Coimisiún na Scrúduithe Stáit State Examinations Commission

Leaving Certificate 2012

Marking Scheme

Higher Level

Design and Communication Graphics

Coimisiún na Scrúduithe StáitState Examinations Commission

Leaving Certificate 2012

Marking Scheme

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 The marking scheme shows one correct solution to each question.

In many cases there are other equally valid methods.

Page 2

1. (a) A particle falls from rest from a point P. When it has fallen a distance 19·6 m a second particle is projected vertically downwards from P with initial velocity 39·2 1m s− . The particles collide at a distance d from P. Find the value of d.

( )

( )

( )

212

212

212

212

2 21 12 2

2 2

212

19·6 02 s

0 2

39·2

2 39·2

4 4 8

1

0 24·9 944·1 m

s ut ft

gt t

d g t

d t gt

g t t gt

t t t t

t

d g t

= +

= +=

= + +

= +

+ = +

+ + = +

=

= + += ×=

5 5 5 5 5

25

Page 3

1. (b) A car, starts from rest at A, and accelerates uniformly at 1 2s m − along a straight level road towards B, where AB = 1914 m. When the car reaches its

maximum speed of 32 1s m − , it continues at this speed for the rest of the journey.

At the same time as the car starts from A a bus passes B travelling towards A with a constant speed of 36 1s m − . Twelve seconds later the bus starts to decelerate uniformly at 0·75 2s m − .

(i) The car and the bus meet after t seconds. Find the value of t.

(ii) Find the distance between the car and the bus after 48 seconds.

( ) ( )

( ){ } ( ) ( )( ){ }

s 40 1984061630

1212364323232512

1914

432 01236

bus

512 3212023 1032

2 car (i)

2

243

21

221

2

22

=+−=

−−+−++−+=

+=

=+×=

+=

==+=+=

+=+=

ttt

ttt

ss

s

ftuts

stst

fsuvftuv

bc

( ) ( ) ( )( )

212

212

3 314 2 4

(ii) car 32 8 0

256

bus

36 40 12 15 8 64

s ut ft

v u ft s ut ft

= += × +=

= + = +

= − − = + − 15 96

distance 256 96 352 m

v s= =

= + =

5, 5

5

5

5

25

Page 4

2. (a) Rain is falling with a speed of 25 1s m − at an angle of °20 to the vertical. A car is travelling along a horizontal road into the rain. The windscreen of the car makes an angle of °32 with the vertical. The car is travelling at 20 1s m −

. Find the angle at which the rain appears to strike the windscreen.

( )

1

25sin20 25cos 20

20 0

25sin20 20 25cos 20

28·55 23·49

28·55tan 3223·49

82·55 .

r

c

rc r c

V i j

V i j

V V Vi j

i j

α −

= −

= − +

= −

= + −

= −

= +

= °

5

5 5 5 20

25 1s m −

Page 5

2 (b) At noon ship A is 50 km north of ship B. Ship A is travelling southwest at 1h km 224 − . Ship B is travelling due west at 1h km 71 − . (i) Find the magnitude and direction of the velocity of B relative to A.

A and B can exchange signals when they are not more than 20 km apart. (ii) At what time can they begin to exchange signals? (iii) How long can they continue to exchange signals?

( ) ( )

==

+=

−−−+−=

−=

−

247tan 25

24 7 24 24 0 71

(i)

1θBA

ABBA

V

jijiji

VVV

2 2

7 (ii) 50sin 50 1425

20 14 2 51 1·2829

50cos 14·28292450 14·2829 33·717125

1·348725

time 13 : 21

(iii) 2 28·5658

1·14225

BC

CE

AD

CD

BC

BCt

CE CD

CEt

θ

θ

= = × =

= − = =

= −

= × − =

= =

=

= × =

= = 6

time 1 h 9 min=

5

5

5 5 5

5

θ B

A

50 E

D

C

30

Page 6

3. (a) A particle is projected with a speed of 98 1m s− at an angle α to the horizontal. The range of the particle is 940·8 m. Find

(i) the two values of α

(ii) the difference between the two times of flight.

