Attempt Mark - Applied Mathematics - Exam, Exams of Applied Mathematics

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Coimisiún na Scrúduithe Stáit

State Examinations Commission

LEAVING CERTIFICATE 2009

MARKING SCHEME

APPLIED MATHEMATICS

HIGHER LEVEL

1. (a) A particle is projected vertically upwards from the point p.

At the same instant a second particle is let fall vertically

from q. The particles meet at r after 2 seconds.

The particles have equal speeds when they meet at r.

Prove that pr  3 rq.

qr pr

gqr gpr

v gpr

v v gpr

v u gpr

v gqr

v u fs

v u

v u g

v g

v u ft

pr 2

qr 0 2

pr 2

qr 0 2

2

2 2

2 2

2

2 2

qr pr

pr g

g g gpr

qr g

g gqr

v u fs

u g

v u g

v g

v u ft

pr 4 16 2

qr 4 0 2

pr 2

qr 0 2

2 2

2

2 2

q r p 5 5 5

or

1. (b) A train accelerates uniformly from rest to a speed v m/s with uniform acceleration

f m/s.

2 It then decelerates uniformly to rest with uniform retardation 2 f m/s.

2

The total distance travelled is d metres.

(i) Draw a speed-time graph for the motion of the train.

(ii) If the average speed of the train for the wh ole journey is 3

d ,

find the value of f.

1

4 4 or 4 4

3 4 or 3 4

3 4

or 9

4

3

3 2

or 3

2

3

3

totaltime

totaldistance averagespeed

3 4

2

2

2 3

2 totaldistance or

2

3

2

totaltime

2

2 tan

tan

2

2 2 2

2

2

2 2

2

3

1 2

2 1 2

2 1 2 1 2

1

1 2

2 2

1 1



 

 

 

  







   

   

   

  

  

f

fd f d fd d

v f d v d

d v

v

d f d

d v

v

d d fd

t t

d d

v fd v

d

f

v

v

d t t

v

d d t t v t t

f

v

f

v

f

v t t

f

v t t

v f

f

v t t

v f

f

v



speed

time

v

t 1 t (^2)

α β

2 (b) The speed of an aeroplane in still air is u km/h.

The aeroplane flies a straight-line course from P to Q, where Q is north of P.

If there is no wind blowing the time for the journey from P to Q is T hours.

Find, in terms of u and T , the time to fly from P to Q if there is a wind blowing

from the south-east with a speed of 4 2 km/h.

cos 4 2 cos 45

time

cos

or sin 4

sin

sin 45

sin

2

2

u

uT

u

s

u

u

u u

u

s uT

α

u

s

3. (a) A straight vertical cliff is 200 m high.

A particle is projected from the top of the cliff.

The speed of projection is 14 10 m/s at an angle
to the horizontal.

The particle strikes the level ground at a distance of 200 m from the foot of the

cliff.

(i) Find, in terms of
, the time taken for the particle to hit the ground.

(ii) Show that the two possible directions of projection are at right angl es

to each other.






directions:areperpendicu lar

tan tan 1 21 2

tan 1 2

tan 2 tan 1 0

200 tan 1001 tan 200

cos

200 tan

14 10 cos

14 10 cos

14 10 sin.

(ii) 14 10 sin. 200

14 10 cos

(i) 14 10 cos. 200

1 2

2

2

2

2

2

1

2 2

1

g

t gt

t

t

4. (a) A light inextensible string pa sses over a small

fixed smooth pulley.

A particle A of mass 10 kg is attached to one

end of the string and a particle B of mass 5 kg

is attached to the other end.

The system is released from rest when B touches

the ground and A is 1 m above the ground.

Find (i) the speed of A as it hits the ground

(ii) the height that B rises above the horizontal ground.



m. 3

height 1

(ii) B 2

or 2.556 m/s 3

A 2

(i) 10 10

2 2

2 2

s

gs

g

v u as

g v

g

v u as

g f

g f

T g f

g T f^5

A

B

1m

4 (b) A mass m 1 kg is at rest on a smooth

horizontal table. It is attached to a

light inextensible string. The string,

after passing over a small fixed pulley

at the edge of the table, passes under a

small moveable pulley C, of mass m (^) 2 kg.

The string then passes over a smooth

fixed pulley and supports a mass of 1 kg.

The system is released from rest.

(i) Find, in terms of m 1 and m 2 , the tension in the string.

(ii) The pulley C will remain at rest if k m m

2 1

Find the value of k.





(ii) Cwillremainatrestif 2 0 or if

1 kg 1

2 1

1 2 1 2

1 2 1 2 1

1 2 1 2

1 2 2

2

1 2 1 2

1 2

1 2 1 2 1 2

2

1

2 2

1

2 2

2 2 2

1 1

k

m m

mm m m

m m mm m

m m mm

mmg mg

p q m g T

m m mm

mmg T

mmg mT mT mmT

mT

m

mT T

mg

T g m

m T mg T

p q m mg T m

T g q

i m T m p 5

m 1

C

1 kg

5 (b) A smooth sphere A, of mass m kg, moving

with speed u , collides with a stationary

identical smooth sphere B.

