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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.


Grade:12th Pass

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
7 years ago
I'll answer first question.
vector perpendicular to plane is coefficients of x y z in the plane equation.



Thanks & Regards
Arun Kumar
IIT Delhi
Askiitians Faculty

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