

The co-ordinates of the points are a (0, 0), b (0, 8), c (12, 8), d (12, 0) and e (9, 6).

Find the co-ordinates of the centre of gravity of the remaining lamina.

co-ordsofc.g.
5. 4 , 5 .2

lamina 60

aed 12 6 36 7, 2

adcb 12 8 96 6, 4

(b) rea: c.g.

(a) 4

2

1

y

y

x

x

x, y

a

q

q q

p

p (^) 10

a

b c

d

e

7 (b) Two light inextensible strings are tied to a particle weighing 50 N.

The other ends of the strings are tied to two points on a horizontal ceiling.

The strings make angles α^ and β^ with the ceiling, as shown in the diagram.

tan α = 3

and tan β = 4

(i) Show on a diagram the forces acting on the particle.

(ii) Write down the two equations that arise from resolving the forces horizontally and vertically.

(iii) Solve these equations to find the tension in each of the strings.

(i)

30 Nand 40 N

sin sin 50

(iii) cos cos

vert sin sin 50

(ii) horiz cos cos

2 1

2

2

1 2

1 2

1 2

1 2

1 2

1 2

T T

T

T

T T

T T

T T

T T

T T

T T

α β

α β

T 1 T 2

8. (a) A particle describes a horizontal circle of radius 0.5 m with u niform angular velocity ω radians per second. Its acceleration is 8 m/s 2. Find (i) the value of ω (ii) the time taken to complete one revolution.

(b) A right circular hollow cone is fixed to a horizontal surface. Its semi-vertical angle is 30^ and its axis is vertical.

A smooth particle of mass 2 kg describes a horizontal circle of radius r cm on the smooth inside surface of the cone.

The plane of the circular motion is 5 cm above the horizontal surface. (i) Find the value of r in surd form. (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 cone. (iv) Calculate the angular velocity of the particle.

(a)

s. 2

(ii) Period

4 rad/s

(i) acceleration 2

2

r

(b)

600 rad/s

( ) cos 30

20 40 N

( ) sin 30 2

cm 3

() tan 30

2

2

iv R mr

R R

iii R g

ii

r

r i

R

2g

30 ^ 5 cm

r

Coimisiún na Scrúduithe Stáit

Marcanna Breise as ucht freagairt trí Gha eilge

Léiríonn an tábla thíos an méid marcanna breise ba chóir a bhronnadh ar iarrthóirí a ghnóthaíonn níos mó ná 75% d’iomlán na marcanna.

N.B. Ba chóir marcanna de réir an ghnáthráta a bhronnadh ar iarrthóirí nach ngnóthaíonn níos mó ná 75% d’iomlán na marcanna don scrúdú. Ba chóir freisin an marc bónais sin a shlánú síos.

Tábla 300 @ 5% Bain úsáid as an tábla seo i gcás na n -ábhar a bhfuil 300 marc san iomlán ag gabháil leo agus inarb é 5% gnáthráta an bhónais.

Bain úsáid as an ngnáthráta i gcás 225 marc agus faoina bhun sin. Os cionn an mharc sin, féach an tábla thíos.

Bunmharc Marc Bónais Bunmharc Marc Bónais 226 11 261 - 266 5 227 - 233 10 267 - 273 4 234 - 240 9 274 - 280 3 241 - 246 8 281 - 286 2 247 - 253 7 287 - 293 1 254 - 260 6 294 - 300 0