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**(3 hours)** $\hspace{70mm}$ **[Max. Marks 80]**

**N.B. :**

**1. Question no. 1 is compulsory.**

**2. Assume suitable data if necessary.**

**3. Attempt any three questions from remaining questions.**

**Q.1**

a) Find the Laplace transform of $\frac{1}{t}e^{-t} sint$ .

b) Find the inverse Laplace transform of $\frac{1}{\sqrt{2s-1}}$ .

c) Find the function f(z)= sinh z and find f'(z)in terms of z.

d) Find the Fourier series of f(x)=x in (0,$2\pi $).

**Q.2**

a) Use laplace transform to prove $\int_{0}^{\infty }{e^{-2t}\frac{\sin ^{2}t}{t}dt}$.

b) If {f(k)} = | 4^{k} , 3^{k} | , find Z{f(k)}.

c) Show that the function u = $\cos$x $\cos$y is a harmonic function find the harmonic conjugate of and the corresponding analytic function.

**Q.3**

a) Find the equation of the line regression of Y on X for the following data

X | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|

Y | 11 | 14 | 14 | 15 | 12 | 17 | 16 |

b) Find the Bilinear transformation which maps the points (1,-1 ,2) on z plane on to (0,2, -i) respectively of w plane.

c) Find the half range sine series for f(x) = $x , 0 \lt x \lt \frac{\pi }{2} $

$\hspace{64mm}$ $\pi -x , \frac{\pi }{2} \lt x \lt \pi $

Hence find the sum of $\sum_{(2n-1)}^{\infty }{\frac{1}{n^{4}}}$ .

**Q.4**

a) Find the inverse Laplace transform by using convolution theorem $\frac{1}{(s-a)(s+a)^{2}}$.

b) Find the inverse Z transform of $\frac{1}{(z-a)^{2}} \vert z\vert \lt a$ , $\frac{1}{(z-3)(z-2)} \vert z\vert \gt 3$.

c) Calculate the coefficient of correlation of X and Y from the following data

X | 8 | 8 | 7 | 5 | 6 | 2 |
---|---|---|---|---|---|---|

Y | 3 | 4 | 10 | 13 | 22 | 8 |

**Q.5**

a) Using Laplace transform evaluate $\int_{0}^{\infty }{e^{-t}(1+2t-t^{2}+t^{3})H(t-1)dt}$.

b) Show that the set of functions cos x cos 2x cos 3x is a set of orthogonal functions over [ - $\pi$ , $\pi$ ] . Hence construct a set of Orthonormal functions.

c) Solve using Laplace transform $(D^{3}-2D^{2}+5D)y=0 $, y(0)=0, y'(0)=0, y''(0)=1.

**Q.6**

a) Find the complex form of Fourier series f(x)= 2x in (0,2$\pi$).

b) If f(z) and $\bar{f(z)}$ prove that f(z) is constant.

c) Fit a curve of form $y=ab^{x}$ from the data

X | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Y | 151 | 100 | 61 | 50 | 20 | 8 |