## Engineering Mathematics-2 - Jun 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.

### Answer any one question from Q1 and Q2

**1 (a)** Solve the following differential equations: $$ i) \ \ \dfrac {dy}{dx} = \cos x \cos y + \sin x \sin y \\ ii) \ (x^2 + y^2 +1) dx - 2xy \ dy =0 $$(8 marks)
**1 (b)** In a circuit containing inductance L, resistance R and voltage E, the current I is given by: $$ E=RI + L \dfrac {dI}{dt} $$ Given:

L=640H, R=250 Ω, E=500 Volts. I being zero when t=0. Find the time that elapses before it reaches 80% of its maximum value.(4 marks)
**2 (a)** Solve$$ x \dfrac {dy}{dx}+y=y^2 \log x $$(4 marks)
**2 (b)** Solve the following: i) A body at temperature 100°C is placed in a room whose temperature is 20°C and cools to 60°C in 5 minutes. Find its temperature after a further interval of 3 minutes.

(ii) A steam pipe 20 cm in diameter is protected with a covering 6 cm thick for which the coefficient of thermal conductivity is k = 0.003 cal/cm deg. sec in steady state. Find the heat lost per hour through a meter length of the pipe, if the surface of pipe is at 200°C and outer surface of the covering is at 30°C.(8 marks)

### Answer any one question from Q3 and Q4

**3 (a)** Find a half range cosine series of f(x) =πx-x^{2} in the interval 0<x<π< a="">

</x<π<><>(5 marks)
**3 (b)** Evaluate: $$ \int^\infty_0 \dfrac {x^3}{3^x} dx $$(3 marks)
**3 (c)** Trace the following curve (any one):

i) y^{2}=x^{5} (2a-x)

ii) r=a sin 2θ(4 marks)
**4 (a)** $$ If \ I_n = \int^{\pi /2}_{\pi /4} \cot^n \theta d \theta \\ prove \ that \ I_n=\dfrac{1}{n-1}- I_{n-2}. Hence \ evaluate \ I_3. $$(4 marks)
**4 (b)** Using differentiation under Integral sign prove that: $$ \int^\infty _{0} \dfrac {e^{-x}-e^{-ax}}{x \sec x}dx = \dfrac {1}{2} \log \left ( \dfrac {a^2+1}{2} \right ) $$ for a>0.(4 marks)
**4 (c)** Find the length of the curve

x=a(θ- sin θ), y=a (1-cos θ) between θ=0 to θ=2 π.(4 marks)

### Answer any one question from Q5 and Q6

**5 (a)** Show that the plane 4x-3y+6z-35=0 is tangential to the sphere x^{2}+y^{2}+z^{2}-z-2z-14=0 and find the point of contact.(5 marks)
**5 (b)** Find the equation of the right circular cone whose vertex is given by (-1, -1, 2) and axis is the line $$ \dfrac {x-1}{2} = \dfrac {y+1}{1} = \dfrac {z-2}{-2} $$ and semi-vertical angle is 45°.(4 marks)
**5 (c)** Find the equation of right circular cylinder of radius 2 and axis is given by:

$$ \dfrac {x-1}{2} = \dfrac {y-2}{-3}= \dfrac {z-3}{6} $$(4 marks)
**6 (a)** Find the equation at the sphere through the circle x^{2}+y^{2}+z^{2}=1, 2x+3y+4z=5 and which intersects the sphere x^{2}+y^{2}+z^{2}+3 (x-y+z)-56=0 orthogonally.(5 marks)
**6 (b)** Find the equation of right circular cone with vertex at origin
making equal angles with the co-ordinate axes and having generator with direction cosines proportional to 1, ?2, 2.(4 marks)
**6 (c)** Obtain the equation of the right circular cylinder of radius 5
where axis is: $$ \dfrac {x-2}{3}= \dfrac {y-3}{1}= \dfrac {z+1}{1} $$(4 marks)

### Attempt any two of the following:

**7 (a)** Change the order of integration in the double integral: $$ \int^5_0 \int^{2+x}_{2-x} f(x,y) dy \ dx $$(6 marks)

### Answer any one question from Q7 and Q8

**7 (b)** Evaluate: $$ \int^2_0 \int^x_0 \int^{2x+2y}_0 e^{x+y+z}dx \ dy \ dz $$(7 marks)
**7 (c)** Find the centroid of the loop of the curve: r^{2}=a^{2} cos 2 θ.(6 marks)

### Attempt any two of the following:

**8 (a)** Evaluate: $$ \int^a_0 \int^{\sqrt{a^2-x^2}}_0 e^{-x^2 - y^2}dx dy. $$(6 marks)
**8 (b)** Evaluate: $$ \iiint \sqrt{1- \dfrac {x^2}{a^2} - \dfrac {y^2}{b^2} - \dfrac {z^2}{c^2}}dx \ dy \ dz $$ through the volume of ellipsoid $$ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2}+ \dfrac {z^2}{c^2}=1 $$(6 marks)
**8 (c)** Prove that the moment of inertia of the area included between the curves y^{2}=a ax and x^{2}=4ay about x-axis is 144/35 Ma^{2} where M is the mass of the area included between the curves.(7 marks)