## Engineering Mathematics-2 - Dec 2015

### 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 the following differential equations:

**1 (a) (i)** $$ xy \dfrac {dy}{dx} = (1-x^2) (1+y^2) $$(4 marks)
**1 (a) (ii)** $$ \cos y - x \sin y \dfrac {dy}{dx} = \sec^2 x. $$(4 marks)
**1 (b)** An e.m.f. 200e^{-5t} is applied to a series circuit containing of 20 ohm resistor and 0.01 F capacitor. Find the charge and current at any time assuming that there is no initial charge on the capacitor.(4 marks)
**2 (a)** Solve: 2y dx+(2 x log x ? xy)dy=0.(4 marks)
**2 (b) (i)** A body starts moving from rest is opposed by a force per unit mass of value cx and resistance per unit mass of value bv^{2}, where x and v are the displacement and velocity of the body at that instant. Show that the velocity of the body is given by: $$ v^2 = \dfrac {c}{2b^2} (1-e^{-2bx}) - \dfrac {cx}{b} $$(4 marks)
**2 (b) (ii)** The inner and outer surface of a spherical shell are maintained at T_{0} and T_{1} temperature respectively. If the inner and outer radii of the shell are r_{0} and r_{1} respectively and thermal conductivity of the shell is k, find the amount of heat loss from the shell per unit time. Find also the temperature distribution through the shell.(4 marks)
**3 (a)** Obtain the first three coefficient in the Fourier cosine series for y using practical harmonic analysis:

x |
y |

0 | 4 |

1 | 8 |

2 | 15 |

3 | 7 |

4 | 6 |

5 | 2 |

**3 (b)**Evaluate: $ \int^5_3 (x-3)^{1/2} (5-x)^{1/2} dx. $(3 marks)

### Trace the following curve (any one):

**3 (c) (i)** ay^{2}=x^{2}(a-x)(4 marks)
**3 (c) (ii)** r=a(1 + \cos \theta).(4 marks)
**4 (a)** $$ \text{If } I_n=\int^{\pi / 2}{0} x^n \cos x \ dx $$ Prove that: $$ I_n = \left ( \dfrac {\pi}{2} \right )^n - n (n-1) I_{n-2} $$(4 marks)
**4 (b)** Show that: $$ \int^{\infty}_0 e^{-x^{2}-2bx} dx = \dfrac {\sqrt{\pi}}{2} e^{b^2} [1-erf (b)] $$(4 marks)
**4 (c)** Find the arc length of the curve (using rectification) r=2a cos θ.(4 marks)
**5 (a)** Find the equation of the sphere which passes through the point (3, 1, 2) and meets X to Y plane in a circle of radius 3 units with centre at (1, -2, 0).(5 marks)
**5 (b)** Find the equation of right circular cone whose vertex is at the origin with axis $ \dfrac {x}{1} = \dfrac {y}{2} = \dfrac {z}{3} $ and has a semi-vertical angle 30°.(4 marks)
**5 (c)** Find the equation of right circular cylinder of radius 2 whose axis passes through (1, 2, 3) and has direction cosines proportional to 2, 1, 2.(4 marks)
**6 (a)** Find the equation of the sphere passing through the circle x^{2}+y^{2}+z^{2}=9, 2x+3y+4z=5 and the point (1, 2, 3).(5 marks)
**6 (b)** Find the equation of right circular cone whose vertex is (1, -1, 1) and axis is parallel to $ x=\dfrac {-y}{2} =-z$ and one of its generators has direction cosines proportional to (2, 2, 1).(4 marks)
**6 (c)** Find the equation of right circular cylinder of radius 4 with axis passing through origin and making equal angles with the co-ordinate axes.(4 marks)

### Attempt any two of the following:

**7 (a)** $$ \text{Evaluate :} \int^{1}_0 dx \int^{\infty}_{1} e^{-y} y^x \log y \ dy. $$(6 marks)
**7 (b)** Evaluate :$$ \iiint (x^2y^2 + y^2z^2+z^2x^2) dx \ dy \ dz. $$ throughout the volume of the sphere x^{2}+y^{2}+z^{2}=a^{2}.(7 marks)
**7 (c)** Find the moment of inertia of one loop of the lemniscate r^{2}=a^{2} cos 2θ about initial line.(6 marks)
**8 (a)** Evaluate: $$ \int^a_0 \int^{\sqrt{a^2 - x^2}}_0 \sin \left \{ \dfrac {\pi}{a^2} (a^2 - x^2 - y^2) \right \}dx \ dy. $$(7 marks)
**8 (b)** Evaluate: $$ \int^\infty_0 \int^\infty_0 \int^\infty_0 \dfrac {dx \ dy \ dz }{(1+x^2 + y^2 + z^2)^2}. $$(6 marks)
**8 (c)** Find the C.G. of the loop of the curve:

y^{2}(a+x)=x^{2}(a-x).(6 marks)