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

Solve the following differential equation, Solve any one question from Q.1(a, b) & Q.2 (a, b)
1(a)(i) $$\dfrac{dx}{dy}=\dfrac{x}{y}+ \cot\left ( \dfrac{x}{y} \right )$$/
4 M
1(a)(ii) $$\dfrac{dx}{dy}=e^{x-y}=4x^3e^{-y}$$
4 M
1(b) A voltage e-at is applied at t = 0 to a circuit containing inductance L, and resistance R. Show that the current at any time t is given by : Provided i = 0 at t = 0 $i=\dfrac{1}{R-aL}\left [ e^{-at} -e^{-\dfrac{Rt}{L}}\right ]$
4 M

2(a) Obtain a differential equation from its general solution
y = c1e4x + c2e-3x,
where c1, c2 are arbitrary constants.
4 M
2(b)(ii) The temperature of air is 30°C. The substance kept in air cools from 100°C to 70°C in 15 minutes. Find the time required to reduce the temperature of the substance upto 40°C.
4 M
Solve
2(b))(i) A body of mass m, falling from rest, is subject to the force of gravity and an air resistance proportional to the square of velocity i.e. kv2, where k is a constant of proportionality. If it falls through a distance x and possesses a velocity v at that instant, show that : where mg = ka2 $x=\dfrac{m}{2k}\log\left [ \dfrac{a^2}{a^2-v^2} \right ]$
4 M

Solve any one question from Q.3(a, b, c) & Q.4 (a, b, c)
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 : $$\int ^1_0 x^m(1-x^n)^p dx.$$
3 M
3(c) Trace the curve (any one) :
(i) x = a(t + sint), y = a(1 cost)
(ii) y2 = x2(1 x).
4 M

4(a) Find the perimeter of cardioid r = a(1 + cosθ).
4 M
4(b) If $$I_n=\int ^{\pi/4}_0 \cos^{2n}x\ dx$$ Prove that : $$I_n=\dfrac{1}{n \cdot 2^{n+1}}+\dfrac{2n-1}{2n}I_{n-1}.$$
4 M
4(c) Evaluate : $$\int ^{\infty}_0 \dfrac{x^4}{4^x}dx.$$
4 M

Solve any one question from Q.5(a, b, c) & Q.6 (a, b, c)
5(a) Find the equation of the sphere which touches the coordinate axes, whose centre is in the positive octant and has radius 4.
5 M
5(b) Find the equation of the cone with vertex at (1, 2, -3), semivertical angle $$\cos^{-1}\left ( \dfrac{1}{\sqrt{3}} \right )$$ and the line : $$\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z+1}{-1}$$ as axis of the cone.
4 M
5(c) Find the equation of the right circular cylinder whose guiding curve is :
x2 + y2 + z2 = 9,
x y + z = 3.
4 M

6(a) Find the centre and radius of the circle of intersection of the sphere x2 + y2 + z2 - 2y - 4z - 11 = 0 by the plane x + 2y + 2z = 15.
5 M
6(b) Obtain the equation of a right circular cone which passes through the point (2, 1, 3) with vertex (2, 1, 1) and axis parallel to the line : $\dfrac{x-2}{2}=\dfrac{y-1}{1}=\dfrac{z+2}{2}$
4 M
6(c) Find the equation of the right circular cylinder whose axis is : $$\dfrac{x-2}{2}=\dfrac{y-1}{1}=\dfrac{z}{3}$$ and which passes through the point (0, 0, 3).
4 M

Attempt any two of the following, Solve any one question from Q.7(a, b, c) & Q.8 (a, b, c)
7(a) Evaluate by changing the order of integration : $\int^{\infty}_0 \int ^x_0xe^{-x^2/y}\ dy\ dx.$
7 M
7(b) Find the volume of solid common to the cylinders :
x2 + y2 = a2
x2 + z2 = a2.
6 M
7(c) Find the moment of inertia of the circular plate r = 2a cosθ about θ = π/2 line.
6 M

8(a) Find the total area of the Astroid :
x2/3 + y2/3 = a2/3.
7 M
8(b) Evaluate: $\iiint _v \sqrt{x^2+y^2}\ dx\ dy\ dz,$
whereV is the volume of the cone x2 + y2 = z2, z > 0 bounded by z = 0 and z = 1 plane.
6 M
8(c) Find centre of gravity of area of the cardioid :
r = a(1 + cosθ).
6 M

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