2

(i) 98cos . 940·89·6

cos

98sin . 4·9 09·698sin 4·9 0

cos

sin 2 0·962 73·74 , 106·26

36·87 , 53·13

t

t

t t

α

α

α

αα

αα

α

=

=

− =

− =

==

= ° °

1

2

2 1

9.6(ii) cos3·8712·00

9·6 cos53·1316·00

4 s.

t

t

t t

=

=

=

=

− =

5 5

5 5 20

Page 7

Q α

20 3 (b) A particle is projected down an inclined plane with initial velocity 20 1s m −

. The line of projection makes an angle of 90° with the inclined plane and the plane is inclined at α to the horizontal. The plane of projection is vertical and contains the line of greatest slope. The particle strikes the plane at Q. The range of the

particle on the inclined plane is 1600 3 .

g Find

(i) the value of α

(ii) the magnitude of the rebound velocity at Q if the coefficient of

restitution is 1 .2

( ) ( ) s m 70

10340

10 340 (ii)

20802120cos20

3408023sin0

60 cos 32in

31600 cos

1600sin

31600 sin0

31600

cos40 0 cos20

0 (i)

1

22

2

2221

221

221

−=

+=

==

−=××−=×−=

=××=×+=

°==

=×

=×+×

=

=

=×−

=

v

v

v v

ggtg v

ggtg v

s

g

gg

g tgt

gr

gt

tgt

r

j

i

j

i

i

j

α

α

ααα

αα

α

α

α

5 5 5

5 5

5 30

Page 8

4. (a) Two particles A and B each of mass m are connected by a light inextensible string passing over a light, smooth, fixed pulley.

Particle A rests on a rough plane inclined

at α to the horizontal, where 5tan .12

α =

Particle B hangs vertically 1 m above the ground. The coefficient of friction between

A and the inclined plane is 1 .2

The system is released from rest.

(i) Find the speed with which B strikes the ground.

(ii) How far will A travel after B strikes the ground?

12

2 2

2 2

(i) sin cos

5 6 213 13

13

2

0 2 113

213

(ii) 220

mg T mfT mg mg mf

mg mgmg mf

gf

v u fsg

gv

v u fsg

α α− =

− − =

− − =

=

= +

= + × ×

=

= +

= 112 13 131 m.11

g s

s

− + × ×

=

5

5 5

5 5

25

A 1 m

B

α

mg μR

T T

mgcosα

R

mgsinα

Page 9

4 (b) Two particles of mass m kg and 2m kg lie

at rest on horizontal rough tables. The coefficient of friction between each

particle and the table it lies on is 2 .3

μ μ <

The particles are connected by a light inextensible string which passes under a smooth movable pulley of mass 4m kg.

The system is released from rest.

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

(ii) If the acceleration of the m kg mass is f, find the acceleration of the 2m kg mass in terms of f.

( )( )

( )

( )

( )( )

12

(i)

2 2

4 2 42 2

4 2 2 2 24 1

5

(ii)

4 1

545 5

2

T mg mf

T mg mp

mg T m f pmf mp

mg T T mg T mgmg

T

mf T mg

mgmg

g gf

mp

μμ

μ μμ

μμ

μ

μ

− =

− =

− = × += +

− = − + −+

=

= −

+= −

= −

= ( )( )

2

4 12 2

52 35 5

3 2

T mg

mgmp mg

g gp

p f g

μμ

μ

μ

−

+= −

= −

= −

5

5

5

5 5

25

m

4m

2m

Page 10

5. (a) Three smooth spheres, A, B and C, of mass 3m, 2m and m lie at rest on a smooth horizontal table with their centres in a straight line. Sphere A is projected towards B with speed 5 1m s .− Sphere A collides directly with B and then B collides directly with C. The coefficient of restitution between the spheres is e.

Show that if 2

53 −>e there will be no further collisions.

( )

( )

( )

1 2

1 2

1

2

2 3 4

PCM 3 5 2 (0) 3 2

NEL 5 0

3 2 3 3

PCM 2 (0) 2

NEL

m m mv mv

v v e

v e v e

m v m mv mv

+ = +

− = − −

= −= +

+ = +

( )

( ) ( )( )

( ) ( )( )

( )( )

3 4 2

23

24

1 3 4

1 3

2

2

1 3

0

2 1 232 1 2 1 13

if 3 2 1 2

3 2 2

3 5 3 53 1 0 2 2

3 5 if 2

v v e v

v v e e e

v v e e e

v v v

v ve e e

- e e e

e e e

v v e

− = − −

= − = + −

= + = + +

< <

<− < + −

< + −

− +− + < < <

− < >

25

5

5

5 5 5

Page 11

5 (b) A smooth sphere P collides with an identical smooth sphere Q which is at rest. The velocity of P before impact makes an angle α with the line of centres at impact,

where 0 90 .α° ≤ < ° The velocity of P is deflected through an angle θ by the collision.

