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 2013

Marking Scheme

Applied Mathematics

Higher Level

Note to teachers and students on the use of published marking schemes

Marking schemes published by the State Examinations Commission are not intended to be standalone documents. They are an essential resource for examiners who receive training in the correct interpretation and application of the scheme. This training involves, among other things, marking samples of student work and discussing the marks awarded, so as to clarify the correct application of the scheme. The work of examiners is subsequently monitored by Advising Examiners to ensure consistent and accurate application of the marking scheme. This process is overseen by the Chief Examiner, usually assisted by a Chief Advising Examiner. The Chief Examiner is the final authority regarding whether or not the marking scheme has been correctly applied to any piece of candidate work. Marking schemes are working documents. While a draft marking scheme is prepared in advance of the examination, the scheme is not finalised until examiners have applied it to candidates’ work and the feedback from all examiners has been collated and considered in light of the full range of responses of candidates, the overall level of difficulty of the examination and the need to maintain consistency in standards from year to year. This published document contains the finalised scheme, as it was applied to all candidates’ work.

In the case of marking schemes that include model solutions or answers, it should be noted that these are not intended to be exhaustive. Variations and alternatives may also be acceptable. Examiners must consider all answers on their merits, and will have consulted with their Advising Examiners when in doubt.

Future Marking Schemes

Assumptions about future marking schemes on the basis of past schemes should be avoided. While the underlying assessment principles remain the same, the details of the marking of a particular type of question may change in the context of the contribution of that question to the overall examination in a given year. The Chief Examiner in any given year has the responsibility to determine how best to ensure the fair and accurate assessment of candidates’ work and to ensure consistency in the standard of the assessment from year to year. Accordingly, aspects of the structure, detail and application of the marking scheme for a particular examination are subject to change from one year to the next without notice.

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 1

Page 2

1. (a) A ball is thrown vertically upwards with a speed of 44·1 .s m 1

Calculate the time interval between the instants that the ball is 39·2 m above the point of projection.

s 718

8 ,1

081

089

8·91·449·23

1

2

221

221

t

tt

tt

tt

tt

atuts

5 5,5 5

20

Page 3

1. (b) A lift ascends from rest with constant acceleration f until it reaches a speed v. It continues at this speed for 1t seconds and then decelerates uniformly to rest with deceleration f.

The total distance ascended is d, and the total time taken is t seconds.

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

(ii) Show that 121 ttfv .

(iii) Show that fdtt 42

1 .

(i)

fdtt

ttfd

tt fttd

vttt

vttdvttvtvttd

tt fvtt

vf

4

4

or (iii)

(ii)

21

21

2

121

121

1121

21

121

141

1141

121

121

5 5

5 5

5 5

30

121 tt

121 tt

1t

v

Page 4

2. (a) Two cars, A and B, travel along two straight roads which intersect at an angle . Car A is moving towards the intersection at a uniform speed of 9 1s m . Car B is moving towards the intersection at a uniform speed of 15 1s m .

At a certain instant each car is 90 m from the intersection and approaching the intersection. (i) Find the distance between the cars when B is at the intersection. (ii) If the shortest distance between the cars is 36 m, find the value of .

.13·53.159cos

0cos159

sin15 cos159

sin15 5cos1

9 (ii)

m 361590990 (i)

1

ABV

jiVVV

jiV

iV

AB

AB

BAAB

B

A

5 5 5 5 5 5

25

B

Blank Page

A

Page 5

2 (b) An aircraft P, flying at 600 ,h km 1 sets out

to intercept a second aircraft Q, which is a distance away in a direction west 30 south, and flying due east at 600 .h km 1

Find the direction in which P should fly in order to intercept Q.

W30 Sor S 60 W 6021cos

2cos2cos401cos2coscos131cos2cosin3

1cossin3

600cos600

sin60030tan

sin600 600cos600

600

sin600 cos600

2

22

22

s

ji

VVV

iV

jiV

QPPQ

Q

P

5 5 5 5 5 25

P 30

600

Q

600

Page 6

3. (a) A particle is projected from a point on horizontal ground.

The speed of projection is u 1s m at an angle to the horizontal. The range of the particle is R and the maximum height reached by the particle is

.34

R

(i) Show that g

uR cossin2 2

.

(ii) Find the value of .

gu

guu

tuR

gut

gttu

cossin2

sin2cos

.cos

sin20.sin (i)

2

221

303

1tan

sinsin32

cos

sin2

sinsin34cossin2

.sin34

sin (ii)

21

22

212

11

1

gug

guu

gu

gttuRg

ut

5 5

5 5 5 5

25

Page 7

3 (b) A plane is inclined at an angle 21tan 1 to the horizontal.

