Question Paper: Engineering Mathematics-2 : Question Paper Dec 2014 - First Year Engineering (Semester 2) | Pune University (PU)
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## Engineering Mathematics-2 - Dec 2014

### First Year Engg (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.

### 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 marks)
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 marks) 2 (a) Solve $$\dfrac {dy}{dx} = \dfrac {y}{x} + \tan \dfrac {y}{x}$$(4 marks) 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 marks)

### 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 marks) 3 (b) $$Evaluate: \ \int^\infty_0 \sqrt[4]{x}e^{\sqrt[-]{x}}dx$$(3 marks) 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 marks)
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 marks) 4 (b) Prove that $$\phi (a) = \int^{\pi/2a}_{\pi/6a} \dfrac{\sin ax}{x} dx$$ is independent of 'a'.(4 marks) 4 (c) Find the length of the arc of cardioide r=a (1-cos ?) which lies outside the circle r=a cos ?.(4 marks)

### 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks)

### Answer any two from Q7

7 (a) Change the order of integration and evaluate: $$\int^\infty_0 \int^{\infty}_x \dfrac {e^{-y}}{y}dx \ dy$$(6 marks) 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 marks) 7 (c) Find the centre of gravity of one loop of the curve r=a sin 2?.(7 marks) 8 (a) $$Evaluate: \iint_R \sin (x^2+y^2)dx \ dy$$ where R is circle x2+ y2=a2.(6 marks) 8 (b) Find the total area included between the two cardiodes r=ac (1+ cos ?) and r=a (1-cos ?).(7 marks) 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 marks)