Engineering Mathematics-2 - Apr 2013

First Year Engineering (Semester 2)

TOTAL MARKS: 50
TOTAL TIME: 2 HOURS (1) Solve Q.1 or Q.2, Q.3 or Q.4, Q.5 or Q.6, Q.7 or Q.8
(2) Assume suitable data, if 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 marks) 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 marks) 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 marks) 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 marks)

Answer any one question from Q3 & Q4

3 (a) Express f(x)=?²-x², -??x?? as a fourier series, where f(x)=f(x+2?)(5 marks) 3 (b) Evaluate: [ int^{infty}_0 dfrac {x^8 - x^{14}}{(1+x)^{24}}dx ](3 marks) 3 (c) Trace the curve (Any one)
(i) y²=x²(1-x)
(ii) r=2sin 3?(4 marks) 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 marks) 4 (b) Evaluate: $$ \int^{\pi}_0 x \sin^5 x \cos^2 x \ dx $$(4 marks) 4 (c) Evaluate: $$ \int^{1}_0 \left [ \dfrac {x^m-1}{\log x} \right ]dx $$(4 marks)

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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 6 (c) Find the equation of a right circular cylinder whose axis is 2(x-1) = y+2 = z and radius is 4.(4 marks)

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 marks) 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 marks) 7 (c) Find the moment of inertia (M.I.) about the line ? =?/2 of the area enclosed by r=a(1-cos ?).(6 marks) 8 (a) Solve any two:
(a) Find by double integration the area between the curve y² x=4a²(2a-x) and its asymptote.(7 marks) 8 (b) Find the volume of the cylinder x²+y²=2ax intercepted between the paraboloid x²+y²=2az and xoy-plane.(6 marks) 8 (c) Find the centre of gravity (C.G.)of one loop of the curve r = a sin?(6 marks)