A particle is projected up the plane with initial speed u 1s m at an angle to the inclined plane. The plane of projection is vertical and contains the line of greatest slope.

Find the value of that will give a maximum range up the inclined plane.

7·31

22tan 2sin2cos2

0cossin22cos2 0

cossin22cos22

5

sin2sin2

5

sinsincos5

sincos

sin5cossin2

0 cossin

0

2

22

221

2

221

221

ddR

gu

ddR

gu

gu

tgtuR

gu

gut

tgt u

rj

5 5 5 5 5 25

Page 8

4. (a) Two particles of masses 6 kg and 7 kg are connected by a light inextensible string passing over a smooth light fixed pulley which is fixed to the ceiling of a lift. The particles are released from rest. Find the tension in the string

(i) when the lift remains at rest (ii) when the lift is rising vertically with constant

acceleration .8g

71·24or 26

1898

666

1049

8

66

8

77 (ii)

63·32or 13

8466

13

66 77 (i)

gT

gfgT

gf

gf g T

gf T g

gT

fgT

gf

f g Tf T g

5

5 5 5 5

25

6 7 6

Page 9

4 (b) A light inextensible string passes over a smooth

fixed pulley, under a movable smooth pulley of mass 3m , and then over a second smooth fixed pulley. A particle of mass 1m is attached to one end of the string and a particle of mass 2m is attached to the other end. The system is released from rest.

Find the tension in the string in terms of . and , 321 mmm

321

213231

321

213231321

321313221321

21

33

21

33

22

11

4114or

44

44

242

22

2

mmm

g

mmmmmmgmmmT

mmmmmmTgmmm

gmmmTmmTmmTmmgmmm

gmTg

mTmTgm

qpmTgmq mg mTp mg mT

5

5

5

5 5

25

1m 3m 2m

Page 10

5. (a) A smooth sphere A, of mass 3m, moving with speed u, collides directly with a smooth sphere B, of mass 5m, which is at rest.

The coefficient of restitution for the collision is e. Find (i) the speed, in terms of u and e, of each sphere after the collision (ii) the value of e if the magnitude of the impulse imparted to each sphere as a

result of the collision is 2mu.

151

11516

18

152

05

183

538

0 NEL

53(0)53 PCM

2

2

1

21

21

e

emu mu

emu mu

mv I

eu v

eu v

u e v vmvmv m um

20 5

5

5

5 5

Page 11

5 (b) A ball is dropped on to a table and it rises after impact to one-quarter of the height of the fall.

(i) Find the value of the coefficient of restitution between the ball and the table.

If sheets of paper are placed on the table the coefficient of restitution decreases by a factor proportional to the thickness of the paper. When the thickness of the paper is 2·5 cm it rises to only one-ninth of the height of the fall. (ii) Find the value of the coefficient of restitution between the ball and this

thickness of paper. (iii) What thickness of paper is required in order that the rebound will be one-

sixteenth of the height of the fall?

cm. 3·33or 3

10

416·0

21

6·0

315·2

215·2

41

16220

2 (iii)

31

9220

2 (ii)

21

4220

2

2

20

2 (i)

2

1

2

22

22

22

1

12

11

22

2

22

2

22

x

xexke

k

keke

e

hgghe

asuv

e

hgghe

asuv

e

hgghe

asuv

ghv

ghv

asuv

5 5 5 5 5 5

30 5

Page 12

6. (a) A rectangular block of wood of mass 20 kg and height 2 m floats in a liquid. The block experiences an upward force of 400d N, where d is the depth, in metres, of the bottom of the block below the surface. Find

(i) value of d when the block is in equilibrium (ii) the period of the motion of the block if it is pushed down 0·3 m from the

equilibrium position and then released.

s. 4·1or 5

5222

52or 20

20

40040020 )(

49·0

20400 )(

T

xmFa

xxdg Fii

d

gd i

5 5 5 5

20 5

2 m

d 2 m

d+x

400 d

20 g

400 (d+x)

20 g

Page 13

6 (b) A vertical rod BA, of length 4l, has one end B fixed to a horizontal surface with the other end A vertically above B. The ends of a light inextensible string, of length 4l, are fixed to A and to a point C, a distance 2l below A on the rod. A small mass m kg is tied to the mid-point of the string. It rotates, with both parts of the string taut, in a horizontal circle with uniform angular velocity . (i) Find the tension in each part of the string in terms of m, l and . (ii) At a given instant both parts of the string are cut.

