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.
