STUPIDSID
1(a)
Find the Laplace transform of te3t sin 4t.
5 M
1(b)
Find half-range cosine series for f(x)=ex,
0
0
5 M
1(c)
Is \( f(z)=\frac{z}{z} \)/ analytic?
5 M
1(d)
Prove that \( \nabla x\left ( \bar{a}x \nabla \log r\right )=2\frac{(\bar{a}.\bar{r})\bar{r}}{r^4} \)/, where \bar{a} is a constant vector.
5 M
2(a)
Find the Z- transform of \(\frac{1}{\left ( z-5 \right )^3} \)/ if |z|<5.
6 M
2(b)
If V=3x2y+6xy-y3, show that V is harmonic & find the corresponding analytic function.
6 M
2(c)
Obtain Fourier series for the function \( f(x)=\left\{\begin{matrix}
1+\frac{2x}{\pi }, -\pi\leq x\leq 0 & \\
\\ 1-\frac{2x}{\pi },0\leq x\leq \pi &
\end{matrix}\right. \)/ hence deduce that \( \frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+......... \)/
8 M
3(a)
Find \( L^{-1}\left [ \frac{(s+2)^2}{(s^2+4s+8)^2} \right ] \)/ using convolution theorem.
6 M
3(b)
Show that the set of functions \(1,\sin \left ( \frac{\pi x}{L} \right ),\cos\left ( \frac{\pi x}{L} \right ),\sin \left ( \frac{2\pi x}{L} \right ),\cos \left ( \frac{2\pi x}{L} \right ),.......... \)/ Form an orthogonal set in (-L,
L) and construct an orthonormal set.
L) and construct an orthonormal set.
6 M
3(c)
Verify Green's theorem for \( \int \left ( e^{2x}-xy^2 \right )dx+\left ( ye^x+y^2 \right )dy \)/ Where C is the closed curve bounded by y2=x&x2=y.
8 M
4(a)
Find Laplace transform of \( f(t)=K\frac{t}{T}for 0/
6 M
4(b)
Show that the vector, \(\bar{F}=\left ( x^2-yz \right )i+\left ( y^2-zx \right )j+\left ( z^2-xy \right )k \)/ is irrotational and hence, find φ such that \bar{F}=∇φ
6 M
4(c)
Find Found series for f(x) in (0,
2π), \(f(x)\left\{\begin{matrix} x,& 0\leq x\leq \pi \\ 2\pi -x, & \pi \leq x\leq 2\pi \end{matrix}\right. \)/ hence deduce that \( \frac{\pi ^4}{96}=\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+.......... \)/
2π), \(f(x)\left\{\begin{matrix} x,& 0\leq x\leq \pi \\ 2\pi -x, & \pi \leq x\leq 2\pi \end{matrix}\right. \)/ hence deduce that \( \frac{\pi ^4}{96}=\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+.......... \)/
8 M
5(a)
Use Gauss's Divergence theorem to evaluate \(\iint_{s}\bar{N}.\bar{F} ds \)/ where\[ \bar{F}=2xi+xyj+zk\] over the region bounded by the cylinder x2<\sup>+y2=4,
z=0,
z=6.
z=0,
z=6.
6 M
5(b)
Find inverse Z- transform of \( f(x)=\frac{z}{\left ( z-1 \right )\left ( z-2 \right )}, |z|>2 \)/
6 M
5(c)
i) Find \(L^{-1}\left [ log\left ( \frac{s+1}{s-1} \right ) \right ] \)/
ii) \( L^{-1}\left [ \frac{s+2}{s^2-4s+13} \right ] \)/
ii) \( L^{-1}\left [ \frac{s+2}{s^2-4s+13} \right ] \)/
8 M
6(a)
Solve (D2+3D+2)y=2(t2+t+1) with y(0)=2 & y'(0)=0.
6 M
6(b)
Find the bilinear transformation which maps the points 0,
i,
-2i of z-plane onto the points -4i,
∞,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
i,
-2i of z-plane onto the points -4i,
∞,
0 respectively of W-plane. Also obtain fixed points oft he transformation.
6 M
6(c)
Find Fourier sine integral of \(\left\{\begin{matrix}
x, &02
\end{matrix}\right. \) /
8 M
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