Engineering Mathematics-2 : Question Paper Dec 2014 - First Year Engineering (Semester 2)

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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	-18.0	-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)

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