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SPPU First Year Engineering (Semester 2)
Engineering Mathematics-2
April 2013
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

Answer any one question from Q1 & Q2
1 (a) Solve the following differential equations. $(i) \ (x^4e^x-2mxy^2)dx+2mx^2 ydy=0 \\ (ii) \ \left ( \tan \dfrac {y}{x}- \dfrac {y}{x} \sec^2 \dfrac {y}{x} \right )dx+\sec^2 \dfrac {y}{x} dy =0$
8 M
1 (b) A constant electromotive force E volts is applied to a circuit containing a constant resistance R ohms in series and a constant inductance L henries. If the initial current is zero, show that the current builds up to half its theoretical maximum in $\dfrac {L \log 2}{R} \ seconds.$
4 M

2 (a) Solve $\left [ \log (x^2+y^2) + \dfrac {2x^2}{x^2+y^2} \right ]dx+ \dfrac {2xy}{x^2+y^2}dy=0$
4 M
2 (b) Solve the following :-
(i) A particle is moving in a straight line with an acceleration $k \left [ x+ \dfrac {a^4}{x^3} \right ]$ directed towards origin. If it starts from rest at a distance 'a' from the origin, prove that it will arrive at origin at the end of time $\dfrac {\pi}{4 \sqrt{k}} .$
(ii) A pipe 10cm in diameter contains steam at 100° C. It is covered with asbestos,5cm thick,for which k=0.0006 and the outside surface is at 30°C .Find the amount of heat lost per hour from a meter long pipe.
8 M

Answer any one question from Q3 & Q4
3 (a) Express f(x)=π2-x2, -π<x>π as a fourier series, where f(x)=f(x+2π)
5 M
3 (b) Evaluate: [ int^{infty}_0 dfrac {x^8 - x^{14}}{(1+x)^{24}}dx ]
3 M
3 (c) Trace the curve (Any one)
(i) y2=x2(1-x)
(ii) r=2sin 3?
4 M

4 (a) show that the length of an arc of the curve
x=log (sec ? + \tan ?)-sin ?, y=cos ? from ?=0 to ?=t is log(sec t).
4 M
4 (b) Evaluate: $\int^{\pi}_0 x \sin^5 x \cos^2 x \ dx$
4 M
4 (c) Evaluate: $\int^{1}_0 \left [ \dfrac {x^m-1}{\log x} \right ]dx$
4 M

Answer any one question from Q5 & Q6
5 (a) Find the equation of the sphere, having its center on the plane 4x - 5y - z = 3 and passing through the circle. x2+y2+z2-2x-3y+4z+8=0, x-2y+z=8.
5 M
5 (b) Find the equation of a right cicular cone, having vertex at the point (0,0,3) and passing through the circle x2+y2=16, z=0.
4 M
5 (c) Find the equation of a right circular cylinder of radius 2,whose axis passes through the point (1,1,-2) and has direction cosines proportional to 2,1,2.
4 M

6 (a) Find the equation of the sphere which is tangential to the plane 4x - 3y + 6z - 35 = 0 at (2,-1,4) and passing through the point (2,-1,-2).
5 M
6 (b) Find the equation of a right circular cone with vertex at origin,the line x = y = 2z as the axis and semi-vertical angle 30°.
4 M
6 (c) Find the equation of a right circular cylinder whose axis is 2(x-1) = y+2 = z and radius is 4.
4 M

Answer any one question from Q7 & Q8
7 (a) Solve any two:
(a) Evaluate $\int^{a}_0 \int^{y}_{y^2/a}\dfrac {ydxdy}{(a-x)\sqrt{ax-y^2}}$
7 M
7 (b) Evaluate $\int \int_y \int \sqrt{x^2+y^2}dxdydz,$ where V is bounded by the surface x2+y2=z2, z ? 0 and the plane z=1.
6 M
7 (c) Find the moment of inertia (M.I.) about the line ? =?/2 of the area enclosed by r=a(1-cos ?).
6 M

8 (a) Solve any two:
(a) Find by double integration the area between the curve y2 x=4a2(2a-x) and its asymptote.
7 M
8 (b) Find the volume of the cylinder x2+y2=2ax intercepted between the paraboloid x2+y2=2az and xoy-plane.
6 M
8 (c) Find the centre of gravity (C.G.)of one loop of the curve r = a sin?
6 M

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