Answer any one question from Q1 and Q2

1 (a)
Solve (any two)

i) (D

ii) d

iii) x

i) (D

^{2}-1)y=x sin x + (1+x^{2}e^{x}ii) d

^{2}y/dx^{2}+y = cosec x (by variation of parameters)iii) x

^{2}d^{2}y/dx^{2}-4x dy/dx+6y=x^{5}
8 M

1 (b)
Find Fourier cosine transform of the function \[ f(x)= \left\{\begin{matrix}
\cos x &0 < x< a \\0
&x>a
\end{matrix}\right. \]

4 M

2 (a)
A resistance of 50 ohms, an inductor of 2 Henry and farad capacitor are all in series with an e.m.f. of 40 volt. Find the instantaneous change and current after the switch is closed at t=0, assuming that at that time the change on the capacitor is 4 Coulomb.

4 M

2 (b)
Solve (any one)

i) Find z transform for f(k)=(1/3)

Find inverse z transform of [Z

i) Find z transform for f(k)=(1/3)

^{|k|}Find inverse z transform of [Z

^{2}/ (Z/14)(Z-1/5)] for |Z|<1/5
4 M

2 (c)
Solve f(k+2)+3 f(k+1)+2f(k)=0 Given f(0)=0, f(1)=1

4 M

Answer any one question from Q3 and Q4

3 (a)
Solve the following differential equation to get y(0.1) dy/dx=x-y

^{2}, y(0)=1.
4 M

3 (b)
Find Lagrange's long interpolating polynomial passing through set of points

Use it to find y at x=1.5 and find ∫

x |
0 | 1 | 2 |

y |
2 | 3 | 6 |

Use it to find y at x=1.5 and find ∫

_{0}^{2}y dx.
4 M

3 (c)
Find the directional derivative of ϕ=3 log (x+y+z) at (1,1,1) in the direction of tangent to the curve x=b sint, y=b cost, z=bt at t=0

4 M

4 (a)
Show that (any one) \[ i) \ \ \nabla^2 \left [ \nabla \cdot (\overline r / r^2) \right ] = 2/r^4 \\ ii) \ \ \nabla (\overline a \cdot \overline r / r^3)= \overline a /r^3 - 3 (\overline a \cdot \overline r)/r^5 \overline r \]

4 M

4 (b)
If φ, ψ satisfy Laplace equation then prove that the vector (φ ∇ ψ - ∇ ψ φ) is solenoidal.

4 M

4 (c)
Use Simpson's 1/3

^{rd}rule to find. \[ \int^{0.6} _0 e^{-x^3} \ dx \] by taking seven ordinates.
4 M

Answer any one question from Q5 and Q6

5 (a)
Find the work done by F=(2x + y

^{2}) i + (3y-4x)j in taking particle around the parabolic arc y=x^{2}from (0,0) to (1,1).
4 M

5 (b)
Apply Stoke's theorem to evaluate \[ \oint_c (4y \ dx + 2 zdy + 6 \ ydz) \] where is curve of intersection of x

^{2}+y^{2}+z^{2}= 2z and z=x+1.
5 M

5 (c)
Evaluate \[ \iint_s (2y\widehat{i} + yzj + 2xz\widehat{k} )\cdot d\overline s \] over the surface of region bounded by y=0, y=3, x=0, z=0, x+2z=6.

4 M

6 (a)
Using Green's Lemma, evaluate \[ \int_c \overline F . d\overline r \ where \ \overline F = 3 y \ i + 2xj\] and c is boundary of region bounded by y=0, y=sinx fpr 0≤x≤π.

4 M

6 (b)
Evaluate \[ \iint_s (z^2-x) dydz - xydxdz + 3zdxdy \]Where us closed surface of region bounded by x=0, x=3, z=0, z=4-y

^{2}.
5 M

6 (c)
Show that \[ \overline E = - \nabla \phi ? 1 /c \dfrac {\partial \overline A} {\partial t}; \ \ \overline H = \nabla \times \overline A\] are solutions of Maxwell's equations \[ i) \ \nabla \cdot \overline H = 0 \\ ii) \ \nabla \times \overline H = 1/c \dfrac {\partial \overline E}{\partial t} \ if \ \nabla \cdor \cdot \overline A + 1/c \dfrac {\partial \phi}{\partial t} \ and \ \nabla^2 \overline A=1/c \dfrac {\partial ^2 A}{\partial t^2} \]

4 M

Answer any one question from Q7 and Q8

7 (a)
Show that the anaytic function with constant amplitude is constant.

4 M

7 (b)
By using Cauchy's integral formula evaluate \[ \oint_c 2Z^2+ z/z^2 -1 \ dx \] where C is the circle |z-1|=1.

5 M

7 (c)
Find the bilinear transformation which maps the points z=1,0,1 on the points W=0,i,3i of w-plane.

4 M

8 (a)
If u=cos hz cosy then find the harmonic conjugate v such that f(z)=u+iv is analytical function.

4 M

8 (b)
Evaluate \[ \int_c \dfrac {12 z-7} {(z-1)^2 (2z+3)} dz \] where c is the circle |z|=2 using Cauchy's residue theorem.

5 M

8 (c)
Show that the transformation w=z+1/z-2i maps the circle |z|=2 into an ellipse.

4 M

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