1. a) Determine a unit vector perpendicular to the plane of A = ˆi − 2ˆj + kˆ r and B = 2ˆi + 3ˆj − kˆ r . (5) b) r1 r and r2 r are unit vectors in the x-y plane making angles a and b with the positive x-axis. By considering r1 . r2 r r , derive cos (a − b) = cos a cos b + sin a sin b (5) 2. a) Determine the unit vector normal to the surface (x − 2)2 + (y + 1)2 + z2 = 9 at the point (2, 1, 4). (5) b) For what value of a is the following vector field irrotational: A ( ) ˆi ( 2) ˆj (1 ) kˆ 3 2 2 = axy − z + a − x + − a xz r (5) 3. a) Evaluate . ( ) 3 r r Ñ r . (5) b) Evaluate . (A r) r r Ñ × if ( A) r Ñ × is zero. (5) 4. a) Evaluate


+ + V (x y z )dx dy dz 2 2 2 using spherical polar coordinates where V is a sphere having its centre at the origin and radius equal to 5 units. (5) b) Determine whether the following coordinate transformation is orthogonal x = 3u1 + u2 − u3 y = u1 + 2u2 − 2u3 z = 2u1 − u2 − u3 (5) 5. a) A conductor along the z-axis carries a current I. The magnetic vector potential due to this current is z I A eˆ 1 ln 2       p r μ = r Using cylindrical coordinates show that the magnetic induction (B A) r r = Ñ× is j        pr μ B = eˆ 2 r I (5) 4 b) Show that A aˆ ( / ) r e iw t−r c = r , where aˆ is a constant vector, satisfies the vector equation: 2 2 2 2 2 2 1 r r r c ¶t ¶ = ¶ ¶ + ¶ ¶ A A A r r r (5) 6. Evaluate

S A ds r r . using the Divergence theorem where A 4 ˆi 2 ˆj kˆ 2 2 = x − y + z r , and S is the region bounded by x2 + y2 = 4, z = 0 and z = 3. (10) 7. Determine the work done in moving a particle in the force field F 3 ˆi (2 )ˆj kˆ 2 = x + xz − y + z r along the curve x = 2t2, y = t, z = 4t2 − t from t = 0 to t = 1. (10) 8. a) Two dice are thrown and it is known that the first dice shows a 6. Find the probability that the sum of numbers showing on the dice is 7. (5) b) Three coins are tossed in a game. Let E be the event that a head appears on the first coin and F be the event that a tail appears on the third throw. Are E and F independent? (5) 9. a) X is the amount of petrol sold in thousands of litres, in a petrol pump every week. X has the probability density f (x) = 6x (1 − x) for 0 £ x £ 1 = 0 otherwise. Find the mean and variance. (5) b) A box contains 20 fuses, of which 5 are defective. If a sample of 3 fuses is chosen from the box at random without replacement, find the probability that x fuses in this sample will be defective. (5) 10. The modulus of rigidity of a wire is p q h = 4 2 r LN The following measurements are made for L, r and q / N r = 1.2 ± 0.05 mm L = 400 ± 2 mm N q = 5.00 ± 0.20 rad N−1 m−1 Obtain the best value of h.

1. a) Determine a unit vector perpendicular to the plane of A = ˆi − 2ˆj + kˆ

r

and B = 2ˆi + 3ˆj − kˆ

r

.

(5)

b) r1

r

and r2

r

are unit vectors in the x-y plane making angles a and b with the positive

x-axis. By considering r1 . r2

r r

, derive

cos (a − b) = cos a cos b + sin a sin b (5)

2. a) Determine the unit vector normal to the surface (x − 2)2 + (y + 1)2 + z2 = 9 at the point

(2, 1, 4). (5)

b) For what value of a is the following vector field irrotational:

A ( ) ˆi ( 2) ˆj (1 ) kˆ 3 2 2 = axy − z + a − x + − a xz

r

(5)

3. a) Evaluate . ( ) 3 r

r

Ñ r . (5)

b) Evaluate . (A r)

r r

Ñ × if ( A)

r

Ñ × is zero. (5)

4. a) Evaluate


+ +

V

(x y z )dx dy dz 2 2 2 using spherical polar coordinates where V is a sphere

having its centre at the origin and radius equal to 5 units. (5)

b) Determine whether the following coordinate transformation is orthogonal

x = 3u1 + u2 − u3

y = u1 + 2u2 − 2u3

z = 2u1 − u2 − u3 (5)

5. a) A conductor along the z-axis carries a current I. The magnetic vector potential due to

this current is

z

I

A eˆ

1

ln

2

 



 



p r

μ

=

r

Using cylindrical coordinates show that the magnetic induction (B A)

r r

= Ñ× is

j  





 



pr

μ

B = eˆ

2

r I

(5)

4

b) Show that A aˆ

( / )

r

e iw t−r c

=

r

, where aˆ is a constant vector, satisfies the vector equation:

2

2

2 2

2 2 1

r r r c ¶t

¶

=

¶

¶

+

¶

¶ A A A

r r r

(5)

6. Evaluate

S

A ds

r r

.

using the Divergence theorem where A 4 ˆi 2 ˆj kˆ 2 2 = x − y + z

r

, and S is the region bounded by

x2 + y2 = 4, z = 0 and z = 3. (10)

7. Determine the work done in moving a particle in the force field F 3 ˆi (2 )ˆj kˆ 2 = x + xz − y + z

r

along the curve

x = 2t2, y = t, z = 4t2 − t

from t = 0 to t = 1. (10)

8. a) Two dice are thrown and it is known that the first dice shows a 6. Find the probability

that the sum of numbers showing on the dice is 7. (5)

b) Three coins are tossed in a game. Let E be the event that a head appears on the first coin

and F be the event that a tail appears on the third throw. Are E and F independent? (5)

9. a) X is the amount of petrol sold in thousands of litres, in a petrol pump every week. X has

the probability density

f (x) = 6x (1 − x) for 0 £ x £ 1

= 0 otherwise.

Find the mean and variance. (5)

b) A box contains 20 fuses, of which 5 are defective. If a sample of 3 fuses is chosen from

the box at random without replacement, find the probability that x fuses in this sample

will be defective. (5)

10. The modulus of rigidity of a wire is

p q

h =

4

2

r

LN

The following measurements are made for L, r and q / N

r = 1.2 ± 0.05 mm

L = 400 ± 2 mm

N

q

= 5.00 ± 0.20 rad N−1 m−1

Obtain the best value of h.