Solve any two question Q.1(a)(i, ii, iii) Solve any one question from Q.1(a,b) & Q.2(a,b,c)

1(a)(i)
\( \frac{d^{2}y}{dx^{2}}-y=\frac{2}{l+e^{x}}\text{(use method of variation of parameters)} \)/

4 M

1(a)(ii)
\[\left ( D^{2} -4\right )y=e^{4x}+2x^{3}\]

4 M

1(a)(iii)
\[\left ( 2x+1 \right )^{2}\frac{d^{2}y}{dx^{2}}-2\left ( 2x+1 \right )\frac{dy}{dx}-12y=24x.\]

4 M

1(b)
Solve the following intergral equation using Fourier transform:\[\begin{align*} \int_{0}^{\infty }f(x)\sin \lambda xd\lambda &=1-\lambda ,0\leq \lambda \leq 1\\
\\\
\ \ \ \ &=0 , \lambda \geq 1.\end{align*}\]

4 M

2(a)
A electrical circuit consists of an inductance 0.1 henry, a registance R of 20 ohms and a condenser of capacitance C of 25 microfarads. If the differential equation of electric circuit is: \( L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{q}{C}=0, \)/ then find the charge q and current i at any time t, given that when t = 0, q = 0.05 columbs and \[i=\frac{dq}{dt}=0\]

4 M

Solve any two question Q.2(b)(i, ii)

2(b)(i)
Find: \(z^{-1}\left \{ \frac{1}{\left ( z-4 \right )\left ( z-5 \right )} \right \} \)/ by inversion integral method.

4 M

2(b)(ii)
Find z transform of: \[f(k)=\left ( k+1 \right )a^{k},k\geq 0.\]

4 M

2(c)
Using z transform, solve the following difference equation:\[\begin{align*} f(k+1)+\frac{1}{2}f(k)\left ( \frac{1}{2} \right )^{k}, k\geq 0\\
\
f(0)=0\end{align*}\]

4 M

Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)

3(a)
The first four moments of a distribution about the value 4 of the variable are -1.5, 17, -30 and 108. Find the central moments, β

_{1}and β_{2}
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3(b)
By the method of least squares, find the straight line that best fits the following data:

x | y |

1 | 14 |

2 | 27 |

3 | 40 |

4 | 55 |

5 | 68 |

4 M

Solve any one question from Q.3(c) (i,ii)

3(c)
There is small chance of 1/1000 for any computer produced to be defective. Determin in a sample of 2000 computers, the probability:

i) no defective and

i) 2 defectives.

i) no defective and

i) 2 defectives.

4 M

Solve any one question from Q.4(a, b, c)

4(a)
Team A has a probability of \( \frac{2}{3}\)/ of wining whenever the team plays a particular game. If team A plays 4 games, find the probability that the team wins:

i) exactly two games and

ii) at least two games.

i) exactly two games and

ii) at least two games.

4 M

4(b)
The lifetime of certain component has a normal distribution with mean 400 hours and standard deviation of 50 hours Assuming a normal sample of 1000 components, determine approximately the number of components whose lifetime lies between 340+465 hours. Given:

Z = 1.2 Area = 0.3849

Z = 1.3 Area = 0.4032

Z = 1.2 Area = 0.3849

Z = 1.3 Area = 0.4032

4 M

4(c)
Calculate the coefficient of correlation for the following data:

x | y |

10 | 18 |

14 | 12 |

18 | 24 |

22 | 6 |

22 | 30 |

30 | 36 |

4 M

Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)

5(a)
Find the directional derivative of a function:\( \phi =2x^{2}+3y^{2}+z^{2}\ \ \text{at}\left ( 2, 1, 3 \right ) \)/ in the direction of (i+j+k).

4 M

5(b)
Show that the vector field: \( \bar{F}=\left ( x+2y+4z \right )i+\left ( 2x-3y-z \right )j+\left ( 4x-y+2z \right )k\)/ is irrotational and hence find a scalar potential function φ such that \[\bar{F}=\nabla\phi .\]

4 M

5(c)
Find the work done by a force field:\(\bar{F}=x^{2}i+\left ( x-y \right )j+\left ( y+z \right )k \)/ along a straight line from (0, 0, 0) to (2, 1, 2).

5 M

6(a)
Find the directional derivative of : \(\phi =4xz^{3}-3x^{2}y^{2}z\ \ \text{at}\ \ (1, 1, 1) \)/ in the direction of a vector 3i-2j+k.

4 M

Solve any one question from Q.6 (b) (i, ii)

6(b)(i)
\( \[\nabla\left ( \frac{\bar{a}.\bar{r}}{r^{3}} \right )=\frac{\bar{a}}{r^{3}}-\frac{3\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{5}}\] \)/ where \( \bar{a} \)/ is a constant vector.

4 M

6(b)(ii)
\[\nabla^{4}\left ( r^{4} \right )=120.\]

4 M

6(c)
Evaluate the integral: \(\int _{C}\bar{F}.d\bar{r} \)/ along the curve x = y = z t from t = 0 to t = 2 where \( \bar{F}=\left ( x^{2}+yz \right )i+\left ( y^{2}+zx \right )j+\left ( z^{2}+xy \right )k \)/

5 M

Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)

7(a)
If \( u=3x^{2}y-y^{3},\)/ find v such that \( f(z)= u +iv \)/ is analytic.

4 M

7(b)
Evaluate : \( \oint \frac{z+4}{\left ( z+1 \right )\left ( z+2 \right )}dz, \)/ where C is the circle |z| < 3.

5 M

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

4 M

8(a)
If \( u = 3x^{2}-3y^{2}+2y,\)/ find v such that f(z) = u +iv is analytic. Determine f(z) in terms of z.

4 M

8(b)
Evaluate: \( \oint _C\frac{Az^{2}+z}{z^{2}-1}dz,\)/ where C is the contour \( |z-1|=\frac{1}{2}. \)/

5 M

8(c)
Find the map of straight line y = x under the transformation \( w=\frac{z-1}{z+1}\)/

4 M

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