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SPPU First Year Engineering (Semester 2)
Engineering Mathematics-2
December 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) $i) \ \dfrac {dy}{dx} = \dfrac {x+y-2}{y-x-4} \\ ii) \ \dfrac {dy}{dx} =x^2 \cos^2 y -x \sin 2y$
8 M
1 (b) An e.m.f. 200 e-5t is applied to a series circuit consisting 20? resistor and 0.01 F capacitor. Find the charge and current at any time, assuming that there is no initial change on capacitor.
4 M

2 (a) Solve $\dfrac {dy}{dx} = \dfrac {y}{x} + \tan \dfrac {y}{x}$
4 M
2 (b) Solve the following

i) Find the orthogonal trajectory to of x2+cy2=1.
A body originally at 85°C cools to 65°C in 25 minutes, the temperature of air being 40°, what will be the temperature of the body after 40 minutes.
8 M

Solve any one question from Q3 & Q4
3 (a) Find the fourier expansion for y in terms x upto first harmonic as given in following table.
 x° 0 30 60 90 120 150 180 210 240 270 300 330 y 10.5 20.2 26.4 29.3 27 21.5 12.5 1.6 -19.2 -18 -15.8 -0.4
5 M
3 (b) $Evaluate: \ \int^\infty_0 \sqrt[4]{x}e^{\sqrt[-]{x}}dx$
3 M
3 (c) Trace the following curve (any one):

i) x=a(t-sin t), y=a(1-cos t)
ii) r=a sin 3?
4 M

4 (a) $If \ l_n = \int^{\pi /4}_0 \cos^{2n}x dx, \ prove \ that l_n = \dfrac {1}{2^{n+1}}+ \dfrac {2n-1}{2n}l_n$
4 M
4 (b) Prove that $\phi (a) = \int^{\pi/2a}_{\pi/6a} \dfrac{\sin ax}{x} dx$ is independent of 'a'.
4 M
4 (c) Find the length of the arc of cardioide r=a (1-cos ?) which lies outside the circle r=a cos ?.
4 M

Solve any one question from Q5 & Q6
5 (a) Find the equation of the sphere tangential to the plane x-2y-2z=7 at (3, -1, -1) and passing through the point (1, 1, -3).
5 M
5 (b) Find the equation of the right circular cone which passes through the point (1, 1, 2) has its axis at the line ?x= -3y=4z and vertex at origin.
4 M
5 (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

6 (a) A sphere s has points (1, -2, 3) and (4, 0, 6) as opposite ends of a diameter. Find the equation of the sphere having the intersection of s with the plane x+y-2z=6=0 as its great circle.
5 M
6 (b) Find the equation of right circular cone whose vertex is (1, 2, 3) and the axis is given by $\dfrac {x-1}{2} = \dfrac {y-2}{-1}= \dfrac {z-3}{4}$ and semi-vertical angle is 60°.
4 M
6 (c) Find the equation of the right circular cylinder of radius 3 whose axis is the line$\dfrac {x-1}{2} = \dfrac {y-3}{2} = \dfrac {z-5}{-1}$
4 M

7 (a) Change the order of integration and evaluate: $\int^\infty_0 \int^{\infty}_x \dfrac {e^{-y}}{y}dx \ dy$
6 M
7 (b) Find the volume of the tetrahydron bounded by the co-ordinate planes and the plane $\dfrac {x}{a}+ \dfrac {y}{b}+ \dfrac {z}{c} =1$
6 M
7 (c) Find the centre of gravity of one loop of the curve r=a sin 2?.
7 M

8 (a) $Evaluate: \iint_R \sin (x^2+y^2)dx \ dy$ where R is circle x2+ y2=a2.
6 M
8 (b) Find the total area included between the two cardiodes r=ac (1+ cos ?) and r=a (1-cos ?).
7 M
8 (c) Find the moment of inertia about x-axis of the area enclosed by the lines x=0, y=0 $\dfrac {x}{a} + \dfrac {y}{b}=1$
6 M

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