## 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 x^{2}+cy^{2}=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 |

**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 x^{2}+ y^{2}=a^{2}.(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)