The coefficient of restitution between the spheres is 1 .3

Show that 2

2 tantan .1 3 tan

αθα

=+

( ) ( )

( )

( )

tan31tan2tan

tantan3tan3tantan

tan3

cossin3

tantan1tantan

sintan

3

cos

0cos31 NEL

0cos PCM

2

2

1

1

21

21

ααθ

θααθα

α

αα

θαθα

αθα

α

α

α

+=

−=+

=

=−

+

=+

=

−−=−

+=+

uuv

u

uv

u v v

mv mv m um

5

5 5 5 5

25

Q P θ

Page 12

6. (a) A particle of mass 0·5 kg is suspended from a fixed point P by a spring which executes simple harmonic motion with amplitude 0·2 m. The period of the motion is 2 seconds.

Find (i) the maximum acceleration of the particle

(ii) the greatest force exerted by the spring correct to one place of

decimals.

2

2

2

2

2

2( ) 2

0·2

5

( )

0·55

2 105·9 N

i

a A

ii F m a

T mg

gT

πω

ω π

ωππ

π

π

=

=

== ×

=

= ×

− = ×

= +

=

5 5 5 5

20

Page 13

6 (b) A particle of mass m kg lies on the top of a smooth fixed sphere of radius 30 cm.

The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B.

(i) Find the speed of the particle at B.

(ii) The horizontal distance, in metres, of the particle from the centre of the

sphere t seconds after it has left the surface of the sphere is 5 .10

kt+

Find the value of k correct to two places of decimals.

( )

( )

( )( )

212

2

2

2

2

1

(i) 0·3 0·3cos

0·6 1 cos

cos0·3

0 0·3 cos

0·3 cos 0·6 1 cos

cos 2 1 cos2cos3

0·3 cos 0·21·4 m s

(ii) 0·3sin 1·4cos

5 20·3 1·43 3

mv mg

v g

mvmg R

R v g

g g

v g gv

x t

α

α

α

α

α αα α

α

α

α α

−

= −

= −

− =

= =

= −

= −

=

= = =

= + ×

= × + ×

5 2·810 3

14 0·9315

t

t

k

×

= + ×

= =

30

5 5 5 5

5 5

B

Page 14

7. (a) A uniform wire ABC is bent at right angles at B. When it is suspended from B the parts AB and BC make angles of 30 and 60 respectively with the vertical. If the mass per unit length of the wire is m and BChAB = find the value of h.

12

12

2

sin 30

sin 60

1 32 2

3

1·316

mgh BC h BC

mg BC BC

h

h

× =

×

× =

=

=

20

5, 5 5

5

60 30

A

B C

Page 15

7. (b) Two rough rings of equal weight W are

a distance d apart on a horizontal rod. A light smooth inelastic string of length 2 connects the rings. Another ring of weight 2W slides on the string. The coefficient of friction between the rough rings and the rod is .μ

Show that the system remains at rest if 2

4 .1 4

d μμ

<+

( )

( )

1

22

1 1

22

22

2 2 2 2 2

2 2 2 2

2

2

2 cos 2cos

cos2

2 2 4

2 22 2 4

44

16 4

1 4 16

41 44

1 4

T WT W

R W TW

d d dR W R

d d dW W W

dd

d d

d

d

d

θθ

θ

μ

μ

μ

μ μμ μ

μμ

μμ

==

= +=

= + −

= + −

= −

= −

+ =

=+

<+

5 5

5, 5 5 5

30

d

μR1

W

d

μR2

2W

W

R1 R2

θ

θθ

T

T

T T

Page 16

8. (a) Prove that the moment of inertia of a uniform circular disc, of mass m and

radius r, about an axis through its centre perpendicular to its plane is 21 .2

m r

{ }

{ }

221

4

0

4

0

3

2

2 M

4 M 2

d M 2 disc theof inertia ofmoment

d 2M element theof inertia ofmoment

d 2M element of mass

areaunit per mass MLet

rm

r

x

xx

xxx

xx

r

r

=

=

=

=

=

=

=

π

π

π

π

π

5 5 5 5

20

Page 17

8. (b) A string is wrapped around a smooth pulley wheel of radius r. A particle of mass m is attached to the string.

