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SPPU First Year Engineering (Semester 2)
Engineering Mathematics-2
May 2014
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 Q1 & Q2
1 (a) Solve the following $i) \ \dfrac {dy}{dx} = \dfrac {x^2 + y^2 +1}{2xy} \\ ii) \ (1-x^2) \dfrac {dy} {dx} = 1+ xy$
8 M
1 (b) An electric circuit contains an inductance of 0.5 henries and a resistance of 100 ohms in series with an e.m.f. of 20 volts. Find the current at any time t, if it, is zero at t=0.
4 M

2 (a) Solve: (2x+3y-1)dx=(6x+9y+6)dy
4 M
2 (b) Solve the following:

i) A bullet is fired sand tank, its retardation is proportinal to square root of its velocity. Show that the bullet will come to rest in time $\dfrac {2\sqrt{v}} {k}$ where V is initial velocity.

ii) A pipe 20 cm in diameter steam at 150°C and is protected with a covering 5cm thick for which k=0.0025. If the temperature of the outer surface of the covering is 40°C, find the temperature half-way through the covering under steady state conditions.
8 M

Solve any one question from Q3 & Q4
3 (a) Find the half range cosine series for the function F(x)=x-x3, 0?x?1
5 M
3 (b) $Evaluate \ int^5_2 (x-2)^2 (5-x)^2 dx$
3 M
3 (c) Trace the curve (any one)
i) x=a(t+sin t), y=a(l-cos t)
ii) x2y2=a2 (y2-x)
4 M

4 (a) $If \ I_n = \int^\infty_0 e^{-x} \sin^n x \ dx$ obtain the relation between In and In2.
4 M
4 (b) Show that $\int^b_a e^{-x^2} dx = \dfrac {\sqrt{\pi}}{2} [erf (b)- erf (a)]$
4 M
4 (c) Find the arclength of one loop of Lemniscate r2=a2= cos 2?
4 M

Solve any one question from Q5 & Q6
5 (a) Find the equation of right circular cylinder of radius a, whose axis passes through the origin and makes equal angles with the coordinate axes.
4 M
5 (b) Lines are drawn from the origin with direction co-sines proportional to (1,2,2), (2,3,6)(3,4,12). Find direction co-sines of the axis of right circular cone through them, and prove that the semivertical angle of cone is $\cos^{-1} \dfrac {1}{\sqrt{3}}$
4 M
5 (c) Find the equation of the sphere which passes through the points (1,-4,3)(1,-5,2)(1,-3,0) and whose centre lies on the plane x+y+z=0
5 M

6 (a) A sphere of constant radius K passes through the origin and meets the axes in A,B,C prove that the centroid of the triangle ABC lies on the sphere g(x2+y2+z2) =4K2
5 M
6 (b) Find the equation of the right circular cone which has its vertex at the point (0,0,10) and whose intersection with the plane XOY is a circle of diameter 10.
4 M
6 (c) Find the equation of the right circular cylinder of radius 3 and axis $\dfrac {x-1}{2} = \dfrac {y-3}{2} = \dfrac {z-5}{-1}$
4 M

7 (a) $Evaluate \ \iint (x+y)^2 dxdy \ over \ the \ area \ bounded \ by \ an ellipse \ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} =1$
7 M
Solve any one question from Q7 & Q8
7 (b) Find the volume of the tetrahydron bounded by the co-ordinate planes and the plane x+y+z=1
6 M
7 (c) Find the moment of inertia about the initial line of the cardiode r=a(1+cos ?)
6 M

8 (b) $Evaluate \ \iiint z(x^2 + y^2)dxdydz$ over the volume of the cylinder x2+y2=1 intercepted by the planes z=2 and z=3.