leaving certificate 2010 marking scheme … material/appliedmaths/pdf/higher... · - award the...

Coimisiún na Scrúduithe Stáit State Examinations Commission

LEAVING CERTIFICATE 2010

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.

1. (a) A car is travelling at a uniform speed of 14 1s m when the driver notices a traffic light turning red 98 m ahead. Find the minimum constant deceleration required to stop the car at the traffic light,

(i) if the driver immediately applies the brake

(ii) if the driver hesitates for 1 second before applying the brake.

2

2

2

22

221

2

2

22

s m 17.1or 6

7168

196

168140

14982140

2

14

0114

(ii)

s m 1

196196

982140

2 (i)

f

f

f

fs uv

s

s

ftuts

f

f

f

fs uv

5 5

5 5 20

1. (b) A particle passes P with speed 20 1s m and moves in a straight line to Q with uniform acceleration.

In the first second of its motion after passing P it travels 25 m.

In the last 3 seconds of its motion before reaching Q it travels 20

13 of PQ .

Find the distance from P to Q.

m 300

65620

2

062018065

25305160605

25305

160605

45420

25305

15120

10

5

112025

2

2

22207

2207

2

2

221

2

2

207

221

21

2

21

221

PQ

t

tt

tttt

ttPQ

tt

ttPQ

ftutsPQ

tt

ttPQ

ftutsPY

f

f

f

ftutsPX

5

5

5

5

5

5 30

P Q YX

t=1 s=25

t=3 s= PQ20

13

2. (a) Two particles, A and B, start initially from points with position vectors

ji

14 6 and ji

2 3 respectively. The velocities of A and B are constant

and equal to ji

3 4 and ji

7 5 respectively.

(i) Find the velocity of B relative to A.

(ii) Show that the particles collide.

South 75.58East directionor

4 slope

4 3or 12 3

2 3

14 6 (ii)

South 75.58East directionor

4 slopeor s m 17 magnitude

4

7 5

3 4 (i)

1

jiji

RRR

jiR

jiR

ji

VVV

jiV

jiV

BAAB

B

A

ABBA

B

A

The particles collide

5

5 5

5 20

2 (b) When a motor-cyclist travels along a straight road from South to North at a constant speed of 12.5 1s m the wind appears to her to come from a direction North 45 East.

When she returns along the same road at the same constant speed, the wind appears to come from a direction South 45 East.

Find the magnitude and direction of the velocity of the wind.

Westdirection

s m 2.51 magnitude

0 5.12

5.12

5.125.12 and

5.12

5.12 0

5.12

5.12 0

1

jiV

yx

yxyx

VV

jyiy

VVV

jyiyV

jiV

jxix

VVV

jxixV

jiV

W

WW

MWMW

WM

M

MWMW

WM

M

5

5

5 5

5

5 30

3. (a) In a room of height 6 m, a ball is projected from a point P. P is 1.1 m above the floor.

The velocity of projection is 28.9 1s m at an angle of 45 to the horizontal.

The ball strikes the ceiling at Q without first striking a wall. Find the length of the straight line PQ.

m 96.10or 59.4

9.48.9

8.9

.45cos28.9

1

012

09.48.99.4

9.4.45sin28.9

22

2

2

221

PQ

tr

t

tt

tt

gtt

i

5

5

5

5 20

P

Q

1.1 m

6 m 28.9

3 (b) A particle is projected up an inclined plane with initial speed 80 1s m . The line of projection makes an angle of 30 with the inclined plane and the

plane is inclined at an angle to the horizontal. The plane of projection is vertical and contains the line of greatest slope.

The particle strikes the plane at an angle of3

2tan 1 .

Find (i) the value of

(ii) the speed with which the particle strikes the plane.

1

22

221

s m 52.9or 720

40320 speed

40

320 (ii)

4.23 4

3tan

tan80340

40

3

2

tan

40

cos

80cos30sin80

tan80340

cos

80sin30cos80

cos

80

cos30sin80 0

0 (i)

v

v

v

v

gg v

gg v

g t

.tg.t

r

j

i

i

j

j

i

j

5 5

5

5

5 5 30

4. (a) Two particles of masses 0.24 kg and 0.25 kg are

connected by a light inextensible string passing over a small, smooth, fixed pulley.

The system is released from rest.

Find (i) the tension in the string

(ii) the speed of the two masses when the 0.25 kg mass has descended 1.6 m.

1

22

sm 0.8

64.0

6.12.020

2 (ii)

N 4.2

2.0

49.001.0

24.024.0

25.0.250 (i)

v

v

fs u v

T

f

f g

f g T

f T g

5

5

5 15

0.25 0.24

4 (b) A smooth wedge of mass 4m and slope 45º

rests on a smooth horizontal surface. Particles of mass 2m and m are placed on the smooth inclined face of the wedge. The system is released from rest. (i) Show, on separate diagrams, the forces acting on the wedge

and on the particles. (ii) Find the acceleration of the wedge.

