STUPIDSID
Solve any one question form Q.1(a, b) &Q.2(a, b, c)Solve any two question from Q.1(a)(i, ii, iii)
1(a)
\( \begin{align*}i)&4\frac{d^2y}{dx^2}+4\frac{dy}{dx}+y=xe^{-x/2}\cos x \\
ii)&x^2\frac{d^2y}{dx^2}-x\frac{dy}{dx}-3y=x^2\log x\\
iii)&\frac{d^2y}{dx^2}-2\frac{dy}{dx}e^x \sin x \end{align*} \)/ (use method of variation of parameters).
8 M
1(b)
Solve the differential equation by using Laplace transform method: \( \frac{dy}{dt}+3y(t)+2\int_{0}^{t}y(t)dt=t\\ \text{given}y(0)=0 \)/
4 M
2(a)
An inductor of 0.25 henry is connected in series with a capacitor of 0.04 farads and a generator having alternative voltage given by 12 sin 10t. Find the charge and current at any time t.
4 M
Solve any one question from Q.2(b) (i, ii)
2(b)
Find the Laplace transform of: \( \frac{d}{dt}\left ( \frac{1-\cos t}{t} \right ).\)/
ii) Find the inverse Laplace transform of: \( \frac{S}{S^4+S^2+1}.\)/
ii) Find the inverse Laplace transform of: \( \frac{S}{S^4+S^2+1}.\)/
4 M
2(c)
Find the Laplace transform of:\[ f(t)=\left ( 1+2t+3t^2 \right )u\left ( t-2 \right )+sin 2t\delta \left ( t-\frac{\pi }{4} \right ).\]
4 M
Solve any one question from Q.3(a,b,c) & Q.4(a,b,c)
3(a)
Find the Fourier transform of: \[f(x)=\left\{\begin{matrix}
1-x^2,|x| &\leq 1 \\
0,|x| &>1.
\end{matrix}\right.\]
4 M
3(b)
Find: \( z^{-1}\begin{Bmatrix}
\frac{1}{\left ( z-\frac{1}{2} \right )\left ( z-\frac{1}{3} \right )} & \\ For |z|>\[\frac{1}{2}.\]
\end{Bmatrix} \)/
4 M
3(c)
Find the constants a and b, so that the surface \( \ax^2-byz=\left ( a+2 \right )x)/
will be orthogonal to the surface \( 4x^2y+z^3=4 \)/
at the point (1, -1, 2).
will be orthogonal to the surface \( 4x^2y+z^3=4 \)/
at the point (1, -1, 2).
4 M
4(a)
Show that the vector field \( f(r)\underset{r}{\rightarrow} \)/ is always irrotational and determine f(r) such that the field is solential also.
4 M
Solve any one question from Q.4(b)(i, ii)
4(b)(i)
Prove the following:\nabla^2\left ( \nabla.\frac{\underset{r}{\rightarrow}}{r^2} \right )=\frac{2}{r^4}
4 M
4(b)(ii)
\( \nabla\times\left ( \underset{a}{\rightarrow} \nabla\frac{1}{r}\right )=\frac{\underset{a}{\rightarrow}}{r^3}-\frac{\left ( \underset{a}{\rightarrow}.\underset{r}{\rightarrow} \right )\underset{r}{\rightarrow}}{r^5}. \)/
4 M
4(c)
Solve the difference equation: \( f\left ( k+2 \right )-3f\left ( k+1 \right )+2f(k)=1\\
\text{where}f(0)= 0, f(1)=3.\)/
4 M
Solve any one question from Q.5(a,b,c) & Q.6(a,b,c)
5(a)
Evaluate: \(\int _C\bar{F}.d\bar{r}\\\text{for}\\\bar{F}=\left ( 2y+3 \right )\bar{i}+xz\bar{j}+\left ( yz-x \right )\bar{k} \)/ along a straight line joining (0, 0, 0) to (3, 1, 1).
4 M
5(b)
Evaluate: \( \iint _S\left ( \nabla\times \bar{F} \right ).\hat{n}dS \)/ where S is the curved surface of the paraboloid \(x^2+y^2=2z \)/ bounded by plane z=2 where \( \bar{F}=3\left ( x-y \right )\bar{i}+2xz\bar{j}+xy\bar{k}.\)/
5 M
5(c)
Evaluate: \( \iint \bar{r}.\hat{n}dS \)/ over the surface of sphere of radius 2 with centre at origin.
4 M
6(a)
Using Green's theorem evaulate: \(\int _C\left [ \cos y\bar{i} +x\left ( 1-\sin y \right )\right\bar{j} ].d\bar{r} \)/ where C is the closed curve \( x^2+y^2=1, z=0. \)/
4 M
6(b)
Prove that: \( \int _C\left ( \bar{a}\times \bar{r} \right ).d\bar{r}=2\bar{a}.\iint _Sd\bar{S} \)/ where C is open surface bouded by closed curve C.
4 M
6(c)
Evaluate: \(\iint _S\bar{F}.\hat{n}dS\\\text{where} \\\bar{F}=\left ( 2x+3z \right )\bar{i}-\left ( xz+y \right )\bar{j} \left ( y^2+2z \right )\bar{K} \)/ and S is surface of sphere with radius 3.
5 M
Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)
7(a)
If \( u=\log \left ( x^2+y^2 \right ), \)/ find v such that \[ f(z) = u + iv\] is analytic. Determine f(z) in terms of z.
5 M
7(b)
Evaluate: \( \oint _C\frac{dz}{z^2}\)/ where C is the circle |z| = 1.
4 M
7(c)
Find the bilinear transformation which maps the points 0, -1, i of z-plane on to the points 2, &infty;,\( \frac{1}{2}\left ( 5+i \right )\)/ of the w-plane.
4 M
8(a)
Show that the map of straight line parallel to x-axis is family of ellipses under the transformation w = sinh (z).
4 M
8(b)
Evaluate: \( \oint _C\frac{z+2}{z^2+1}dz \)/ where C is the circle | z -i| = \[\frac{1}{2}\].
4 M
8(c)
Find analytic function \( f(z) = u + iv \\\text{where} u= r^3 \cos 3\theta +r\sin \theta . \)/
5 M
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