Question Paper: Engineering Maths 2 : Question Paper Jan 2014 - First Year Engineering (C Cycle) (Semester 2) | Visveswaraya Technological University (VTU)
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Engineering Maths 2 - Jan 2014

First Year Engineering (C Cycle) (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) Choose the correct answer for the following:

( i) Suppose the equation to be solved is of the form, y=f(x, φ) then differentiating x we get equation of the form,
$$ (a) \ \phi \left (x,p, \dfrac{dp}{dy} \right )= 0 \$$b) \ \phi \left ( y, p , \dfrac {dp}{dx} \right )= 0 \$$c) \ \phi (x,yp)=0 \$$d)\ \phi (x,y,0)= 0 $$
(ii) The general solution of the equation p2-3p+2=0 is,
(a) (y+x-c)y+2x-c)
(b) (y-x-c)(y-2x-c)=0
(c) (-y-x-c)(y-2x-c)=0
(y-x-c)(y+x-c)=0
(iii) Clairaut's equation is of the form,
(a) x=py+f(p)
(b) y=p2+f(p)
(c) y=px+f(p)
(d) None of these
(iv) Singular solution of y=px+2p2 is,
(a) y2+8y=0
(b) x2-8y=0
(c) x2+8y-c=0
(d) x2+8y=0
(4 marks)
1 (b) Solve p2+2p cosh x+1=0.(4 marks) 1 (c) Find singular solution of p=sin(y-xp).(6 marks) 1 (d) Solve the equation y2(y-xp)=x4p2 using substitution $$ X=\dfrac {1}{x} and Y=\dfrac {1}{y} $$(6 marks) 2 (a) Choose the correct answer for the following:

(i) A second order linear differential equation has,
(a) two arbitary solution
(b) One arbitary solution
(c) no arbitary solution
(d) None of these
(ii) If 2, 4i and -4i are the roots of A.E of a homogeneous linear differential equation then its solution is,
$$ (a) \ e^x+ e^x (\cos 4x+\sin 4x) \\ (b) \ C_1 e^{2x}+ C_2 \cos 4x + C_3 \sin 4x \$$c) \ C_1 e^{2x} + C_2 e^x \cos 4x+C_3 e^x \sin 4x \$$d) \ C_1 e^{2x}\cos 4x+ C_2e^{2x}\sin 4x $$
(iii) P.I. of (D+1)2 y=e-x+3 $$ (a)\ \dfrac {x^2}{2} \\ (b) \ x^3 e^x \\ (c) \ \dfrac {x^3}{3} e^{-x=3} \\ (d)\ \dfrac {x^2}{2}e^{-x+3}$$
(iv) Particular integral of f(D)y=eax V(x) is, $$ (a)\ \dfrac {e^{ax}V(x)}{f(D)} \\ (b) \ e^{ax}= \dfrac {1}{f(D)}[V(x)] \\ (c) \ e^{ax} \dfrac {1}{f(D+a)}[V(x)] \\ (d) \ \dfrac {1}{f(D+a)} [e^{ax}V(x)] $$
(4 marks)
2 (b) $$ Solve \ \dfrac {d^3y}{dx^3}- 3 \dfrac {d^2y}{dx^2}+ 3 \dfrac {dy}{dx}- y = 0 $$(4 marks) 2 (c) Solve y"-3y'+2y=2 sin x cos x(6 marks) 2 (d) Solve the system of equation, $$ \dfrac {dx}{dt}- 2y = \cos 2t, \ \dfrac{dy}{dt} + 2x =\sin 2t $$(6 marks) 3 (a) Choose the correct answer for the following:

(i) In x2y"+ xy'-y=0 if et=x then we get x2y" as,
(a) (D-1)y
(b) (D+1)y
(c) D(D+1)y
(d) None of these
(ii) In second order homogeneous differential equation P0(x)y"+P1(x)y'+P2(x)y=0 x=a is a singular point if,
(a) P0(a)>0
(b) P0(a)?0
(c) P0(a)=0
(d) P0(a)<0
(iii) The general solution of $$ x^2 \dfrac {d^2 y}{dx^2}+ x\dfrac{dy}{dx}-y = 0 \ is, \\ (a) \ y=C_1x-C_2 \dfrac {1}{x} \\ (b) \ C_1x + C_2 \dfrac {1}{x} \\ (c) \ C_1x+C_2 x \\ (d) \ C_1 x- C_2 x $$
(iv) Frobenius series solution of second order linear differential equation is of the form,
$$ (a) \ x^{m} \sum^{\infty}_{r=0}a_rx^r \\ (b) \ \sum^{\infty}_{r=0}a_rx^r \\ (c) \ \sum^{\infty}_{r=a}a_rx^{m-r} \\ None \ of \ these $$
(4 marks)
3 (b) Solve y"+a2y=sec ax by the method of variation of parameters.(4 marks) 3 (c) $$ Solve \ x^2 \dfrac {d^2 y}{dx^2}+ 4x \dfrac {dy}{dx}+ 2 y = e^x $$(6 marks) 3 (d) Obtain the series solution of $$ \dfrac {dy}{dx}- 2xy=0 $$(6 marks) 4 (a) Choose the correct answer for the following:

(i) PDE of az+b=a2x+y is, $$ (a) \ \dfrac {\partial z}{\partial x} \cdot \dfrac{\partial z}{\partial y}= 1 \\ (b) \dfrac {\partial z} {\partial x} \cdot \dfrac {\partial z}{\partial y} = 0 \\ (c) \ \dfrac {\partial z}{\partial x} + \dfrac {\partial z}{\partial y} = 0 \\ (d)\ \dfrac{\partial z}{\partial x}+ \dfrac {\partial z}{\partial y}=1 $$
(ii) The solution of PDE Zxx=2 y2 is,
(a) z=x2+xf(y)+ g(y)
(b) z=x2y2+xf(y)+g(y)
(c) z=x2y2+f(x)+g(x)
(d) z=y2+xf(y)+g(y)
iii) The subsidiary equations of (y2+z2)p+x(yq-z)=0 are, $$ (a)\ \dfrac {dx}{p}= \dfrac {dy}{q} = \dfrac {dz}{R} \\ (b) \ \dfrac {dx}{y^2 + z^2} = \dfrac {dy}{x} = \dfrac {dz}{xz} \\ (c) \ \dfrac {dx}{y^2 + z^2} = \dfrac {dy}{xy} = \dfrac {dz}{xz} \\ (d) \ None \ of \ these $$ (iv) In the method of seperation of variable to solve xzn+zt=0 the assumed solution is of the form,
(a) X(x)Y(x)
(b) X(y)Y(y)
(c) X(t)Y(t)
(d) X(x)T(t)
(4 marks)
4 (b) $$ Solve \ \dfrac {\partial ^3 z}{\partial x^2 \partial y}= cos (2x+3y)$$(4 marks) 4 (c) Solve xp-yq=y2-x2(6 marks) 4 (d) Solve 3ux+2uy=0 by the seperation of variable method given that u=4e-x when y=0(6 marks) 5 (a) Choose the correct answer for the following:

$$ \int^{1}_0 \int^{x^2}_0 e^{y/x}dy dx = \_\_\_\_\_\_\_ \\ (a) \ 1 \ \ (b) \ -1/2 \ \ (c) \ 1/2 \ \ (d) \ None\ of \ these $$ (ii) The integral $$ \iint_R f(x,y) dxdy $$ by changing to polar form becomes, $$ (a) \ \iint_R \phi (r, \theta) drd\theta \\ (b) \ \iint_R f(r, \theta)drd\theta \\ (c) \ \iint_R f(r,\theta)rdrd\theta \\ (d)\ \iint_R \phi (r, \theta)rdrd \theta $$ (iii) For a real positive number n, the Gamma function ?(n)= _________ $$ (a) \ \int^{\infty}_0 x^{n-1}e^{-x}dx \\ (b) \ \int^1_0 x^{n-1}e^{-x}dx \\ (c) \ \int^{x}_0 x^ne^{-x}dx \\ (d) \ \int^1 _0 x^n e^{-x}dx $$
(iv) The Beta and Gamma functions relation for B(,n)= _______ $$ (a) \ \dfrac {\Gamma (m )\Gamma (n)} {\Gamma (m-n)} \\ (b) \ \dfrac {\Gamma (m)\Gamma (n)}{\Gamma (m+n)} \\ (c) \ \Gamma (m)\Gamma(n) \\ (d) \ \Gamma(mn) $$
(4 marks)
5 (b) By changing the order of integration evaluate, $$ \int^a_0 \int^{\sqrt{x/a}}_{x/a}(x^2 + y^2)dydx, \ a>0 $$(4 marks) 5 (c) $$ \displaystyle Evaluate \ \int^a_0 \int^{x}_0 \int^{x-y}_0 e^{x+y+z}dzdydx $$(6 marks) 5 (d) Express the integral $$ \int^1_0 \dfrac{dx}{\sqrt{1-x^n}}$$ in terms of the Gamma function, Hence evaluate $$ \int^1_0 \dfrac {dx}{\sqrt{1-x^{2/3}}} $$(6 marks) 6 (a) Choose the correct answer for the following:

