Leaving Certificate 2011 Applied Mathematics Marking Scheme, Exams of Applied Mathematics

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Coimisiún na Scrúduithe Stáit

State Examinations Commission

Leaving Certificate 2011

Marking Scheme

Ordinary Level

APPLIED MATHEMATICS

1. The points P and Q lie on a straight level road. A car passes P with a speed of 10 m s^1 and accelerates uniformly for 6 seconds to a speed of 22 m s^1. The car then decelerates uniformly to a speed of 18 m s^1 and travels 80 m during this deceleration. The car now maintains a constant speed of 18 m s^1 for 3 seconds and then passes Q. Find (i) the acceleration (ii) the deceleration (iii) | PQ |, the distance from P to Q (iv) the average speed of the car, correct to one decimal place, as it moves from P to Q.

17. 7 m s

(iv)

230 m

183 0 54 m

80 m

96 m.

(iii)

1 ms

(ii) 2

2 ms

(i)

1

2

3

2 2 1

2

1

2

1 1

2 2

1

2

2 2

2 2

2







t

s v

f

v u t

PQ

s

s ut at

s

s

s

s ut at

f

f

v u fs

f

f

v u ft

2. Ship A is 126 km due west of ship B. A is moving at a constant speed of 50 km h^1

in the direction east α north where tan α = 7

B is moving due north at a constant speed of 48 km h^1.

Find (i) the velocity of A in terms of

 i and

 j

(ii) the velocity of B in terms of

 i and

 j

(iii) the velocity of A relative to B in terms of

 i and

 j.

Ship A intercepts ship B after t hours.

Find (iv) the value of t (v) the distance each ship travels in this time t.

   

432 km

B 48 9

450 km

(v) A 50 9

9 h

(iv)

14 i 0 j

14 i 48 j 0 i 48 j

(iii) V V V

(ii) V 0 i 48 j

14 i 48 j

(i) V 50 cos i 50 sin j

AB A B

B

A

s

s

t

^ 

α A B

km h^1 48 km h^1

4. (a) A particle of mass 3 kg is connected to a particle of mass 5 kg by a taut, light, inextensible string which passes over a smooth light pulley at the edge of a rough horizontal table.

The coefficient of friction between the 3 kg mass and the table is 3

The system is released from rest.

(i) Show on separate diagrams the forces acting on each particle. (ii) Find the common acceleration of the particles. (iii) Find the tension in the string.

(i)

(ii)

ms^2 4

or 8

^ 

g a

g g a

g R a

T R a

g T a

(iii)

31. 25 N

T  g  a

3 kg

5 kg

T

5 g

T

R

μR

3 g

4 (b) Masses of 8 kg and 2 kg are connected by a light inelastic string which passes over a smooth light pulley as shown in the diagram.

The 8 kg mass lies on a smooth plane which is inclined at 30  to the horizontal.

The 2 kg mass hangs vertically.

The system is released from rest.

(i) Find the common acceleration of the masses. (ii) Find the tension in the string.

(i)

2 ms^2

8 cos 60 8

^ 

a

a

g g a

g T a

T g a

(ii)

24 N

T  g  a 25

R 1

8 g

T T

2 g

2 kg

8 kg

6. (a) Particles of weight 5 N, 8 N, 3 N and 1 N are placed at the points (4, 1), (–3, p ), ( p , q ) and (15, 4), respectively.

The co-ordinates of the centre of gravity of the system are (1, p ).

Find (i) the value of p (ii) the value of q.

(b) A quadrilateral lamina has vertices A , B , C and D.

The co-ordinates of the vertices are A (0, 0), B (18, 0), C (6, 3), D (0, 9).

Find the co-ordinates of the centre of gravity of the lamina.

(b) 18 3 27 8, 1

(a) 1

2 1

2 1

y

y

x

x

ABCD x, y

ACD

ABC

q

q

p

p (^) 10

A

D

C

B

7. (a) A uniform beam, AB , is held in a horizontal position by two vertical inelastic strings attached at points C and D respectively.

The weight of the beam is 45 N. The length of the beam is 2 m.

A particle of weight 10 N is attached at A and a particle of weight 15 N is attached at B. | AC | = | BD | = 2

m.

Calculate the tension in each of the strings.

       

1 2

1 2

2 3 2 2 1 1

1 2

T T

T T

T T

T T

A B

C D

A B

C D

T 1 T 2

8. (a) A particle describes a horizontal circle of radius 2 m with uniform angular velocity ω radians per second. Its speed is 8 m s^1. Find (i) the acceleration of the particle (ii) the time taken to complete one revolution.

(b) A smooth particle of mass 3 kg is attached by a light inelastic string to a fixed point P. The particle describes a horizontal circle of radius 0·5 m on the smooth surface of a horizontal table.

The centre of the circle is vertically below P.

The string makes an angle α with the vertical, where tan α = 3

The speed of the particle is 2 m s^1. Find (i) the tension in the string (ii) the reaction force between the particle and the table.

(a)

s 2

(ii)

32 ms

(i)

2

2



T

a r

v r

(b)

30 12 N

(ii) cos 3

30 N

(i) sin

(^12)

2

 ^ 

R R

R T g

T

T

r

mv T

P

α

9. (a) A solid sphere, of radius 14 cm, floats at rest in water. 75% of the sphere lies below the surface of the water. Find the weight of the sphere, correct to the nearest Newton.

(b) A solid cone of radius 10 cm and height 12 cm has relative density 7. It is completely immersed in a liquid of relative density 0·9. The cone is held at rest by a light inelastic vertical string which is tied to a fixed support. The upper surface of the cone is horizontal. Find the tension in the string, correct to the nearest Newton.

[ Density of water = 1000 kg m^3 ].

(a)

    86 N

3 3

4

4

3

W

W Vg

s

W

s

W

B W

(b)     

     

77 N

2 3 1

2 3

1

T

T B W

W

B

water