(i)

2sm 67.2or 11

3

833

42

1

445sin45sin 4

2

1

45sin45cos

2

45sin245cos2 2 )(

gf

mfmfmg

mfmfmgmfmg

mf RSm

mfmgS

mfS mg m

mfmgR

mf R mgm ii

5

5

5 5

15 35

45º

m

2m

4m

45º

S

R

4mg

T mg 2mg

SR

5. (a) A sphere, of mass m and speed u, impinges directly on a stationary sphere of mass 3m. The coefficient of restitution between the spheres is e.

(i) Find, in terms of u and e, the speed of each sphere after the collision.

(ii) If 4

1e , find the percentage loss in kinetic energy due to the collision.

% 3.70100128

45

KEin loss Percentage

128

45

128

19 KEin Loss

128

19or

512

76

16

53

16

3after K.E.

before K.E.

16

5 and

16

4

1 )(

4

1

4

31

0 NEL

303 PCM )(

221

2

222

21

22

2

21

2

21

222

1212

1

221

21

2

1

21

21

mu

mu

mumumu

mumu

um

um

vmmv

mu

u v

u v

e ii

eu v

eu v

u e v v

mv mv) m( umi

25

5

5

5

5 5

(b) A smooth sphere, of mass m, moving with

velocity ji

2 6 collides with a smooth sphere, of mass km, moving with velocity ji

4 2 on a smooth

horizontal table.

After the collision the spheres move in parallel directions. The coefficient of restitution between the spheres is e.

(i) Find e in terms of k.

(ii) Prove that 3

1k .

3

1

423

14k2

k3

1 (ii)

4k2

k3

42623

1

4262

1

246

2

42

equal are slopes directions parallel

1

246

1

426

26 NEL

26 PCM (i)

12

21

2

1

21

21

k

kk

e

e

ekkkek

ekk

k

ke

vv

vv

k

kev

k

ekkv

e v v

kmv mv km m

5

5

5

5

5 25

km m

4

2

2

6

6. (a) A particle of mass m kg lies on the top of a

smooth sphere of radius 2 m. The sphere is fixed on a horizontal table at P.

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

Find (i) the speed of the particle at B

(ii) the speed of the particle on striking the table at Q.

11

212

121

2212

121

1

32

21

221

2

2

s m 8

3

42

3

4

cos22

at energy Total at energy Total )(

s m 3

4

cos

cos22cos2

cos22

cos2 0

2

cos )(

gv

mgg

mmv

mgmvmv

B Qii

gv

mggm

mgmv

gvR

mvR mgi

5

5

5

5 5 25

P Q

B

P Q

α

R

mg

6 (b) A particle moves with simple harmonic motion of amplitude 0.75 m. The period of the motion is 4 s. Find (i) the maximum speed of the particle

(ii) the time taken by the particle to move from the position of

maximum speed to a position at which its speed is half its maximum value.

s. 3

2 s.

3

2

3

11time

2sin

4

3

8

33

3

1

2cos

4

3

8

33

cos

8

33

4

3

216

3

16

3 )(

s m 8

3

4

3

2

2

42

4 Period )(

2222

2222

max21

1

max

t

tt

t

tax

x

x

xav

vii

av

i

25

5 5

5

5 5

7. (a) One end of a uniform ladder, of weight W, rests against a smooth vertical wall, and the other end rests on rough horizontal ground. The coefficient of friction between the ladder and the ground is μ. The ladder makes an angle with the horizontal and is in a vertical plane which is perpendicular to the wall.

Show that a person of weight 3W can safely climb to the top of the ladder if

tan8

7 .

tan8

72

7tan4

2

7tan

cos3cossin

: about moments

4

4 vertical

horizontal

2

21

2

2

1

12

WW

WR

WWR

c

WR

WR

RR

25

5

5

5 5

5

∟c

W

3W R1

R2

μ R1

7. (b) Two uniform smooth spheres each of weight W

and radius 0.5 m, rest inside a hollow cylinder of diameter 1.6 m.

The cylinder is fixed with its base horizontal.

(i) Show on separate diagrams the forces

acting on each sphere.

(ii) Find, in terms of W, the reaction between the two spheres.

(iii) Find, in terms of W, the reaction between the

lower sphere and the base of the cylinder.

in cos 54

53 s

W

WW

WR

WR

WR

WR

2R

R

sinR B Sphere (iii)

4

5

5

4

sin A Sphere (ii)

3

3

23

2

2

2

5 5 5

5

5

1.6

0.5

0.5

0.6 θ

1

A

B

R2 R4

W

W

R1 R2

R3

25

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 2

2

1rm .

