Answer any one question from Q1 and Q2

1 (a)
Solve any two:

i) (D

ii) (D

i) (D

^{2}+6D+9) y=x^{-3}e^{-3x}ii) (D

^{2}-2D+2) y =e^{x}tan x ( by variation of parameters method) \[ iii) \ \ x^2 \dfrac {d^2 y}{dx^2}-3x \dfrac {dy}{dx} + 5y =x^2 \sin (\log x) \]
8 M

1 (b)
Find the Fourier sine cosine transforms of e

^{-mx}m > 0.
4 M

2 (a)
The currents x and y in the coupled circuits are given by:

(LD + 2R)x - Ry=E

(LD + 2R)y- Rx =0.

Find the general values of x and y in terms of t.

(LD + 2R)x - Ry=E

(LD + 2R)y- Rx =0.

Find the general values of x and y in terms of t.

4 M

2 (b)
Find the inverse z-transform (any one): \[ i) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ (by \ inversion \ integral \ method) \\ ii) \ F(z) = \dfrac{z}{\left ( z- \frac {1}{4}\right )\left ( z- \frac {1}{5} \right )}, \ |z|> \dfrac {1}{4}. \]

4 M

2 (c)
Solve the difference equation:

f(K+1)- f(K)=1, K≥0, f(0)=0.

f(K+1)- f(K)=1, K≥0, f(0)=0.

4 M

Answer any one question from Q3 and Q4

3 (a)
The first four moments about 44.5 of a distribution are -0.4, 2.99, -0.08 and 27.63. Calculate moments about means, coefficients of Skewness and Kurtosis.

4 M

3 (b)
The incidence of certain disease is such that on the average 20% of workers suffer from it. If 10 workers are selected at random, find the probability that:

i) exactly 2 workers suffer from disease.

ii) not more than 2 workers suffer.

i) exactly 2 workers suffer from disease.

ii) not more than 2 workers suffer.

4 M

3 (c)
Find the directional derivative of:

ϕ=4xz

at (2, -1, 2) along a line equally inclined with coordinate axes.

ϕ=4xz

^{3}- 3x^{2}y^{2}zat (2, -1, 2) along a line equally inclined with coordinate axes.

4 M

4 (a)
A random sample of 200 screws is drawn from a population which represents size of screws. If a sample is normally distributed with a mean 3.15 cm and S.D. 0.025 cm, find expected number of screws whose size falls between 3.12 cm and 3.2 cm.

[Give: For z=12, area =0.3849; for z=2, area = 0.4772]

[Give: For z=12, area =0.3849; for z=2, area = 0.4772]

4 M

4 (b)
show that (any one): \[ i) \ \nabla \cdot \left ( \dfrac {\overline a \times \overline r}{r} \right ) = 0 \\ ii) \ \nabla^4 (r^2 \log r) = \dfrac {6}{r^2} \]

4 M

4 (c)
A fluid notaion is given by: \[ \overline v = (y \sin z - \sin x)\widehat i + (x \sin z + 2yz)\widehat j + (xy \cos z + y^2)\widehat h \] Is the motion irrotational? If so, find the scalar velocity potential.

4 M

Answer any one question from Q5 and Q6

5 (a)
Find the work done by the force: \[ \overline F (x^2 - yz)i + (y^2 -zx)j + (z^2 - xy)k \] in taking a particle from (1,1,1) to (3, -5, 7).

4 M

5 (b)
Use divergence theorem to evaluate: \[\iin_s (y^2z^2 i + z^2x^2j+x^2y^2k) \cdot overline {ds} \] where s is the upper half of the sphere x

^{2}+y^{2}+z^{2}=9 above the xoy plane.
5 M

5 (c)
Apply Stoke's theorem to evaluate: \[\int_c (4y \ dx + 2z \ dy \ + 6y \ dz \] Where C is the curve x

^{2}+y^{2}+z^{2}=6z, z=x+3.
4 M

6 (a)
Find the work done in moving a particle from (0, 1, -1) to \[ \left ( \dfrac {\pi}{2}, -1, 2 \right ) in a force field: \[ \overline F (y^2 \cos x + z^3)i + (2y \sin x -4)j+(3xz^2 + 2)k. \]

4 M

6 (b)
Evaluate: \[ \iint_s [(x+y^2)i-2x \ j + 2yz \ k] \cdot \overline {ds} \] Where s is the plane 2x+y+2z-6=0 considered as one of the bounding planes of the tetrahedron x=0, y=0, z=0, 2x+y+2z=6.

5 M

6 (c)
Verify Stoke's theorem for: F= -y

^{3}i + x^{3}j and closed curve c is the boundary of the circle x^{2}+y^{2}=1.
4 M

Answer any one question from Q7 and Q8

7 (a)
Find the condition under which:

u=ax

u=ax

^{3}+ bx^{2}y+cxy^{2}+dy^{3}is harmonic.
4 M

7 (b)
Evaluate: \[ \oint_c \dfrac {4z^2 + z}{z^2 -1}dz, \] where C: |z-1|=3.

5 M

7 (c)
Show that: \[ w= \dfrac {z-i} {1-iz} \] \] maps upper half of z-plane onto interior of unit circle in w-plane.

4 M

8 (a)
Find the harmonic conjugate of:

u=r

u=r

^{3}cos 3θ + r sin θ.
4 M

8 (c)
Evaluate: \[ \oint_c \dfrac {\sin 2z}{\left ( z+\frac {\pi}{3} \right )^4}dz, \] where C: |z|=2.

5 M

8 (c)
Find the bilinear transformation which maps the points 1, 0, i of the z-plane onto the points \[ \infty, -2, -\dfrac {1}{2} (1+i) of the w-plane.

4 M

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