The direction of motion of A, before impact,

makes an angle
with the line of centres at

impact and just touches sphere B, as shown in the diagram.

The coefficient of restitution between the spheres is 5

(i) Show that
 30.

(ii) Find the direction in which each sphere travels after the collision.

(iii) Find the percentage loss in kinetic energy due to the col lision.



%KElost 100 13. 5 %

KElost

KEafter

(iii) KEbefore

directionofB alonglineofcentres

velocityofB 0

directionofA tan

velocityofA

and 20

NEL

(ii) PCM

(i) sin

2 2

1

2 400

27

2 400

27

2 2 2

2

1

2 2

1

1

1 2

1 2

1 2



mu

mu

mu

u u u m

mu

j

u i

j

u i

u

u v

u v

u v v

m mv mv

u m

r

r

u

A

B

6. (a) The distance, x , of a particle from a fixed point, o , is given by

x  a cos
 t   

where a ,  and  areconstants.

(i) Show that the motion of the particle is simple harmonic.

A particle moving with simple harmonic motion starts from a point 5 cm from

the centre of the motion with a speed of 1 cm/s.

(ii) The period of the motion is 11 seconds. Find the maximum speed of

the particle, correct to two decimal places.







3. 03 cm/s.

sin

cos

1 , 0 1 sin

5 , 0 5 cos

or 11

( ) Period 11

S.H.M.about 0.

cos

sin

() cos

max

2

2

2

2 2

2

2

v a

a a

a a

a a

a

a

a

v t a

x t a v a x

ii

x

x

x a t

x a t

i x a t

7. (a) A uniform rod of length 2 m and of mass

34 kg, is suspended by two vertical strings.

One string is attached to a point 20 cm from

one end and can just support a mass of 17 kg

without breaking; the second string is

attached 30 cm from the other end and can

just support a mass of 20.74 kg without breaking.

A mass of 3.4 kg is now attached to the rod.

Find the length of the section ab of the rod within which the 3.4 kg mass can

be attached without breaking either string.

















1. 15 mor 15 cm

1 m

Takemomentsabout :

1. 15 m

Takemomentsabout :

1

2

ab

x

g g x g

F g x g

d

x

g gx g

F gx g

c

a b

c

d

3.4 g 34 g

F 1 F 2

x

7 (b) A uniform rod of length 2 p and weight W rests with

its lower end a in contact with a smooth

hemispherical bowl, of radius p.

The axis of the bowl is vertical.

The upper end of the rod projects beyond

the rim of the bowl as shown in the diagram.

The inclination of the rod to the horizontal is .

The point b on the rod is in contact with the rim of the bowl.

ab  2 p cos .

(i) Find, in terms of W , the reaction at b.

(ii) Show that cos   2 cos 2 .






cos 2 cos 2

cos 2 2 cos 2

cos 2 cos sin 2 sin 2 cos 2

sin 2 2 cos 2

sin cos 2

vert : cos sin 2

2 cos 2

sin

cos 2

sin

horiz : sin cos 2

(ii)

2 cos cos

(i) Momentsabout :

W

W W

R S W

R W

S

R S

W

R

R p W p

a

a

b

a

b

θ

α θ W

R

S

8 (b) Three equal uniform rods, each of length 2  and mass m , form the sides of an

equilateral triangle abc.

(i) Find the moment of inertia of the frame abc about an axis through a

perpendicular to the plane of the triangle.

The triangular frame abc is attached to a smooth hinge at a about which it can

rotate in a vertical plane. The frame is held with ab horizontal, and c below

ab , and then released from rest.

(ii) Find the maximum angular velocity of the

triangle in the subsequent motion.



 









(ii) GaininKE LossinPE

I

(i) 2

2 2

3

(^221) 2

1

3

1

2 2

1

2

2 2

2 2

2 2 3

(^21) 3

(^24) 3

4

(^222)

g

g

m mg d

h d

mgh

I Mgh

m

m m

m md

m m m md

d

d

a (^) b

c

a (^) b

c

d

2 

9. (a) A uniform cylindrical piece of wood 12 cm long floats in water with its axis

vertical and 10 cm of its length immersed.

Oil of relative density 0.75 is poured on to the water until the top of the

cylinder is in the surface of the oil.

Find the depth of the layer of oil.












8 cm.

Let thedepthofoil

relativedensityofrod

water

12

12 12

6

5

12

12

6

5

12

12

10





h

h h

W

W W

B B W

h

s

W

s

W

B W

h h

h h

oil water

oil

water

h