The axis of rotation of the wheel is horizontal, perpendicular to the wheel, and passes through the centre of the wheel.

The moment of inertia of the wheel about the axis is I. The particle is released from rest and moves vertically downwards. (i) Find, in terms of I, m and r, the tension in the string.

(ii) If the acceleration of the particle is ,5g find the mass of the pulley

wheel in terms of m.

( )

mMMrmr

mrImr

gmrI

mgr f

mrImgIT

mrImrmg

mfmgTmfTmg

mrImgr

mrImgf

mgmrIf

mghfhmrfhI

fhfsuv

mghmvrvI

mghmvI

845

5 (ii)

1

22

202

(i)

2212

22

2

2

2

2

2

2

2

2

2

21

221

22

221

2

2

21

2212

21

=

=

+=

=+

=

+=

+−=

−==−

+=

+=

=

+

=+

+=+=

=+

=+ω

30

5 5 5 5 5 5

m

Page 18

9. (a) Stainless steel is an alloy of iron, chromium and nickel. A piece of stainless steel consists of 70% iron, 20% chromium and 10% nickel by volume.

The relative densities of iron, chromium and nickel are 7·8, 7·2 and 8·9

respectively.

Find the relative density of stainless steel.

7800 0·7 7200 0·2 8900 0·1

7790

7·79

I C N SSm m m m

V V V V

s

ρ

ρ

+ + =

× + × + × =

=

=

5 5, 5 5

20

Page 19

9 (b) A uniform rod, of length 2 m and weight W, is freely hinged at a point P. The rod of relative density 0·756 is free to move about a horizontal axis through P. The other end of the rod is immersed in a liquid of relative density 0·9. The point P is 0·4 m above the surface of the liquid. The rod is in equilibrium and is inclined at an angle of θ to the vertical. Find (i) the length of the immersed part of the rod (ii) the value of .θ

2

2

2

0·9(i) 0.756

1sin 2 sin2

0·9 20·756 2

1·68 22

4 3·36 0

1·2 m.

(ii)

x

x

W B

xW B

W xW

xx

x x

x

θ θ

×=

× = × −

× = −

= −

− + =

=

0·4 1 cos 2 1·2 2

60

θ

θ

= =−

= °

30

5 5, 5 5 5 5

P

θ 0·4 m

P

θ 0·4 m B

W x

Page 20

10. (a) Newton’s law of cooling states that ‘the rate of cooling of a body is proportional to the difference between the temperature of a body and the temperature of its surroundings.’ If θ is the difference between the temperature of a body and the temperature of its surroundings then

.d kdtθ θ= −

A body cools from 80 C to 60 C in 10 minutes. The temperature of the surroundings is maintained at 20 C. Find (i) the value of k

(ii) the temperature of the body after a further 15 minutes.

[ ] [ ]

40 10

60 0

40 10

60 0

(i)

1

1

ln

ln 40 ln 60 106010 ln40

1 ln1·5 0·0405

10

(ii)

d kdt

d k dt

d k dt

kt

k

k

k

θ θ

θθ

θθ

θ

= −

= −

= −

= −

− = −

=

= =

[ ] [ ] 20 15

40 0

0·6075

ln

20ln 15 40

2040

20 21·7884

41·8

θ kt

k

e

θθ

θ

θ

θ

−

−

= −

− = −

− =

− =

= °

5

5 5 5

5

25

Page 21

10 (b) A particle of mass m is fired horizontally through a block of resistive gel. The resistance to motion is mkv2 N when v m s–1 is the speed. The particle enters

the gel at a speed of 1000 m s–1 and 1100

seconds later exits the gel at a speed

of 10 m s–1.

(i) Show that k = 99 .10

(ii) Use the fact that 1 ln(9900 1)

9900 1 9900dt t C

t= + +

+ or otherwise to

show that the length of the block of gel is 10 ln100 m.99

[ ]

2

10 0·01

2 1000 0

10 0·1

0 1000

( )

1

1

1 1 0·0110 1000

9910

( )

dvi kvdt

dv k dtv

ktv

k

k

ii

= −

= −

− = − − + = − ×

=

[ ]

2

2

10

1000 0

10

1000 0

9910

1 9910

99ln 10

10 99ln 1000 10

10 ln10099

x

x

dvv kvdx

v

dv dxv

xv

x

x

= −

= −

= −

= −

= −

=

5 5 5 5

5

25

Page 22

Page 23

Page 24