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SPPU First Year Engineering (Semester 2)
Engineering Mathematics-2
May 2017
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

Solve any one question from Q.1(a,b) & Q.2(a,b)
1(a) Solve the following differential equations:\begin{align*}& i)x^{4}\frac{dy}{dx}+x^{3}y=sec(xy)\\ &ii)\frac{dy}{dx}=\frac{1+y^{2}+3x^{2}y}{1-2xy-x^{3}}\end{align*}
8 M
1(b) A body starts moving from rest is opposed by a force per unit mass of value cx and resistance per unit mass of a value bv2 where x and v are the displacement and velocity of the particle at that instant show that the velocity of the particle is given by:$v^{2}=\frac{c}{2b^{2}}\left ( 1-e^{-2bx} \right )-\frac{cx}{b}$
4 M

2(a) Solve: $\frac{dy}{dx}+x\sin 2y=x^{3} \cos^{2} y.$
4 M
2(b) Solve the following :
i) Water at temperature 100°C cools in 10-minutes to 88°C in a room of temperature 25°C. Find the temperature of water after 20 minutes.
ii) A resistance of 100Ω, an inductance of 0.5 henry are connected in a series with battery of 20 volts. Find the current in a circuit as a function of time t .
8 M

Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)
3(a) Find Fourier series to represent the function f(x) = x in -π < x< π and f(x) = f(x+2π)
5 M
3(b) Evaluate:$\int_{0}^{\infty }\sqrt{y.}e^\sqrt{y}\ dy$
3 M
Solve any one question from Q.3(c) (i,ii)
3(c)(i) Trace the curve:$y^{2}\left ( x^{2}-1 \right )=x$
4 M
3(c)(ii) $r = a(1+\cos \theta ).$
4 M

Solve any one question from Q.4(a, b, c)
4(a) If : \begin{align*}I_{n}=\int_{\pi /4}^{\pi /2}\cot ^{n}\ \theta\ d\theta ,\\ \text{prove that}:\\ I_{n}=\frac{1}{n-1}-I_{n-2}. \end{align*}/
4 M
4(b) $$\text{Prove that}:\\ \\ \int_{0}^{1}\frac{x^{a}x^{b}}{\log x}dx=\log \left ( \frac{a+1}{b+1} \right ),a>0, b>0$$/
4 M
4(c) Find the legth of the arc of cycloid $$x = a \left ( \theta +\sin \theta \right ), y=a\left ( 1-\cos \theta \right )$$/ between two consecutive cups.
4 M

Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)
5(a) Find the centre and radius of the circle which is an intersection of the sphere $$x^{2}+y^{2}+z^{2}-2y-4z-11=0$$/ by the plane $x+2y+2z=15.$
5 M
5(b) Find the equation of the right circular cone which passes through the point (1, 1, 2) & has its axis along the line 6x = -3y = 4x and vertex at (0,0,0).
4 M
5(c) Find the equation of a right circular cylinder of radius 2 whose axis passes through (1, 2, 3) and has direstion cosines proportional to 2, -3, 6.
4 M

Solve any two question from Q.6 (a, b, c)
6(a) Show that the plane 4x-3y+6z-35 = 0 is tangential to the sphere $$x^{2}+y^{2}+z^{2}-y-2z-14=0.$$/
5 M
6(b) Find the equation of a right circular cone whose vertex is at ( 1, 2, 3) and axis has direction ratios (2, -1, 4) and semivertical angle 60°.
4 M
6(c) Find the equation of the right circular cylinder of radius 3 whose axis is the line $\frac{x-1}{2}=\frac{y-3}{2}=\frac{z-5}{-1}.$
4 M

Solve any two question from Q.7(a, b, c) Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)
7(a) Evaluate $$\iint \frac{x^{2}y^{2}dxdy}{x^{2}y^{2}}$$/ where R is annulus between $$x^{2}+y^{2}=4, x^{2}+y^{2}=9.$$/
6 M
7(b) Evaluate $$\iiint \left ( x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2} \right )dxdydz$$/ throughout the volume of sphere$$x^{2}+y^{2}+z^{2}= a^{2}.$$/
7 M
7(c) Find the moment of inertia of one loop of lemniscate $$r^{2}=a^{2}\cos 2\theta$$/
6 M

Solve any two question from Q.8(a, b, c)
8(a) Find the total area included between the two cardioids $r = a\left ( 1+\cos \theta \right )\ \ \text{and}\ r =a\left ( 1-\cos \theta \right ).$
6 M
8(b) Find the volume cut-off from the paraboloid $$x^{2}+\frac{y^{2}}{4}+z=1$$/ by the plane z=0.
7 M
8(c) Find the C.G. of an area of the cardioid:$r =a\left ( 1+\cos \theta \right ).$
6 M

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