Find the time (in terms of l ) which elapses before the mass strikes the horizontal surface.

gt

gt

atut s

gmT

gmT

mTT

m

mrTT

mgTT

mgTT

6

03

(ii)

2

3

60sin60sin

2

60cos60cos

60 (i)

221

221

22

21

221

2

221

21

21

30 5

5 5 5

5

5 5 5

4l

A

B

C

4l

A

B

C

α T1

T2 mg

Page 14

7. (a) Two forces 5 N and 12 N are inclined at an angle as shown in the diagram.

They are balanced by a force of 15 N.

Find the acute angle .

18·62

82·117

4667·012056cos

cos12025144225

cos512251215 222

20 5

5, 5 5 5 5

15 N 12 N

5 N

15

12

5

Page 15

7. (b) Two uniform rods AB and BC, of length 1 and weight W, are hinged at B and rest in equilibrium on a smooth horizontal plane.

A weight W is attached to AB at a distance b from A as shown in the diagram. A light inextensible string AC of length 2q prevents the rods from slipping.

(i) Find the reaction at A and the reaction at C.

(ii) Show that the tension in the string is 2

1

2 1

q b W

q

.

2

2

2

21

2

1

21

2

23

21

2

23

21

2

121

12

1

122

2

sin (ii)

24

3

22

2

coscos2 (i)

qqbWT

qTqbW

qTWqqbW

TqWqR

bWR

WRR

bWR

bqWqWqWqR

qbWqWqWq R

5 5 5 5 5 5 5

30 5

A

B

C 2q

W b

W W

R1 R2

α

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

mr

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

5 5 5 5

20 5

Page 17

8. (b) A uniform circular lamina, of mass 8m and radius r, can turn freely about a horizontal axis through P perpendicular to the plane of the lamina.

Particles each of mass m are fixed at four points which are on the circumference of the lamina and which are the vertices of square PQRS. The compound body is set in motion.

Find (i) the period of small oscillations of the compound pendulum

(ii) the length of the equivalent simple pendulum.

35

3522 (ii)

352

12202

2

20

22288

1228 (i)

2

2

2222221

rL

gr

gL

gr

mgrmr

MghIT

mr

rmrmrmrmrmI

mgrrmgrmgrmgrmgMgh

30 5

5,5 5,5 5 5 5

P Q

R S

P

Q

R

S

8mg mg

mg

mg

mg

Page 18

9. (a) 1V cm3 of liquid A of relative density 0·8 is mixed with 2V cm3 of liquid B of relative density 0·9 to form a mixture of relative density 0·88.

The mass of the mixture is 0·44 kg.

Find the value of 1V and the value of .2V

332

331

11

21

12

12

2121

cm 400or m 0004·0

cm 100or m 0001·0

44·04880

44·0880

4

8020

880900800

V

V

VV

VV

VV

VV

VVVV

mmm MBA

5 5 5 5 5

25 5

Page 19

9 (b) Liquid C of relative density 0·8 rests on liquid D of relative density 1·2 without mixing. A solid object of density floats with part of its volume in liquid D and

the remainder in liquid C.

The fraction of the volume of the object immersed in liquid D is a

a2 .

Find the value of .a

400

400800

8001200800

1800

12001

1

1

1200800

1200008

2

121

2

21

2121

a

VVVV

V

VV

gVgVgVV

BBW DC

25 5

5 5, 5 5 5 5

Page 20

10. (a) If

072 dxdyx

and 1y when 7,x find the value of y when 14x .

(b) A particle starts from rest at O at time .0t It travels along a straight line with acceleration 224 16 m st , where t is the time measured from the instant when the particle is at O. Find

(i) its velocity and its distance from O at time 3t

(ii) the value of t when the speed of the particle is 80 .s m 1

(c) Water flows from a tank at a rate proportional to the volume of water remaining in

the tank. The tank is initially full and after one hour it is half full.

After how many more minutes will it be one-fifth full?

5·1

11417

71

14171

1 7

7

7

7 (a)

14

7

1

14

7

2

1

2

2

y

y

y

xy

dxx dy

dxxdy

dxdyx

y

y

5 5 5

5

10 5

Page 21

t

tt

tt

tt

ttvb

s

tts

dtt ds

v

ttv

dtt dv

t dtdvb

s

v

s3

10

02103

02043

161280

1612 (ii) )(

m 36

84

16t)-2(1

s m 60

1612

16)-24(

1624 (i) )(

2

2

2

30

23

3

0

2

0

1

30

2

3

0

0

min 3·791

h 322·22ln5ln

ln5

ln

693·0or 2ln

ln2

ln

1

)(

1

1

0

V

21

tt

t

k tVV

k

kVV

dtk dV V

kV dtdVc

V

5 5 5 5

20 5

5 5 5 5 5 5

20 5

Page 22

 

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Page 23

 

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