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 particular integral of (D2-2D+1)y=cosh x
2 M

2 $Solve \ x^2\dfrac {d^2y}{dx^2}+4x \dfrac {dy}{dx}+2y=0$
2 M

3 Find the directional derivative of ?=xyz at (1,1,) in the direction of$\overrightarrow {i}+ \overrightarrow{j}+ \overrightarrow{k}$
2 M

4 If $\overrightarrow{A} \ and \ \overrightarrow{B}$ are irrotational, prove that $\overrightarrow{A}\times \overrightarrow{B}$ is solenoidal.
2 M

5 Find the image of the line x=k under the transformation w=1/z
2 M

6 Find the fixed points of mapping $w=\dfrac {6z-9}{z}$
2 M

7 Evaluate $\int_c \dfrac {3z^2 + 7z+1}{z+1}dz, \ where \ C \ is |z|=\dfrac {1}{2}$
2 M

8 Find the residue of $\dfrac {1-e^{2z}}{z^4} \ at \ z=0$
2 M

9 Find the Laplace transform of $\dfrac {t}{e^t}$
2 M

10 Verify initial value theorem for the function f(t)=ae-bt
2 M

Answer any one question from Q11 (a) & Q11 (b)
11 (a) (i) Solve the differential equation $\dfrac {d^2y}{dx^2}+2\dfrac {dy}{dx}+ y=\dfrac {e^{-x}}{x^2}$ by the method of variation of parameters.
8 M
11 (a) (ii) $Solve : \ (3x+2)^2 \dfrac {d^2y}{dx^2}+3 (3x+2)\dfrac {dy}{dx} -36y = 3x^2 +4x+1$
8 M
11 (b) (i) Solve the simultaneous differential equations: $\dfrac {dx}{dt}+5x-2y=t; \ \dfrac {dy}{dt}+2x+y=0$
8 M
11 (b) (ii) $Solve \ x^2 \dfrac {d^2 y}{dx^2}+ 4x \dfrac {dy}{dx}+2y=x^2 + \dfrac {1}{x^2}$
8 M

Answer any one question from Q12 (a) & Q12 (b)
12 (a) Verify Stokes' Theorem for the vector field $\overrightarrow{F}= (2x-y)\overrightarrow {i}-yz^2 \overrightarrow{j}-y^2z\overrightarrow{k}$ over the upper half surface x2+y2+z2-1, bounded by its projection on the xy-plane.
16 M
12 (b) Verify divergence theorem for $\overrightarrow{F}=x^2 \overrightarrow{i}+z\overrightarrow{j} + yz\overrightarrow{k}$ over the cube formed by the plane x=±1, y=±1, z=±1.
16 M

Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i) Prove that the function u-ex(x cos y -y sin y) satisfies Laplace's equation and find the corresponding analytic function f(z)=u+iv.
8 M
13 (a) (ii) Find the Bilinear transformation which maps z=0, z=1, z=∞ into the points w=i, w=1, w=-i.
8 M
13 (b) (i) Find the image of |z-2i|=2 under the transformation w=1/z.
8 M
13 (b) (ii) If f(z) is an analytic function of z, prove that $\left ( \dfrac {\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )|f(z)|^2=4 |f(z)|^2$
8 M

Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i) Expand the function $f(z)= \dfrac {z^2-1}{z^2+5z+6}$ in Laurent's series for |z|>3
8 M
14 (a) (ii) Evaluate $\int_c \dfrac {\sin \pi z^2+ \cos \pi z^2 }{(z+1)(z+2)}dz$ where C is |z|=3
8 M
14 (b) (i) Evaluate $\int^{\infty}_0 \dfrac {x^2 dx}{(x^2 +a^2)(x^2+b^2)}, \ a>0,b>0$
8 M
14 (b) (ii) Evaluate $\int^{2\pi}_0 \dfrac {\cos 3\theta}{5-4\cos \theta}d\theta$ using contour integration.
8 M

Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i) $Find \ L\left [ t^2e^{-3t} \sin 2t \right]$
8 M
15 (a) (ii) Find the Laplace transform of the square-wave function (or Meoander function) of period α defined as $f(t)= \left\{\begin{matrix}1 & when & 0<t<\frac{a}{2} \\-1, &when & \frac {a}{2} <t<a\end{matrix}\right.$
8 M
15 (b) (i) Using convolution theorem find the inverse Laplace transform of $\dfrac {4}{(s^2+2s+5)^2}$
8 M
15 (b) (ii) Solve y''+5y'+6y=2 given y'(0)=0 and y(0)=0 using Laplace transform.
8 M

More question papers from Mathematics 2