vector

420 Vectors
9. Find div F and curl F where F = grad (x 3 + y 3 + z 3 – 3xyz). (R.G.P.V. Bhopal Dec. 2003)
Ans. div F = 6(x + y + z), curl F = 0
10. Find out values of a, b, c for which v = (x + y + az) i + (bx + 3y – z) j + (3x + cy + z) k
is irrotational.
Ans. a = 3, b = 1, c = –1
11. Determine the constants a, b, c, so that F = (x + 2y + az) i + (bx – 3y – z) j + (4x + cy + 2z) k is
irrotational. Hence find the scalar potential such that F = grad .
(R.G.P.V. Bhopal, Feb. 2005) Ans. a = 4, b = 2, c = 1
Choose the correct alternative:
Potential =
12. The magnitude of the vector drawn in a direction perpendicular to the surface
x 2 + 2y 2 + z 2 = 7 at the point (1, –1, 2) is
2 2
x 3y
2
z 2xy yz 4zx
2 2
2
3
(i)
(ii) (iii) 3 (iv) 6 (A.M.I.E.T.E., Summer 2000) Ans. (iv)
3
2
13.If u = x 2 – y 2 + z 2 and V xi yj zk
then ( uV ) is equal to
(i) 5u (ii) 5| V | (iii) 5( u | |) (iv) 5( u | |) (A.M.I.E.T.E., June 2007)
14.A unit normal to x 2 + y 2 + z 2 = 5 at (0, 1, 2) is equal to
1
(i) ( )
5 i j k
1
(ii) ( )
5 i j k
1
(iii) ( 2 )
5 j
k 1
(iv) ( )
5 i j k
V
V
(A.M.I.E.T.E., Dec. 2008)
15.The directional derivative of = x y z at the point (1, 1, 1) in the direction i is:
1
(i) –1 (ii)
1
(iii) 1 (iv)
Ans. (iii)
3
3
(R.G.P.V. Bhopal, II Sem., June 2007)
16.If r xi y j zk and r = | r | then (r) is:
(i) (r) r
(ii)
()
r r
r
(iii)
()
r r
r
(iv) None of these
Ans. (iii)
(R.G.P.V. Bhopal, II Semester, Feb. 2006)
17. If
r = xi yj zk is position vector, then value of (log r) is (U.P., I Sem, Dec 2008)
r
r
(i)
(ii)
(iii) – r (iv) none of the above. Ans. (ii)
2
3
r
r
r
18. If r xi y j zk and | r |
= r, then div r is:
(i) 2 (ii) 3 (iii) –3 (iv) –2 Ans. (ii)
(R.G.P.V. Bhopal, II Semester, Feb. 2006)
2
2
2
19. If V xy i 2yx zj 3yz k then curl V at point (1, –1, 1) is
(i) ( j 2 k)
(ii) ( i 3 k)
(iii) ( i 2 k)
(iv) ( i 2 j k)
(R.G.P.V. Bhopal, II Semester, Feb 2006)
Ans. (iii)
20. If A is such that A = 0 then A is called
(i) Irrotational (ii) Solenoidal (iii) Rotational (iv) None of these
(A.M.I.E.T.E., Dec. 2008)
21. If F is a conservative force field, then the value of curl F is
(i) 0 (ii) 1 (iii) F (iv) –1 (A.M.I.E.T.E., June 2007)