Time duration: 3Hours $\hspace{50mm}$ Total marks: 80

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N.B:

1) Question No. 1 is compulsory

2) Attempt any Three (03) Questions from remaining Five (05) Questions.

Q.1

A) Find Laplace transform of sin $\sqrt{t}$. $\hspace{20mm}$ 5

B) Prove that u = -r$^{3}$sin3? is harmonic function. Also find the conjugate function of u. $\hspace{20mm}$ 5

C) Find a Fourier series to represent f(x) = $(\frac{\pi -x}{2})^{2}$ in (0, 2$\pi$) hence deduce that $\hspace{20mm}$ 5

$\frac{\pi ^{2}}{6}=\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+ \ldots \ldots \ldots \ldots ..$

D) Find the acute angle between the surface x$^{2}$+y$^{2}$+z$^{2}$=9 and z=x$^{2}$+y$^{2}$-3 at (2,-1,2). $\hspace{20mm}$ 5

Q2

A) Prove that $J_{(-3/2)}(x)= -\sqrt{\frac{2}{\pi x}} .(\frac{\cos x}{x}+\sin x) $. $\hspace{20mm}$ 6

B) Find the Bilinear transformation which maps the points z= 1, i, -1 onto the points w = i, 0,$\infty$. $\hspace{20mm}$ 6

C) Obtain the Fourier series for f(x) = |x| in (-$\pi$, $\pi$) $\hspace{20mm}$ 8

Hence deduce that $\frac{\pi ^{2}}{8}=\frac{1}{1}+\frac{1}{9}+\frac{1}{25}+ \ldots \ldots \ldots \ldots $

Q.3

A) Find inverse Laplace transform of (i) tanh$^{-1}$ $(\frac{2}{S}) $ (ii) e$^{- 4s}$ . $\frac{S}{(S+4)^{3}}$ $\hspace{20mm}$ 6

B) Find the image of rectangular region bounded by x = 0, x = 3, y = 0, y = 2 under the $\hspace{20mm}$ 6

bilinear transformation w = z + (1+i).

C) Prove that y = $\sqrt[]{x}$ . J$_{n}$(x) is a solution of the equation, $X^{2}\frac{d^{2}y}{dx^{2}}+(x^{2}-n^{2}+\frac{1}{4})y=0$. $\hspace{20mm}$ 8

Q.4

A) Find Complex form of Fourier Series of coshax in (-a, a). $\hspace{20mm}$ 6

B) Use Gauss's Divergence theorem to evaluate, $\iint_{S}{\overrightarrow{N }}. \overrightarrow{F} ds$ where $ \overrightarrow{F}$ = 4xi +3yj -2zk and S $\hspace{20mm}$ 6

is the surface bounded by x=0, y=0, z=0 and 2x +2y+z=4.

C) Solve using Laplace transform( D$^{2}$ +2D+1)y =3te$^{-t}$, given y(0)=4 and y'(0)=2. $\hspace{20mm}$ 8

Q.5

A) Find half range cosine series for $f(x)=\bigg\{ \begin{gathered} x , 0 \lt x \lt \frac{\pi }{2} \\ \pi -x , \frac{\pi }{2} \lt x \lt \pi \\ \end{gathered}$ $\hspace{20mm}$ 6

B) Find inverse Laplace transform of $\frac{S}{(S^{2}+4s+13)^{2}}$ using convolution theorem. $\hspace{20mm}$ 6

C) Prove that $\overrightarrow{F }=(y^{2} cosx+ z^{3})i+(2 y\sin x-4)j+(3xz^{2}+2)k $ is a conservative field. $\hspace{20mm}$ 8

Find: (i) Scalar Potential for (ii)The work done in moving an object in this field from (0,1,-1) to $(\frac{\pi }{2},-1,2)$.

Q.6

A) Find Laplace transform of e$^{-4t}$ $\int_{0}^{t}{u sin3u du}$. $\hspace{20mm}$ 6

B) Use Stoke's Theorem to evaluate $\int_{C}{\overrightarrow{F}. \overrightarrow{dr}} $where $\overrightarrow{F}=(2x-y)i-yz^{2}j-y^{2}zk$ and S is $\hspace{20mm}$ 6

the surface of the hemisphere x2+y2+z2=a2 lying above the XY - plane.

C) Express the function f(x) = $\bigg\{ \begin{gathered} 1 , |x|\lt1 \\ 0 , \vert x\vert \gt 1 \\ \end{gathered} $ as Fourier integral. Hence evaluate $\hspace{20mm}$8

$\int_{0}^{\infty }{\frac{sinw . sinwx}{w} dw}$