Misreading - Applied Mathematics - Exam, Exams of Applied Mathematics

Download Misreading - Applied Mathematics - Exam and more Exams Applied Mathematics in PDF only on Docsity!

Coimisiún na Scrúduithe Stáit

State Examinations Commission

LEAVING CERTIFICATE 2010

MARKING SCHEME

APPLIED MATHEMATICS

HIGHER LEVEL

1. (a) A car is travelling at a uniform speed of 14

1 m s

 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

2 2

2 2

1

2

2

2 2

or 1. 17 ms 6

(ii)

1 ms

(i) 2





f

f

f

v u fs

s

s

s ut ft

f

f

f

v u fs

1. (b) A particle passes P with speed 20

1 m s

 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

of PQ.

Find the distance from P to Q.

   

   

   

 

   

300 m

2

2

2 2 20

7

2 20

7

2

2

2 2

1

2

2 20

7

2 2

1

2

1

2 2

1

2 2

1

PQ

t

t t

t t t t

PQ t t

t t

PQ t t

PQ s ut ft

t t

PQ t t

PY s ut ft

f

f

f

PX s ut ft

P X Y Q

t =

s =

t =

s = 20 PQ

13

2 (b) When a motor-cyclist travels along a straight road from South to North at

a constant speed of 12.

1 m s

 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.

 

 

direction West

magnitude 1 2.5ms

and 12. 5 12. 5

1



V i j

x y

x y x y

V V

yi y j

V V V

V yi yj

V i j

xi x j

V V V

V xi x j

V i j

W

W W

W WM M

WM

M

W WM M

WM

M

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 9. 8 2

1 m s



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.

4. 9 5 or 10. 96 m

9. 8 2 cos 45.

9. 8 2 sin 45. 4. 9

2 2

2

2

2 2

1

PQ

r t

t

t t

t t

t gt

i



P

Q

1.1 m

9. (^82) 6 m

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

2 2

0.8ms

(ii) 2

2. 4 N

(i) 0 .25 0. 25

 

v

v

v u fs

T

f

g f

T g f

g T f

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

rests on a smooth horizontal surface.

Particles of mass 2 m 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 )

 

 

   

2 or 2. 67 ms 11

4 sin 45 sin 45 4

cos 45 sin 45

( ) 2 2 cos 45 2 sin 45

 

g f

mg mf mf

mg mf mg mf mf

m S R mf

S mg mf

m mg S mf

R mg mf

ii m mg R mf

m

2 m

4 m

S

R

4 mg

T mg 2 mg

S

R

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

velocity i j

6  2 collides with a smooth

sphere, of mass km , moving with

velocity i j

2  4 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

k .

   

 

 

2 4k

3 k

(ii) 1

2 4k

3 k

paralleldirections slopesareequal

NEL 6 2

(i) PCM 6 2

2 1

1 2

2

1

1 2

1 2

k

k k

e

e

e k k ek

k

k ek

k

e k

v v

v v

k

e k v

k

k ek v

v v e

m km mv kmv 5

m km

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.

 

   

 

1 1

2

(^21) 2 1

1

2 2

(^21) 2 1

1

1

3

2

2

1

2 2

1

2

2

8 ms

2 2 cos

( ) Totalenergyat Totalenergyat

ms 3

cos

2 cos 2 2 cos

2 2 cos

0 2 cos

() cos





v g

mg

g mv m

mv mv mg

ii Q B

g v

m g mg

mv mg

R v g

mv i mg R

P Q

B

P Q

α

R

mg

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 3 W can safely climb to

the top of the ladder if

8 tan

     

 

 

8 tan

4 tan

tan

sin cos 3 cos

momentsabout :

vertical 4

horizontal

2

2

1 2

2

1

2 1

W

W

W

R

R W W

c

R W

R W

R R

c

W

3 W

R 1

R 2

μ R 1



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.

cos in 5 4 5

3

   s 

W

W W

R W

W

R

R W

R W

R 2

R

(iii) SphereB R sin

(ii) SphereA sin

3

3

3 2

2

2

2

θ

A

B

R 2

R 4

W

W

R 1

R 2

R 3

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 2 M a  b .

(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 .

 

 

 

 

   

2 2

2 2

(^21)

2 2

2 2

(^21) 2

1

2 2

(^21) 2

1

2 2

4 4

2 2

4

1

3 1

sin 30 2

(ii) GaininKE LossinPE

(i) momentofinertiaofannulus 2

a b

ga

a Ma Mg

Ma b

a I Ma Mg

I Mv Mgh

Ma b

a b

a b

M

x M

M x dx

a

b

a

b



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

3 3

3

1250 kgm

1. 488

  

g g

W B

g

g

W Vg

g

g

B Vg