2. Answer the following questions :2x5=10
(a) Show that lim sin does not exist. (b) For what values of a, A and B are perpendicular if Ã=ai-2j+k and
B = 2ai +aj - 4k?
(c) What is a Wronskian? How is it used to find the linear dependence of two functions?
(d) Show that B is perpendicular to A,if | B|#0 and B = dà dt
(e) Evaluate using the property of Dirac delta function : [x8(x-4)dx
3. Answer following:any five questions fromthe 4x5=20
(a) What do you mean by linearly dependent and linearly independent solutions of a homogeneous equation? If y₁(x) = sin 3x and y₂(x) = cos 3x are two solutions of y'+9y = 0, then show that y₁(x) and y₂(x) are linearly independent solutions.
(b) If z(x+y) = x² + y², then show that მz dx dy 1 az əz ax ay
(c) Solve the differential equation
d²y dx2 -3 dy +2y = 0 dx
Hence find the solution for d²y -3 dy + 2y = e³x dx² dx
(d) What is directional derivative? Find the directional derivative of =x²-2y² + 4z² at (1, 1, -1) in the direction 2i+j-k.
(e) State Bayes' theorem of probability. 6 cards are drawn from a pack of 52 cards. What is the probability that 3 will be red and 3 black?S State Green's theorem in a plane. Starting from Green's theorem, show that the area bounded by a closed curve is given by1 (xdy - y dx)
4. Answer any following three questions from the 6x3=18
(a) What are complementary function and particular integral of a differential equation? Solve the differential equation
d²y 4 dy dx +4y=x²
if y(0)=0 and y'(0) = -—-.
(b) Define line integral and surface integral. Find the total work done in moving a particle in a force field given by F=3xyi-5zj+10xk along a curve x = 1² +1, y = 21², z=t³ from t=1 to t = 2
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