(i) The scalar surface integral of $$ \overrightarrow{f} $$ over s, where s is a surface in a three-dimensional region R is given by, $$ \int \overrightarrow{f}.nds= \_\_\_\_\_\_\_ $$ by using Gauss divergence theorem $$ (a) \ \iiint_v \nabla\cdot \overrightarrow{f}dV \\ (b)\ \iint_s \nabla\cdot \overrightarrow{t}dx dy \\ (c) \ \iiint_v \nabla \cdot \overrightarrow{F}dV \\ (d) \ None \ of \ these $$ (ii) If all the surface are closed in a region containing volume V then the following theorem is applicable.
(a) Stroke's theorem
(b) Green's theorem
(c) Gauss divergence theorem
(d) None of these
(iii) The value of $$ \int \left \{ (2xy-x^2)dx + (x^2 + y^2)dx \right \} $$ by using Green's theorem is,
(a) Zeron (b) One (c) Two (d) Three
(iv) $$ \iint_s f.nds = \_\_\_\_\_\_\_ $$ where f=xi+yj+2k and S is the surface of the sphere x2y2+z2=a2
(a) 4πa (b) 4πa2 (c) 4πa3 (d) 4π
(4 marks)
6 (b) Find the work done by a force f=(2y-x2)i+ 6yzj-8xz2k from the point (0, 0, 0) to the point (1, 1, 1) along the straight-line joining these points.(4 marks) 6 (c) If C is a simple closed curve in the xy-plane, prove by using Green's theorem that the integral $$ \int_C \dfrac {1}{2} (xdy-ydx) $$ represent the area A enclosed by . Hence evaluate $$ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} = 1 $$(6 marks) 6 (d) Verify Stoke's theorem for $$ \overrightarrow{f} = (2x-y)i - yz^2 j- y^2 zk $$ for the upper half of the sphere x2+y2+z2=1(6 marks) 7 (a) Choose the correct answer for the following:

(i) L[tn]= ________
$$ (a) \ \dfrac {n}{s^{n+1}} \\ (b) \ \dfrac {n}{s^{n-1}} \\ (c) \ \dfrac {n!}{s^{n-1}} \\ (d) \ \dfrac {n!}{s^{n+1}} $$
(ii) L[e-3t]= _______
$$ (a) \ \dfrac {3}{s-3} \\ (b) \ \dfrac {3}{s+3} \\ (c) \ \dfrac {1}{s+3} \\ (d) \ \dfrac {1}{s-3} $$
iii) L{f(t-a)H(t-a)} is equal to, $$ (a) \ \dfrac{3!}{(s+2)^4} \\ (b) \ \dfrac{3!}{(s-2)^4} \\ (c) \ \dfrac{3}{(s-2)^4} \\ (d) \ \dfrac{3}{(s-2)} $$
(iv) L{δ(t-1)}= _______
(a) e-s (b) e5 (c) eaS (d) e-aS
(4 marks)
7 (b) Evaluate L{sin3 2t}(6 marks) 7 (c) Find L{f(t)} given that $$f(t)= \begin{cases}2 &3>t>0 \\t &t>3 \end{cases}$$(6 marks) 7 (d) Express $$f(t) = \begin{cases}t^2 &2>t>0 \\4t &4\ge t>2 \\8 &t>4 \end{cases}$$ in terms of unit step function and hence find their Laplace transform.(4 marks) 8 (a) Choose the correct answer for the following:

(i) L-1 {cos at}= _______ $$ (a)\ \dfrac {s}{s^2 + a^2} \\ (b) \ \dfrac {s}{s^2 - a^2} \\ (c) \ \dfrac {1}{s^2 + a^2} \\ (d) \ \dfrac {1}{s^2 - a^2} $$ (ii) L-1 {F (s-a)}= ________
(a) etf(t)
(b) eatf(t)
(c) e-atf(t)
(d) None of these
$$ L^{-1} \left \{ \cot^{-1} \left ( \dfrac {2}{s^2} \right ) \right \} = \_\_\_\_\_\_ \\ (a) \ \dfrac {\sin t}{t} \\ (b) \ \dfrac {\sinh at}{t} \\ (c) \ \dfrac{\sin at }{t} \\ (d) \ \dfrac {\sinh t}{t} $$
(iv) For the function f(t)=1, convolution theorem condition,
(a) Not satisfied
(b) Satisfied with some condition
(c) Satisfied
(d) None of these
(4 marks)
8 (b) Find the inverse Laplace transform of $$ \dfrac {2s^2 - 6s + 5}{(s-1)(s-2)(s-3)} $$(4 marks) 8 (c) Find $$ L^{-1} \left(\dfrac{s}{(s-1)(s^2 + 4)}\right) $$ using convolution theorem(6 marks) 8 (d) Solve differential equation y"(t) + y = F(t) where $$F(t)= \begin{cases} 0 & 1>t>0 \\2 &t>1 \end{cases}$$ Given that y(0)=0=y'(0)(6 marks)

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