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

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

4 M

1(a)(ii)
\[ \frac{d^2y}{dx^2}+\frac{dy}{dx}=\frac{1}{1+e^x}\]

4 M

1(a)(iii)
\[ \left ( 1+x \right )^2\frac{d^2y}{dx^2}+\left ( 1+x \right )\frac{dy}{dx}+y=2\sin \log \left ( 1+x \right )\]

4 M

1(b)
Apply Gauss elimination method to solve: \[\begin{align*} &x+4y-z=5\\
&x-y-6z=-12\\
&3x-y-z=4.\end{align*}\]

4 M

2(a)
The deflection of a strut with one end built in (x=0) and other supported and sunjected to end thrust P satisfies the equation \( \frac{d^2y}{dx^2}+a^2y=\frac{a^2R}{P}(1-x).\)/ Given that \( \frac{dy}{dx}=y=0 \)/ When x=0 and y=0, when x=l. Prove that: \(y=\frac{R}{P}\left [ \frac{sin ax}{a}-l\ \cos a x+1-x \right ] \)/ where al= tan al.

4 M

2(b)
Apply Runge-Kutta method of fourth order to solve \(10\frac{dy}{dx}=x^2+y^2; \ \ y(0)=1 \)/ for x=0.1 taking h=0.1.

4 M

2(c)
Solve the following system by using traingularisation method:\[\begin{align*}&x_1+x_2-x_3=2
\\&2x_1+3x_2+5x_3=-3
\\&3x_1+2x_2-3x_3=6 \end{align*}
\]

4 M

Solve any one Q.4b(i, ii) Solve any one question Q.3 (a,b,c) and Q.4(a,b,c)

3(a)
Claculate the coeffcient of correlation between the marks obtained by 8 students in Mathematics and Statistics:

Mathematics Marks | Statistic Marks |

25 | 08 |

30 | 10 |

32 | 15 |

35 | 17 |

37 | 20 |

40 | 23 |

42 | 24 |

45 | 25 |

4 M

3(b)
If mean and variance of a binomial distribution are 4 2 respectively, find probability of:

i) exactly 2 successes and

ii) less than 2 success.

i) exactly 2 successes and

ii) less than 2 success.

4 M

3(c)
Find the directional derivative of the function \( \phi = x^2-y^2+2x^2 \)/ at the point, P(1, 2, 3) in the direction of line PQ, where Q the point (5,0,4).

4 M

4(a)
If the first four moment of a distribution about the value 5, are eqaul -4, 22, -117 and 560, determine the central moment and β>sub>1 and β

_{2}.
4 M

4(b)(i)
Attempt any one:

i) \[\nabla^2(r^n \log r)=\left [ n(n+1)\log r +2n+1 \right ]r^{n-2}\]

i) \[\nabla^2(r^n \log r)=\left [ n(n+1)\log r +2n+1 \right ]r^{n-2}\]

2 M

4(b)(ii)
Evaluate \[\nabla.\left [ r^\nabla\left ( \frac{1}{r^3} \right ) \right ].\]

2 M

4(c)
Show that the vector field given by

\[\bar{A}=\left [ x^2+xy^2\right ]i+\left ( y^2+x^2y \right )j

\[\bar{A}=\left [ x^2+xy^2\right ]i+\left ( y^2+x^2y \right )j

4 M

Attempt any two. Solve any one question Q.5&Q.6

5(a)
Evaluate \( \int _c\bar{F}.d\bar{r}\ \ \text{where} \bar{F}=\left ( 2x+y^2 \right )i+\left ( 3y-4x \right )\bar{j} \)/ along the parabolic are y

^{2}=x joining (0,0) to (1,1).
6 M

5(b)
Evaluate \( \int \int _s\bar{F}.d\bar{S},\ \ \text{where} \bar{F}=yz\bar{i}+zx\bar{j}+xy\bar{k}\)/ and S is the part of the surface of the sphere \(x^2+y^2+z^2=1 \)/ which lies in the first octant.

