SPPU Engineering Mathematics-2 - June 2015 Exam Question Paper

SPPU First Year Engineering (Semester 2)
Engineering Mathematics-2
June 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Answer any one question from Q1 and Q2

1 (a) Solve the following differential equations:

i) \frac{d y}{d x} = \cos x \cos y + \sin x \sin y i i) (x^{2} + y^{2} + 1) d x - 2 x y d y = 0

$i) \ \ \dfrac {dy}{dx} = \cos x \cos y + \sin x \sin y \\ ii) \ (x^2 + y^2 +1) dx - 2xy \ dy =0$

8 M

1 (b) In a circuit containing inductance L, resistance R and voltage E, the current I is given by:

E = R I + L \frac{d I}{d t}

$E=RI + L \dfrac {dI}{dt}$ 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 M

2 (a) Solve

x \frac{d y}{d x} + y = y^{2} \log x

$x \dfrac {dy}{dx}+y=y^2 \log x$

4 M

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 M

Answer any one question from Q3 and Q4

3 (a) Find a half range cosine series of f(x) =πx-x² in the interval 0

5 M

3 (b) Evaluate:

\int_{0}^{\infty} \frac{x^{3}}{3^{x}} d x

$\int^\infty_0 \dfrac {x^3}{3^x} dx$

3 M

3 (c) Trace the following curve (any one):
i) y²=x⁵ (2a-x)
ii) r=a sin 2θ

4 M

4 (a)

I f I_{n} = \int_{π / 4}^{π / 2} \cot^{n} θ d θ p r o v e t h a t I_{n} = \frac{1}{n - 1} - I_{n - 2} . H e n c e e v a l u a t e I_{3} .

$If \ I_n = \int^{\pi /2}_{\pi /4} \cot^n \theta d \theta \\ prove \ that \ I_n=\dfrac{1}{n-1}- I_{n-2}. Hence \ evaluate \ I_3.$

4 M

4 (b) Using differentiation under Integral sign prove that:

\int_{0}^{\infty} \frac{e^{- x} - e^{- a x}}{x \sec x} d x = \frac{1}{2} \log (\frac{a^{2} + 1}{2})

$\int^\infty _{0} \dfrac {e^{-x}-e^{-ax}}{x \sec x}dx = \dfrac {1}{2} \log \left ( \dfrac {a^2+1}{2} \right )$ for a>0.

4 M

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 M

Answer any one question from Q5 and Q6

5 (a) Show that the plane 4x-3y+6z-35=0 is tangential to the sphere x²+y²+z²-z-2z-14=0 and find the point of contact.

5 M

5 (b) Find the equation of the right circular cone whose vertex is given by (-1, -1, 2) and axis is the line

\frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 2}{- 2}

$\dfrac {x-1}{2} = \dfrac {y+1}{1} = \dfrac {z-2}{-2}$ and semi-vertical angle is 45°.

4 M

5 (c) Find the equation of right circular cylinder of radius 2 and axis is given by:

\frac{x - 1}{2} = \frac{y - 2}{- 3} = \frac{z - 3}{6}

$\dfrac {x-1}{2} = \dfrac {y-2}{-3}= \dfrac {z-3}{6}$

4 M

6 (a) Find the equation at the sphere through the circle x²+y²+z²=1, 2x+3y+4z=5 and which intersects the sphere x²+y²+z²+3 (x-y+z)-56=0 orthogonally.

5 M

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 M

6 (c) Obtain the equation of the right circular cylinder of radius 5 where axis is:

\frac{x - 2}{3} = \frac{y - 3}{1} = \frac{z + 1}{1}

$\dfrac {x-2}{3}= \dfrac {y-3}{1}= \dfrac {z+1}{1}$

4 M

Attempt any two of the following:

7 (a) Change the order of integration in the double integral:

\int_{0}^{5} \int_{2 - x}^{2 + x} f (x, y) d y d x

$\int^5_0 \int^{2+x}_{2-x} f(x,y) dy \ dx$

6 M

Answer any one question from Q7 and Q8

7 (b) Evaluate:

\int_{0}^{2} \int_{0}^{x} \int_{0}^{2 x + 2 y} e^{x + y + z} d x d y d z

$\int^2_0 \int^x_0 \int^{2x+2y}_0 e^{x+y+z}dx \ dy \ dz$

7 M

7 (c) Find the centroid of the loop of the curve: r²=a² cos 2 θ.

6 M

Attempt any two of the following:

8 (a) Evaluate:

\int_{0}^{a} \int_{0}^{\sqrt{a^{2} - x^{2}}} e^{- x^{2} - y^{2}} d x d y .

$\int^a_0 \int^{\sqrt{a^2-x^2}}_0 e^{-x^2 - y^2}dx dy.$

6 M

8 (b) Evaluate:

∭ \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}}} d x d y d z

$\iiint \sqrt{1- \dfrac {x^2}{a^2} - \dfrac {y^2}{b^2} - \dfrac {z^2}{c^2}}dx \ dy \ dz$ through the volume of ellipsoid

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

$\dfrac {x^2}{a^2} + \dfrac {y^2}{b^2}+ \dfrac {z^2}{c^2}=1$

6 M

8 (c) Prove that the moment of inertia of the area included between the curves y²=a ax and x²=4ay about x-axis is 144/35 Ma² where M is the mass of the area included between the curves.

7 M

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