Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 Find the unit normal vector to the surface x2+y2=z at (1, -2, 5)
2 M

2 Prove that curl(grad ϕ)=0
2 M

3 Solve the equation $\dfrac {d^2y}{dx^2}+2 \dfrac {dy}{dx}+ y=0$
2 M

4 Find the particular integral of the equation (D2-9)y=e-3x
2 M

5 $Find \ L \left [\dfrac {\sin t}{t} \right ]$
2 M

6 Evaluate $L^{-1} \left [ \dfrac {1}{s^2+6s+13} \right ]$
2 M

7 Is the function f(z) = z analytic?
2 M

8 Find the invariant points of f(z)=z2
2 M

9 Evaluate $\int_c \dfrac {z}{z-2} dz$ where C is (a) |z|=1, (b)|z|=3
2 M

10 State Cauchy's residue theorem.
2 M

Answer any one question from Q11 (a) & Q11 (b)
11 (a) Verify Gauss divergence theorem for $\overrightarrow{F}=x^2\overrightarrow{i}+y^2\overrightarrow{j}+z^2\overrightarrow{k}$ taken over the cube bounded by the planes x=0, y=0, z=0, x=1, y=1 and z=1.
16 M
11 (b) (i) Find the value of n such that the vector $r^n\overrightarrow{r}$ is both solenoidal and irrotational.
8 M
11 (b) (ii) Verify Stokes' theorem for $\bar{F}= \left ( x^2-y^2 \right )\bar{i}+ 2xy\bar{j}$ in the rectangular region of z=0 plane bounded by the lines x=0, y=0, x=a and y=b.
8 M

Answer any one question from Q12 (a) & Q12 (b)
12 (a) (i) Solve (D2-4D+3)y=cos 2x+2x2
8 M
12 (a) (ii) $Solve \ \dfrac {d^2y}{dx^2}+a^2y=\tan ax$ using method of variation of parameters.
8 M
12 (b) (i) Solve $(x^2D^2-xD+1)y=\left ( \dfrac {\log x}{x} \right )^2$
8 M
12 (b) (ii) Solve the simultaneous equations $\dfrac {dx}{dt}+2y=-\sin t \ and \ \dfrac {dy}{dt}-2x=\cos t$
8 M

Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i) Find the Laplace transform of f(t), where $f(t)=\left\{\begin{matrix}\sin wt &for &0<t<\frac {\pi}{w} \\0 & for & \frac{\pi}{w} <t<\frac {2\pi}{w}\end{matrix}\right.$ and f(t+2π)=f(t)
8 M
13 (a) (ii) Using convolution theorem find the inverse Laplace transform of $\dfrac {s^2}{(s^2+a^2)(s^2+b^2)}$
8 M
13 (b) (i) Find the Laplace transform of f(t)=ie-3tcos 2t
8 M
13 (b) (ii) Using Laplace transform, solve $\dfrac {d^2y}{dt^2}+4y=\sin 2t,$ given y(0)=3 and y(0)=4
8 M

Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i) Prove that the real and imaginary parts of an analytic functions are harmonic functions.
8 M
14 (a) (ii) Find the bilinear transformation that maps 1, i and -1 of the z-plane onto 0,1 and ∞ of the w-plane.
8 M
14 (b) (i) Show that v=e-x(x cos y + y sin y) is harmonic functions. Hence find the analytic function f(z)=u+iv
8 M
14 (b) (ii) Find the image of |z+1|=1 under the map w=1/z
8 M

Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i) Obtain the Laurent's series expansion of $f(z)= \dfrac {z^2-1}{(z+2)(z+3)}$ in 2<|z|<3
8 M
15 (a) (ii) Evaluate $\int^{2\pi}_0 \dfrac {d\theta}{13+5 \sin \theta}$
8 M
15 (b) (i) Using Cauchy's residue theorem evaluate $\int_c \dfrac {(z-1)}{(z-1)^2 (z-2)}dz$ where C is |z-i|=2
8 M
15 (b) (ii) Evaluate by using contour integrating $\int^\infty_0 \dfrac {dx}{(1+x^2)^2}$
8 M

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