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# 1. a) Determine a unit vector perpendicular to the plane of A = ˆi − 2ˆj + kˆrand B = 2ˆi + 3ˆj − kˆr.(5)b) r1rand r2rare unit vectors in the x-y plane making angles a and b with the positivex-axis. By considering r1 . r2r r, derivecos (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 xzr(5)3. a) Evaluate . ( ) 3 rrÑ r . (5)b) Evaluate . (A r)r rÑ × if ( A)rÑ × is zero. (5)4. a) Evaluate  + +V(x y z )dx dy dz 2 2 2 using spherical polar coordinates where V is a spherehaving its centre at the origin and radius equal to 5 units. (5)b) Determine whether the following coordinate transformation is orthogonalx = 3u1 + u2 − u3y = u1 + 2u2 − 2u3z = 2u1 − u2 − u3 (5)5. a) A conductor along the z-axis carries a current I. The magnetic vector potential due tothis current iszIA eˆ1ln2  p rμ=rUsing cylindrical coordinates show that the magnetic induction (B A)r r= Ñ× isj   prμB = eˆ2r I(5)4b) Show that A aˆ( / )re iw t−r c=r, where aˆ is a constant vector, satisfies the vector equation:222 22 2 1r r r c ¶t¶=¶¶+¶¶ A A Ar r r(5)6. EvaluateSA dsr r.using the Divergence theorem where A 4 ˆi 2 ˆj kˆ 2 2 = x − y + zr, and S is the region bounded byx2 + 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 + zralong the curvex = 2t2, y = t, z = 4t2 − tfrom 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 probabilitythat 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 coinand 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 hasthe probability densityf (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 fromthe box at random without replacement, find the probability that x fuses in this samplewill be defective. (5)10. The modulus of rigidity of a wire isp qh =42rLNThe following measurements are made for L, r and q / Nr = 1.2 ± 0.05 mmL = 400 ± 2 mmNq= 5.00 ± 0.20 rad N−1 m−1Obtain the best value of h. Arun Kumar IIT Delhi
7 years ago
vector perpendicular to plane is coefficients of x y z in the plane equation.

Thanks & Regards
Arun Kumar
IIT Delhi