STUPIDSID
Solve any one question form Q.1(a)(i,ii,ii) Solve any one question from Q.1(a,b) & Q.2(a, b, c)
1(a)
\( \begin{align*}i)& \left ( D^2-4D+3 \right )y=x^3e^{2x}\\
ii)&\left ( 1+x \right )^2\frac{d^2y}{dx^2}+\left ( 1+x \right )\frac{dy}{dx}+y=2\sin \left [ \log \left ( 1+x \right ) \right ]\\
iii)&\left ( D^2-1 \right )y=\frac{2}{1+e^x} \end{align*} \)/ using method of variation of parameters.
8 M
1(b)
Solve using Laplace transforms: \( \frac{d^2y}{dx^2}+y=t,\\
\text{given}y(0)=1, y'(0)=-2. \)/
4 M
2(a)
A circuit consists of an induactance L and condenser of capacity C in series. An e.m.f. Σsin n t is applied to it at time t =0, the initial chrage and initial current being zero, find the current flowing in the circuit at any time \( t\ \text{for}\frac{1}{\sqrt{LC}}\neq n. \)/
4 M
Solve any one question form Q.2(b)(i,ii)
2(b)(i)
Find:\( L\left [ \frac{e^{-at}-e^{-bt}}{t} \right ]. \)/
4 M
2(b)(ii)
Find:\( L^{-1}\left [ \frac{s+7}{s^2+2s+2} \right ].\)/
4 M
2(c)
Find Laplace transform of: \( L\left [ \sin t\ U\left ( t-4 \right ) \right ]. \)/
4 M
Solve any one question from Q.3(a, b,c) & Q.4(a, b,c)
3(a)
Solve the integral equation:\( \int_{0}^{\infty }f(x)\cos \lambda x\ dx= e^{-\lambda }
\ \text{where}\ \lambda >0\)/
4 M
Solve any one question form Q.3(b)(i,ii)
3(b)(i)
Find z-transform of \(f(k)=\frac{2^k}{K}, k\geq 1. \)/
4 M
3(b)(ii)
Find inverse z-transform of: \( F(z)=\frac{1}{\left ( z-a \right )^2},\ |z| < a.\)/
4 M
3(c)
If directional derivative of:\( \phi =ax^2y+by^2z+cz^2x\)/ at (1,1,1) has maximum magnitude 15 in the direction parallel to \( \frac{x-1}{2}=\frac{y-3}{-2}=\frac{z}{1},\)/ hence find the values of a, b, c.
4 M
Solve any one question fromQ.4(a)(i, ii)
4(a)(i)
\( \nabla.\left [ r^\nabla\left ( \frac{1}{r^n} \right ) \right ]=\frac{n\left ( n-2 \right )}{r^{n+1}} \)/
4 M
4(a)(ii)
\( \nabla^2f(r)=f"(r)+\frac{2}{r}f'(r). \)/
4 M
4(b)
Find the values of the constant scalars a, b, c if the vector point function: \( \bar{V}=\left ( x+2y+az \right )i+\left ( bx-3y+z \right )j+\left ( 4x+cy+2z \right )k \)/
4 M
4(c)
Obtain f(k), given that:\( f_{k+2}-4f_k=0, k\geq 0,f(0)=0,f(1)=2. \)/
4 M
Solve any two question from Q.5(a,b,c) Solve any one question from Q.5(a, b, c) &Q.6(a, b,c)
5(a)
Using Green's theorem, show that the area bounded by a simple closed curve C is given by: \( \frac{1}{2}\int \left ( xdy-ydx \right) .\)/ Hence find the area of the ellipe \[x=a\cos \theta , y= b\sin \theta .\]
6 M
5(b)
Use the divergence theorem to evaluate:\( \iint_s \left ( y^2z^2i+z^2x^2j+x^2y^2k \right ).\overline{dS} \)/
where S is the upper half of the sphere x2+y2+z2=9 above the xoy plane.
where S is the upper half of the sphere x2+y2+z2=9 above the xoy plane.
6 M
5(c)
Verify Stokes' theorem for:\( \bar{F}=\left ( y-z+2 \right )i+\left ( yz+4 \right )+xzk \)/ over the surface x=0, y=0, z=0, x=2, y=2.
7 M
Solve any two question from Q.6(a, b,c)
6(a)
Evaluate \( \int_{c}\bar{f}.d\bar{r}\ \text{where}\\
\bar{f}=\left ( 5xy-6x^2 \right )i+\left ( 2y-4x \right )j\)/ and c is the arc of the curve in the xoy plane, y=x3 from (1, 1) to (2, 8)
6 M
6(b)
Evlauate \( \iint _s\bar{F}.\overline{ds}\ \text{where}\\
\bar{F}=yzi+zxj+xyk \)/ and s is the part of the surface of the sphere \(x^2+y^2+z^2=1 \)/ which lies in the first octant.
6 M
6(c)
Use Storke's theorem to evaluate:\( \int _c\left ( 4yi+2zj+6yk \right ).d\bar{r} \)/ where c is the curve of intersection of \( x^2+y^2+z^2=2z\ \text{and} x=z-1. \)/
7 M
Solve any one question from Q.7(a, b,c) & Q.8(a, b,c)
7(a)
If \(v=\frac{-y}{x^2+y^2}, \)/ find u such that, u+iv is analytic function.
4 M
7(b)
Evlauate:\( \oint _c\frac{z+4}{z^2+2z+5}dz,\)/ where c is circle |z-2i| = 3 / 2.
5 M
7(c)
Find the bilinear transformation which maps points 0, -1, ∞ of z-plane onto -1, -(2+i) pf W-plane.
4 M
8(a)
Find the condition satisfied by a, b, c and d under which,\( u=ax^3+bx^2y+cxy^2+dy^3\)/is harmonic function.
4 M
8(b)
Evlauate:\( \int_{0}^{2\pi }\frac{d\theta }{5-3\cos \theta } \)/ using Cauchy's theorem.
5 M
8(c)
Find the image of st. Line y=x under the transformation: \[W=\frac{z-1}{z+1}.\]
4 M
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