MORE IN Engineering Maths 3
SPPU Civil Engineering (Semester 3)
Engineering Maths 3
December 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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.
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 β2.
4 M
4(b)(i) Attempt any one:
i) $\nabla^2(r^n \log r)=\left [ n(n+1)\log r +2n+1 \right ]r^{n-2}$
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
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 y2=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 $$\bar{F}=\left ( 2x-y+z \right )\bar{i}+\left ( x+y-z^2 \right )\bar{j}+\left (2x-3y+4z \right )\bar{k}.$$/
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) = y0 sin3$$\left ( \frac{\pi }{l} \right )$$/. 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.
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
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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