221

4

0

4

0

3

2

4 M 2

4 M 2

M 2 disc theof inertia ofmoment

2M element theof inertia ofmoment

2M element of mass

areaunit per mass MLet

rm

r

x

dxx

xxdx

xdx

r

r

5

5 5

5 20

8. (b) An annulus is created when a central hole of radius b is removed from a uniform circular disc of radius a. The mass of the annulus (shaded area) is M.

(i) Show that the moment of inertia of the annulus about an axis through

its centre and perpendicular to its plane is

2

22 baM .

(ii) The annulus rolls, from rest, down an incline of 30˚. Find its angular

velocity, in terms of g, a and b, when it has rolled a distance 2

a.

22

2

212

22

2

212

21

2212

21

22

44

22

4

1

31

3

422

1

30sin2

PEin Loss KEin Gain (ii)

2

4 2

4 2

2 annulus of inertia ofmoment (i)

ba

ga

aMgaM

baM

aMgaMI

MghMvI

baM

ba

ba

M

xM

dxxM

a

b

a

b

30

5

5

5

5 5

5

a

b

9. (a) State the Principle of Archimedes.

A buoy in the form of a hollow spherical shell of external radius 1 m and internal radius 0.8 m floats in water with 61% of its volume immersed.

Find the density of the material of the shell. Principle of Archimedes

3

33

3

m kg 1250488.0

610

3

4610

3

4488.0

3

4488.0

8.03

41

3

4

3

4610

13

4

100

611000

gg

BW

g

g

VgW

g

g

VgB

5

5

5 5 5 25

1

0.8

9 (b) A uniform rod, of length 1.5 m and weight W, is freely hinged at a point P. The rod is free to move about a horizontal axis through P. The other end of the rod is immersed in water. The point P is 0.5 m above the surface of the water. The rod is in equilibrium and is inclined at an angle of 60 to the vertical. Find (i) the relative density of the rod (ii) the reaction at the hinge in terms of W.

5

25

3

5

3

3 )(

9

5

33

53

1 and

60sin4

360sin

4

5

:about moments2

12

160cos5.1

part immersed oflength )(

31

WR

WRW

WRB

W

s

W Bii

s

Ws

Ws

W

s

WB

WB

P

x

x

xi

25

5

5 5

5 5

P

60 0.5 m

10. (a) Solve the differential equation

2xyxdx

dyy

given that y = 0 when x = 0.

e y

e y

x y

C

xy

C x y

dxxdyy

y

y

yx

dx

dy

xyxdx

dyy

x

x

1

1

2

11ln

2

1

0

0 ,0

2

11ln

2

1

1

1

2

22

22

22

2

2

2

5

5 5

5 20

10 (b) The acceleration of a cyclist freewheeling down a slight hill is

22 s m 0006.012.0 v

where the velocity v is in metres per second. The cyclist starts from rest at the top of the hill. Find (i) the speed of the cyclist after travelling 120 m down the hill

(ii) the time taken by the cyclist to travel the 120 m if his average speed is 2.65 1s m .

s 3.45

12065.2

time

distance speed average )(

s m 18.5

155.10006.012.0

12.0

144.00006.012.0

12.0ln

1200006.012.0

12.0ln

0012.0

1

12012.0ln0012.0

10006.012.0ln

0012.0

1

0006.012.0ln0012.0

1

0006.012.0

0006.012.0 )(

1

144.02

2

2

2

120 0

0

2

120

0 2

0

2

tt

ii

v

ev

v

v

v

x v

dx dv v

v

v dx

dvvi

v

v

5

5 5

5

5

5 30

Marcanna Breise as ucht freagairt trí Ghaeilge

Ba chóir marcanna de réir an ghnáthráta a bhronnadh ar iarrthóirí nach ngnóthaíonn níos mó ná 75% d’iomlán na marcanna don pháipéar. Ba chóir freisin an marc bónais sin a shlánú síos. Déantar an cinneadh agus an ríomhaireacht faoin marc bónais i gcás gach páipéir ar leithligh. Is é 5% an gnáthráta agus is é 300 iomlán na marcanna don pháipéar. Mar sin, bain úsáid as an ngnáthráta 5% i gcás iarrthóirí a ghnóthaíonn 225 marc nó níos lú, e.g. 198 marc 5% = 9·9 bónas = 9 marc. Má ghnóthaíonn an t-iarrthóir níos mó ná 225 marc, ríomhtar an bónas de réir na foirmle [300 – bunmharc] 15%, agus an marc bónais sin a shlánú síos. In ionad an ríomhaireacht sin a dhéanamh, is féidir úsáid a bhaint as an tábla thíos.

Bunmharc Marc Bónais

226 11

227 – 233 10

234 – 240 9

241 – 246 8

247 – 253 7

254 – 260 6

261 – 266 5

267 – 273 4

274 – 280 3

281 – 286 2

287 – 293 1

294 – 300 0

HL BACK 4/8/10 17:34 Page 1

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