Answer any one question from Q1 and Q2

1 (a)
Solve any two of the following: \[ i) \ \dfrac {d^2 y}{dx^2} - 3 \dfrac {dy}{dx} + 2y = x^2 + \sin x \\ ii) \ \dfrac {d^3 y}{dx^3}-7 \dfrac {dy}{dx} - 6y = e^{2x} (1+x) \\ iii) \ \dfrac {d^2 y}{dx^2}+ \dfrac {1}{x} \dfrac {dy}{dx} = A + B \log x \]

8 M

1 (b)
Find the Fourier transform of f(x)=e

^{-|x|}.
4 M

2 (a)
A body of weight W=3N stretches a spring to 15 cm. If the weight is pulled down 10 cm below the equilibrium position and then given a downward velocity 60 cm/sec. Find the subsequent motion of the weight.

5 M

2 (b)
Find \[ i) \ \ L^{-1} \left \{ \dfrac {1}{s^3 (s^2+1)} \right \} \ \ OR \\ ii) \ \ L^{-1} \left \{ \dfrac {1}{(s+1)(s^2+1)} \right \} \]

3 M

2 (c)
Using Laplace transform method, solve the differential equation \[ \dfrac {d^2 y}{dt^3} - 3 \dfrac {dy}{dt}+2y = 12 e^{-2t} \] with conditions y(0)=2, y'(0)=6.

4 M

Answer any one question from Q3 and Q4

3 (a)
Find the coefficient of correlation between x and y from the table.

x |
1 | 3 | 4 | 6 | 8 | 9 | 11 | 14 |

y |
1 | 2 | 4 | 4 | 5 | 7 | 8 | 9 |

4 M

3 (b)
Prove that (any one): \[ i) \ \nabla ^4 e^r =e^r + \dfrac {4}{r}e^r \\ ii) \ \nabla \cdot \left [ r\nabla \dfrac {1}{r^n} \right ] = \dfrac {n(n-2)}{r^{n+1}} \]

4 M

3 (c)
Show that the vector field \[ \overline F = (y^2 \cos x + z^2) i + 2y \sin x \ j+2xzk \] is irrotational and find scalar potential ϕ such that F=∇ϕ.

4 M

4 (a)
If the probability that an individual suffers a bad reaction from a certain injection is 0.001, determine the probability that out of 2000 individual, more than 2 individuals will suffers a bad reaction.

4 M

4 (b)
Find the directional derivative of div (x

^{5}i+y^{5}j+z^{5}k) at (1,2,3) in the direction of outward normal to surface x^{2}+y^{2}+z^{2}=9 at (1,2,3).
4 M

4 (c)
Calculate the first four moments about Mean for the following data:

x |
1 | 3 | 4 | 6 | 8 | 9 | 11 | 14 |

y |
1 | 2 | 4 | 4 | 5 | 7 | 8 | 9 |

4 M

Answer any one question from Q5 and Q6

5 (a)
Evaluate: \[ int_c \overline F \cdot d \ \overline r \] where 'c' is any square with sides of length 5 and \[ \overline F (2x^2-y) \overline i + (\tan y-e^y + 4x) \overline j. \]

4 M

5 (b)
Evaluate: \[ \iint_s (z^2-x)dydz - xydxdz + 3zdxdy \] where 's' is the surface of closed region bounded by z=4 -y

^{2}and planes x=0, x=3, z=0.
5 M

5 (c)
Evaluate: \[ \iint_s (\nabla \times \overline F) \cdot d\overline s \ for \ \overline F = y \overline i + z \overline j + x \overline k \] over the surface x

^{2}+y^{2}=1 z, z≥0.
4 M

6 (a)
Evaluate: \[ \int_c \overline F. d \overline r, \ where \ \overline F = y\overline i + xz^3 \overline j - zy^2 \overline k \] and 'c' is the circle x

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

6 (b)
Evaluate: \[ \int_c \overline F. d \overline\] by using Stokes theorem for \[ \overline F=4y \overline i - 4x\overline j + 3\overline k,\] where 's' is a disk of radius 1 lying on the plane z=1 and 'c' is its boundary.

5 M

6 (c)
If n is the unit outward drawn normal to any closed surface 's' of area s then evaluate \[ \iiint_v \nabla \cdot \overline n dv. \]

4 M

Answer any one question from Q7 and Q8

7 (a)
If a string of length 4 cm is initially at rest in its equilibrium position and each of its point is given velocity v(x) such that \[ v(x) = \left\{\begin{matrix}
3x, & 0\le x \le 2 \\
3(4-x),& 2 \le x \le 4
\end{matrix}\right. \] Obtain the displacement y(x,t) at any time t.

7 M

7 (b)
Solve \[ \dfrac {\partial u}{\partial t} = \dfrac {\partial ^2 u}{\partial x^2} \ if \\ i) \ \ u \ is \ finite \ for \ all \ t, \\ ii) \ u(0,t)=0 \\ iii) \ u(l,t)=0 \\ iv) \ u(x,0) = \dfrac {3x}{l}, \ 0\le x\le l. \]

6 M

8 (a)
Use Fourier transform to solve \[ \dfrac {\partial u}{\partial t} = \dfrac {\partial ^2u}{\partial x^2}, \ 0< x < \infty , \ t>0 \] where u(x,t) satisfies the conditions: \[ i) \ \ \left ( \dfrac {\partial u}{\partial x} \right )_{x=0} = 0 , \ t>0 \\ ii) \ u(x,0)=\left\{\begin{matrix}
x, &01
\end{matrix}\right. \\ iii) \ \big |u(x,t)\big | < M. \]

7 M

8 (b)
An infinitely long uniform metal plate is enclosed between lines y=0 and y=1 for x>0. The temperature is zero along the edges y=0, y=l and at infinity. If the edge x=0 is kept at a constant temperature u

_{o}, find the temperature distribution u(x, y).
6 M

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