Answer any one question from Q1 and Q2

1 (a)
Solve (any two): \[ i) \ \ (D^2 +9) y=x^3 - \cos 3x \\ ii) \ (D^2 + 2D + 1 ) y =e^{-x} \log x \\ iii) \ (2x-1)^2 \dfrac {d^2y}{dx^2}- 6 (2x-1) \dfrac {dy}{dx}+ 16 =8 (2x +1)^2. \]

8 M

1 (b)
Find Fourier sine transform of f(x)=e-x cos x, x>0.

4 M

2 (a)
A resistance of 50 ohms, an inductor of 2H and a 0.005 Farad capacitor are connected in series with e.m.f. of 40 volts and an open switch. Find the instantaneous charged and current after the switch is closed at t=0, assuming that at the time charge on capacitor is 4 Coulomb.

4 M

2 (b)
Solve (any one):

i) Find z-transform of \[ f(k) = \dfrac {\sin ak}{k}, \ k >0 \] ii) Find inverse z-transform of \[ \dfrac {3z^2+2z} {z^2+3z+2}, 1< |z| <2 \]

i) Find z-transform of \[ f(k) = \dfrac {\sin ak}{k}, \ k >0 \] ii) Find inverse z-transform of \[ \dfrac {3z^2+2z} {z^2+3z+2}, 1< |z| <2 \]

4 M

2 (c)
Solve difference equation:

f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.

f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.

4 M

Answer any one question from Q3 and Q4

3 (a)
The first four moments of a distribution about the value 5 ae 2, 20, 40 and 50. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.

4 M

3 (b)
A manufacture of electronic goods has 4% of his product defective. He sells the articles in packets of 300 and guarantees 90% good quality. Determine the probability that a particular packet will violet the guarantee.

4 M

3 (c)
Find the directional derivative of xy

^{2}+yz^{3}at (2, -1, 1) along the line 2(x-2)=(y+1)=(z-1).
4 M

4 (a)
in an intelligence test administered to 1000 students the average score was 42 and standard deviation 24. Find the number of students with score lying between 30 and 54.

(Given: For z=0.5, area = 0.1915).

(Given: For z=0.5, area = 0.1915).

4 M

4 (b)
Prove (any one): \[ i) \ \nabla ^2 \left ( \dfrac {\overline a \cdot \overline b}{r} \right ) =0 \\ ii) \ \nabla \times \left ( \dfrac {\overline a \times \overline r}{r} \right ) = \dfrac {\overline a}{r} + \dfrac {(\overline a \cdot \overline r)\overline r}{r^3} \]

4 M

4 (c)
Show that F = r

^{2}r is conservative. Obtain the scalar potential associated with it.
4 M

Answer any one question from Q5 and Q6

5 (a)
Evaluate: \[ \int_c \overline F \cdot d\overlin r \\ where \ \overline F = (2x+y^2) \overlien i + (3y ? 4x) \overline j \ and \] C is the parabolic arc y=x

^{2}joining (0, 0) and (1, 1).
4 M

5 (b)
Using Strokes theorem, evaluate: \[ \int_c (x+y) dx + (2x ? z) dy + (y+z) dz) \] Where C is the curve given by

x

x

^{2}+ y^{2}+ z^{2}- 2ax =0, x+y=2a.
5 M

5 (c)
Use divergence theorem to evaluate: \[ \iint_s (x\overline i - 2y^2 \overline j + z^2 \overline k) \cdot d\overline s \] where s is the surface bounded by the region x

^{2}+ y^{2}=1 and z=0 and z=1.
4 M

6 (a)
Apply Green's theorem to evaluate: \[ \int_c (2x^2 - y^2) dx + (x^2 + y^2) dy \] where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle x

^{2}+ y^{2}=16.
4 M

6 (b)
Using Strokes theorem, evaluate: \[ \iint_s (\nabla \times \overline F) \cdot d\overline \\ where \ \overline F = 3 y \overline i - xz\overline j + yz^2 \overline k \] and 's' is the surface of the paraboloid 2z=x

^{2}+y^{2}bounded by z=2.
5 M

6 (c)
Show that: \[ \iiint_v \dfrac {2}{r} dv = \iint_s \dfrac {\overline s \cdot \widehat n } {r} ds \]

4 M

Answer any one question from Q7 and Q8

7 (a)
Find the imaginary part of the analytic function whose real part is x

^{3}-3xy^{2}+3x^{2}- 3y^{2}.
4 M

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

4 M

7 (c)
Find the bilinear transformation which maps the points

z=-1, 0, 1 on to the points w=0, i, 3i respectively.

z=-1, 0, 1 on to the points w=0, i, 3i respectively.

4 M

8 (a)
Show that analytic function f(z) with constant amplitude is constant.

4 M

8 (b)
Evaluate the following integral using residue theorem: \[ \oint_c \dfrac {4-3z}{z(z-1) (z-2) } dz \] where C is the circle: \[ |z|=\dfrac {3}{2} \]

4 M

8 (c)
Find the image of the straight line y=3x under the transformation \[ w=\dfrac {z-1} {z+1} \]

5 M

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