Solve any one question from Q1 and Q2

1 (a) (i)
(D

^{2}-2D-3) y=3e^{-3x}sin e^{-3x}+ cos (e^{-3x}).
4 M

1 (a) (ii)
(D

^{2}-2D+2) y=e^{x}tan x. (By variation of parameters).
4 M

1 (a) (iii)
\[ x^2 \dfrac {d^2 y}{dx^2}-2x \dfrac {dy}{dx} - 4y=x^2 \]

4 M

1 (b)
Find the Fourier transform of e

^{-|x|}and hence show that: \[ \int^{\infty}_{-\infty} \dfrac {e^{i\lambda x}}{1+\lambda^2} d \lambda = \pi e^{-|x|} \]
4 M

2 (a)
An unchanged condenser of capacity C charged by applying an e.m.f. of value \( \dfrac {t}{\sqrt{LC}} \) through the LEDs of inductance L and of negligible resistance. The charge Q on the place of condenser satisfied the differential equation: \[ \dfrac {d^2Q}{dt^2} + \dfrac {Q}{LC} = \dfrac {E}{L}\sin \dfrac {t}{\sqrt{LC}} \] Prove that the charge at any time t is given by: \[ Q= \dfrac {EC}{2} \left [ \sin \dfrac {t}{\sqrt{LC}} - \dfrac {t}{\sqrt{LC}} \cos \dfrac {t}{\sqrt{LC}} \right ] \]

4 M

2 (b)
Find the Inverse Z-transform (any one): \[ i) \ F(z) = \dfrac {z+2}{z^2 - 2z+1} \ \text {for }|z|>1. \\
ii) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ \text {(Use inversion integral method).}\]

4 M

2 (c)
Solve the following difference equation to find {f(k)}: \[ f(k+1)+ \dfrac {1}{4} f(k) = \left ( \dfrac {1}{4} \right )^k, \ k\ge 0, \ f(0)=0 \]

4 M

Solve any one question from Q3 and Q4

3 (a)
The first four moments of distribution about the value 4 are -1.5, 17, -30 and 108. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.

4 M

3 (b)
If the probability that a concrete cube fails is 0.001. Determine the probability that out of 1000 cubes:

i) exactly two

ii) more than one cubes will fail.

i) exactly two

ii) more than one cubes will fail.

4 M

3 (c)
Show that: \[\overline F= ( y \sin z - \sin x) \overline i (x \sin z +2 yz)\overline j + (xy\cos z+y^2)\overline k \] is irrotational and hence find scalar function ϕ s.t. F=∇ϕ.

4 M

4 (a)
Find the directional derivative of ϕ=4xz

^{3}-3x^{2}y^{2}z at (2, -1, 2) along a line equally inclined with co-ordinate axes.
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4 (b)
For a solenoidal vector field F, show that:

curl curl curl curl F=∇

curl curl curl curl F=∇

^{4}F.
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4 (c)
The regression equations are:

8x+10y+66=0 and 40x-18y=214.

The value of variance of x is 9. Find.

i) The mean values of x and y

ii) The correlation coefficient between x and y

iii) The standard deviation of y.

8x+10y+66=0 and 40x-18y=214.

The value of variance of x is 9. Find.

i) The mean values of x and y

ii) The correlation coefficient between x and y

iii) The standard deviation of y.

4 M

Solve any one question from Q5 and Q6

5 (a)
Find the work done in moving a particle once round the ellipse: \[ \dfrac {x^2}{16}+ \dfrac {y^2}{4}=1, \ z=0 \] under the field of force given by: \[ \overline{F} = (2x-y+z)\overline i + (x+y-z^2) \overline j + (3x-2y + 4z) \overline{k} \]

4 M

5 (b)
Evaluate: \[ \iint_s (\nabla \times \overline{F} ) \cdot \widehat{n} \ dS \] where \[ \overline F = (x^3 - y^3) \overline {i} - xyz \overline j + y^2 \overline {k} \] and S is the surface x

^{2}+4y^{2}+z^{2}-2x=4 above the plane x=0.
4 M

5 (c)
Evaluate: \[ \iint_s \overline {F} \cdot \overline {dS} \] using divergence theorem, where \[ \overline F= x^3 \overline i + y^3 \overline j+z^3 \overline k \] and S is the surface of sphere x

^{2}+y^{2}+z^{2}=a^{2}.
5 M

6 (a)
If \( \overline F = x^2 \overline i + (x-y) \overline j + (y+z) \overline k \) displaces a particle from A(1, 0, 1) to B(2, 1, 2) along the straight line AB, find work done.

4 M

6 (b)
Evaluate: \[ \int_C (e^x dx + 2ydy - dx) \] where C is the curve x

^{2}+y^{2}=4, z=2.
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6 (c)
Evaluate: \[ \int_s \overline F \cdot \overline {dS} \] using Gauss divergence theorem, where: \[ \overline F = 2xy\overline i + yz^2 \overline j + xz\overline k \] and S is the region bounded by:

x=0, y=0, z=0, y=3, x+2z=6.

x=0, y=0, z=0, y=3, x+2z=6.

5 M

Solve any one question from Q7 and Q8

7 (a)
Show that u=y

^{3}-3x^{2}y is harmonic function. Find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.
5 M

7 (b)
Using Cauchy's integral formula, evaluate: \[ \int_C \dfrac {2z^2 + z +5}{(z-3 /2)^2}dz \] where C is \( \dfrac {x^2}{4} + \dfrac {y^2}{9} = 1. \)

4 M

7 (c)
Find the bilinear transformation which maps the points z=1, i, -1, onto the points w=0, 1, ∞.

4 M

8 (a)
If f(z) is an analytic function v

^{2}=u, then show that f(z) is constant function.
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8 (b)
Using residue theorem evaluate: \[ \int_C \dfrac {z}{z^4 +13z^2 + 36} dz \] where 'C' is the circle \( z| = \dfrac {5}{2}. \)

5 M

8 (c)
Find the map of the circle |z=i|=1 under the transformation \( w=\dfrac {1}{w} \) into w-plane.

4 M

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