Answer any one question from Q1 and Q2

1 (a)
Solve any two of the following:

i) (D

ii) (D

iii) x

i) (D

^{2}+4)y=cos 3x. cos xii) (D

^{2}+6D+9) y=e^{3x}/x^{2}..... ( by variation of parameters method)iii) x

^{2}(d^{3}y/dx^{3})+2x^{2}(d^{2}y/dx^{2})+2y=20 (x+1/x)
8 M

1 (b)
Obtain f(k) given that,

f(k+2)+5f(k+1)+6f (k)=0, k≥0 f(0)=0, f(1)=2 by using Z transform.

f(k+2)+5f(k+1)+6f (k)=0, k≥0 f(0)=0, f(1)=2 by using Z transform.

4 M

2 (a)
An emf E sin (pt) is applied at t=0 to a circuit containing a condenser 'C' and Inductance 'L' in series. The current 'x' satisfies the equation. \[ L (dx/dt) + \dfrac {1} {2} \int x \ dt =E \sin (pt) \ Where \ = \dfrac {-dq}{dt}. \ If \ p^2 = \dfrac {1} {LC} \] and initially the current x and change q is zero then show that current in the circuit at any time is E/2L t sin (pt).

4 M

2 (b)
Solve the integral equation \[ \int^\infty_0 f(x) \cos \lambda x \ dx = \left\{\begin{matrix} 1-\lambda & 0 \le \lambda \le 1\\ 0 & \lambda > 1 \end{matrix}\right. \] And hence show that \[ \int^\infty_0 \dfrac {\sin^2 z}{z^2} dz = \pi/2 \]

4 M

2 (c)
Attempt any one

i) Find the z-transform of f(k)=e

ii) Find the inverse z-transform of \[ \dfrac {z(z+1)}{z^2 - 2z +1}, \ |z| >1 \]

i) Find the z-transform of f(k)=e

^{-2k}cos (5k+3)ii) Find the inverse z-transform of \[ \dfrac {z(z+1)}{z^2 - 2z +1}, \ |z| >1 \]

4 M

Answer any one question from Q3 and Q4

3 (a)
The first four moments of a distribution about 2 are 1, 2.5, 5.5 and 16. Calculate the first four moments about the mean, A, M, S, D, β

_{1}and β_{2}.
5 M

3 (b)
In a certain examination 200 students appeared. Average marks obtained were 50% with standard 5%. How many students do you expect to obtain more than 60% of marks, supposing that the marks are distributed normally ?( Given z=2 ; A = 0.4772).

4 M

3 (c)
Find the directional derivatives of:

ϕ xy

ϕ xy

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

4 (a)
Calculate the coefficient of correlation for the following data.

x |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

y |
9 | 8 | 10 | 12 | 11 | 13 | 14 | 16 | 15 |

5 M

4 (b)
Prove the following (any one): \[ \nabla^4 r^4 =120 \\ \nabla \cdot \left [ r^\nabla \dfrac {1}{r^5} \right ] = \dfrac {15}{r^6} \]

4 M

4 (c)
Such that \[ \overline {F}= (x^2-yz)\overline{i} (y^2 - zx)\overline J + (z^2 - xy)\overline k \] is irrotational. Also find ϕ such that F=∇ϕ.

4 M

Answer any one question from Q5 and Q6

5 (a)
Find the work done in moving a particle along \[ x=a \cos \Theta, \ y=a \sin \Theta, \ z=b\Theta, \ from \ \Theta = \dfrac {\pi}{4} \ to \ \Theta = \dfrac {\pi}{2} \] under a field of force given by \[ \overline F = -3a \sin^2 \Theta \cos \Theta \widehat{i} +a ( 2 \sin \Theta - 3 \sin \Theta)\widehat J +b \sin 2 \Theta \widehat {k} \]

4 M

5 (b)
Evaluate \[ \iint_s ( yz\widehat{i}+ zx\widehat{j}+xy\widehat {k}).d\overline s, \] where s is the curved surface of the cone x

^{2}+y^{2}=z^{2}, z=4
4 M

5 (c)
Using Strokes Theorem to evaluate \[ \int_c (4y \widehat i + 2z\widehat j + 6y\widehat k). d \overline k \] where C is the curve of intersection of x

^{2}+ y^{2}+ z^{2}= 2z and x=z-1
4 M

6 (a)
A vector field is given by \[ \overline F = (2x - \cos y)\widehat i + x (4+\sin y)\widehat j, \ evaluate \ \int_c \overline F.d\overlien r, \] where C is the ellipse \[ \dfrac {x^2}{a^2}+ \dfrac {y^2}{b^2}=1, \ z=0 \]

4 M

6 (b)
Prove that \[ \iiint_v \dfrac {1}{r^2}dv = \iint_s \dfrac {1}{r^2}\overline r . d\overline s, \] where s is closed surface enclosing the volume v. Hence evaluate \[ \iint_s \dfac { xi + yj + z \widehat k}{r^2}\cdot d\overline s, \]k where s is the surface of the sphere x

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

6 (c)
\[ If \ \overline E = \nabla \phi \ and \ \nabla^2 \phi = - 4 \pi p. \] Prove that \[ \iint_s \overline E\cdot d\overline s = - 4\pi \iiint_v \rho \ dv \]

4 M

Answer any one question from Q7 and Q8

7 (a)
Find the value of p such that the function. f(x) = r

^{2}cos 2 Θ + i r^{2}sin p Θ becomes analytical function.
4 M

7 (b)
Evaluate \[ \oint_c \dfrac {z^2 + \cos ^2 z}{\left ( z - \frac {\pi}{4} \right )^3}dz\] where C is a circle x

^{2}+y^{2}=1
5 M

7 (c)
Find the bilinear transformation which maps 1, i, -1 from z plane into i, 0, -i from the w plane.

4 M

8 (a)
Determine the analytic function f(z) whose real part is \[ U=x^3 ? 3xy^2+3x^2-3y^2+1 \]

4 M

8 (b)
Prove \[ \oint _c \left [ \dfrac {\sin \pi z^2 + 2z}{(z-1)^2 (z-2)} \right ]dz \] where C is a circle x~~2+y~~

^{2}=16

5 M

8 (c)
Show that the transformation ω=sin z transforms the straight lines x=c of z plane into hyperbola in the ω plane.

4 M

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