## 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}-x^{2}, -??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) y^{2}=x^{2}(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. x^{2}+y^{2}+z^{2}-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 x^{2}+y^{2}=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 x^{2}+y^{2}=z^{2}, 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 y^{2} x=4a^{2}(2a-x) and its asymptote.(7 marks)
**8 (b)** Find the volume of the cylinder x^{2}+y^{2}=2ax intercepted between the paraboloid x^{2}+y^{2}=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)