Group-A
(Answer any two out of three)
1.(a) Solve the following homogeneous linear equation :
(i)
3
(ii)
4
(iii)
( 6.5 + 6)
+ 2y = 0.
+ 6y = 0.
+ y = 0.
(b) Solve the following Bernoulli differential equation :
2. (a) Evaluate the initial value problem
(b) Solve the following equation (i)
3. Solve
6
2k
+ y = xy3.
+ 25y = 0. y'(0) = -3 , y(0) = -1 (6 + 6.5)
+ k2y =
(ii)
= tanx with variation of parameter.
+ y = cosx
12.5
Group-B
(Answer any one out of two)
4. Solve the following equations:
(i) (D2 + D +1) y = sin2x
(ii) (D2 – 3D + 2) y =
(iii) (D2 9)y = 6
+x
(4 + 4 + 4.5)
5. (a) Define gradient , divergence and curl.
(2 + 4 + 6.5)
(b) Find the projection of the vector ⃗ = ̂ 2 ̂ + ̂ on the vector ⃗⃗ = 4 ̂ 4 ̂ + 7 ̂ .
(c) A particle moves along a curve whose parametric equations are x =
, y = 2cos3t , z = 2sin2t
(i) Determine it’s velocity & acceleration.
(ii) Find the magnitude of the velocity and acceleration at t = 0.
Group-C
(Compulsory)
6. (a) If
(x,y,z) = xy2z and ⃗ = xz ̂
xy2 ̂ + yz2 ̂ . Find
(4.5 + 4 +4)
( ⃗ ) at the point (2, 1, 1 ) .
(b) If ⃗ = x2z ̂
2y3z2 ̂ + xy2z ̂ . Find div ⃗ at the point (1,
,1)
(c) If ⃗⃗ = xz3 ̂
2x2yz ̂ + 2yz4 ̂ . Find curl ⃗⃗ at the point (1,
,1)