6 M

5(c)
Evaluate \( \iint _s\nabla\times \bar{F}.\hat{n}dS\)/ for the surface of the paraboloid \( z=4-x^2-y^2,z\geq 0 \ \ \text{and}{\bar{F}}=y^2\bar{i}+z\bar{j}+xy\bar{k}.\)/

7 M

Attempt any two:

6(a)
Attempt any two:

Find the work done in moving a particle once round the ellipse\( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, z=0 \)/ under the field of force given by \( \bar{F}=\left ( 2x-y+z \right )\bar{i}+\left ( x+y-z^2 \right )\bar{j}+\left (2x-3y+4z \right )\bar{k}. \)/

Find the work done in moving a particle once round the ellipse\( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, z=0 \)/ under the field of force given by \( \bar{F}=\left ( 2x-y+z \right )\bar{i}+\left ( x+y-z^2 \right )\bar{j}+\left (2x-3y+4z \right )\bar{k}. \)/

6 M

6(b)
Show that \(\int _c\left [ \bar{u}\times \left ( \bar{r} \times \bar{v}\right ) \right ].d\bar{r}=-\left ( \bar{u} \times \bar{v}\right ).\iint_sd\bar{S}, \)/ where S is the open surface bounded by closed curve C and \( \bar{u}\ \text{and}\ \bar{v}\)/ are constant vectors.

6 M

6(c)
Evaluate: \(\iint _s\left ( y^2z^2\bar{i}+z^2x^2\bar{j}+x^2y^2\bar{k} \right ).d\bar{S}, \)/ where s is the surface of the sphere \( x^2+y^2+z^2=4 \)/ in the positive octant.

7 M

Solve any one question fromQ.7(a, b, c) and Q.8(a,b,c)

7(a)
A tightly stretched string with fixed end point x =0 and x=l is initially in a position given by y(x, o) = y

_{0}sin^{3}\( \left ( \frac{\pi }{l} \right )\)/. If it is released from this position, find the displacement y at any distance x from one end and at any time t.
7 M

7(b)
Solve the one-dimensional heat flow equation \( \frac{\partial u}{\partial t}=c^2\frac{\partial^2 u}{\partial x^2}\)/ subject to the following conditions:

i) u (0, t) =0

ii) u (l, t) = 0 for all t,

iii) u(x,0) = x, 0≤x≤l/2

\(l-x,\frac{1}{2}\leq x\leq l. \)/

iv) u(x, t) is bounded.

i) u (0, t) =0

ii) u (l, t) = 0 for all t,

iii) u(x,0) = x, 0≤x≤l/2

\(l-x,\frac{1}{2}\leq x\leq l. \)/

iv) u(x, t) is bounded.

6 M

7(c)
A thin sheet of metal is bounded by x-axis and lines x=0 and x=l and stretched to infinity in y direction has its upper and lower faces perfectly insulated and its vertical edges and edge at infinty are maintained at constant temperature0°C, while the temperature on the short edge y=0 is maintained at 100°C. Find the steady state temperature u(x, y).

6 M

8(a)
A string is stretched tightly between x=0 and x=l and both ends are given displacement y = a sin pt perpendicular to the string. If the string satisfies the differential eqaution \( \frac{\partial ^2y}{\partial x^2}=\frac{1}{c^2}\frac{\partial ^2u}{\partial t^2},\)/ prove that the oscillations of the string are given by: \( y=a\
sec\frac{pl}{2c}\cos \left ( \frac{px}{c}-\frac{pl}{2c} \right )\sin pt.\)/

7 M

8(b)
Solve the following one-dimensional heat flow equation: \( \frac{\partial u}{\partial t}=c^2\frac{\partial^2 u}{\partial x^2} \)/ subject to the following conditions:

i) u (0, t) =0

ii) u ( l, t) = 0 for all t,

iii) u (x, 0) = x, 0

i) u (0, t) =0

ii) u ( l, t) = 0 for all t,

iii) u (x, 0) = x, 0

6 M

8(c)
A thin metal plate bounded by the x-axis and the lines x=0 and x=1 and stretching to infinity in y-direction has its upper and lower faces perfectly insulated and its vertical edges and edge at infinity are maintained at the constant temperature 0°C, while over the base temperature of 50°C is maintained. Find the steady state temperature u(x, y).

6 M

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