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:
(i) The general solution of the equation x2p2+3xyp+2y2=0 is _______ $$ (a) \ (y^2 x-c) (xy-c)=0 \\ (b) \ (x-y-c)(x^2 + y^2 - c)=0 \\ (c) \ (xy - c) (x^2 y-c)=0 \\ (d) \ (y-x-c) (x^2 + y^2 + c) = 0 $$
ii) The given differential equation is solvable for y, if it is possible to express y in terms of _______
(a) y and p
(b) x and p
(c) x and y
(d) y and x
iii) The singular solution of Clairaut's equation is ________
(a) y=xg(x)+f[g(x)]
(b) y=cx+f(c)
(c) cy + f(c)
(d) y g2 (x) + f [g(x)]
iv) The singular solution of the equation y=px-log p is ________ $$ (a) \ y^2 = 4ax \\ (b) \ x=1-\log x \\ (c) \ y=1 - \log \left ( \dfrac {1}{x} \right ) \\ (d) \ x^2 = y \log x $$(4 marks)
1 (b) Solve p2-2p sin h x-1=0(4 marks)
1 (c) Solve y=2px+ tan-1 (xp2)(6 marks)
1 (d) Obtain the general solution and singular solution of Clairaut's equation is (y-px)(p-1)=p(6 marks)
2 (a) Choose the correct answer:
(i)The complementary function of [D4+4]x=0 is ________ $$ (a)\ x= e^{-4}[c_1 \cos t + c_2 \sin t] + e^1 [c_3 \cot t+ c_4 \sin t] \\ (b) \ x= [c_1 \cos t + c_2 \sin t] + [c_3 \cos t + c_4 \sin t] \$$c) \ x= [ c_1 + c_2 t]e^{-t} \$$d) \ x= [c_1 + c_2 t]e^{t} $$
Find the particular integral of (D3-3D2+4)y=e2x is _______ $$ (a)\ \dfrac {x^2 e^{2x}}{6} \\ (b) \ \dfrac {x^2 e^{3x}}{6} \\ (c) \ \dfrac {x^2e^x}{6} \\ (d) \ \dfrac {x^2 e^{4x}}{6} $$
$$ iii) \ Roots \ of \ \dfrac {d^2 y}{dx^2} + 4 \dfrac{dy}{dx} + 5y =0 \ are \ \_\_\_\_\_\_\_\_ \\ (a) \ 2\pm i \\ (b) \ 3\pm i \\ (c) \ 2 \pm 2i \\ (d) \ -2 \pm i $$ iv) Find the particular integral of (D3 + 4D)y=sin 2x is ________ $$ (a) \ \dfrac {x \sin x}{8} \\ (b) \ \dfrac {-x \sin x}{8} \\ (c) \ \dfrac {-x \sin 2x}{8} \\ (d) \ \dfrac {x \sin 2x}{8} $$(4 marks)
2 (b) $$ Solve \ \dfrac {d^2 y}{dx^3} + 6 \dfrac {d^2 y}{dx^2}+ 11 \dfrac {dy}{dx} + 6y = e^x +1 $$(4 marks)
2 (c) $$ Solve \ \dfrac {d^2 y}{dx^2} - 4y = \cos h (2x-1)+3^x $$(6 marks)
2 (d) $$ Solve \ \dfrac {dy}{dx} + y = z e^{x}, \ \dfrac {dz}{dx}+ z = y+e^x $$(6 marks)
3 (a) Choose the correct answer:
(i) The Wronskian x and x ex is ________
(a) ex
(b) e2x
(c) e-2x
(d) e-x
(ii) The complementary function of x2y"-xy'-3y=x2 log x is ________
(a) c1 cos (log x) + c2 sin (log x)
(b) c1 x-1+c2x
(c) c1x+c2x3
(d) c1 cosx +c2 sin x
iii) To transform (1+x)2y"+ (1+x)y'+y=2 sin log(1+x) into a linear differential equation with constant coefficient ________
(a) (1+x)=et
(b) (1+x)=e-t
(c) (1+x)2=et
(d) (1-x)2=et
iv) The equation a0(ax+b)2y"+a1(ax+b)y'+a2y=ϕ(x) is _________
(a) Simulataneous equation
(b) Cauchy's linear equation
(c) Legendre linear equation
(d) Euler's equation(4 marks)
3 (b) Using the variation of parameters method to solve the equation y"+2y'+y=e-x log x.(6 marks)
3 (c) $$ Solve \ x^2 \dfrac {d^2 y}{dx^2}- (2m-1) x \dfrac {dy}{dx} + (m^2 + n^2) y = n^2 x^m \log x $$(6 marks)
3 (d) Obtain the Frobenius method solve the equation $$ x\dfrac {d^2y}{dx^2} + \dfrac {dy}{dx}- y =0 $$(6 marks)
4 (a) i) Partial differential equation by eliminating a and b from the relation Z=(x-a)2+(y-b)2 is ________
(a) p2q2=4z
(b) pq=4z
(c) r=4z
(d) t=4
ii) The Lagrange's linear partial differential equation Pp+Qq=R the subsidiary equation is _______ $$ (a) \ \dfrac {dx}{R} = \dfrac {dy}{P} = \dfrac {dz}{Q} \\ (b) \ \dfrac {dx}{P} = \dfrac {dy}{Q} = \dfrac {dz}{R} \\ (c) \ \dfrac {dx}{Q} = \dfrac {dy}{R} = \dfrac {dz}{P} \\ (d) \ \dfrac {dx}{P} + \dfrac {dy}{Q} + \dfrac {dz}{R} $$ iii) By the method mof separation of variable we seek a solution in the form is _______
(a) x=x+y
(b) z= x2+y2
(c) x=z+y
(d) x=x(x)y(y)
iv) The solution of $$ \dfrac{\partial^2 z}{\partial x^2}= \sin (xy) \ is \$$a) \ z= -x^2 \sin (xy) + y\ f(x)+ \phi (x) \\ (b) \ \dfrac {-\sin (xy)}{y}+ x \ f(y)+ \phi (y) \$$c) \ z= \dfrac {-\sin xy}{x^2} + y f(x) + \phi (x) \$$d) \ None \ of \ these $$
(4 marks)
4 (b) From the partial differential equation of all sphere of radius 3 units having their centre in the xy-plane.(4 marks)
4 (c) $$ Solve \ x(y^2 + z) p-y (x^2 +z) = z (x^2 - y^2) $$(6 marks)
4 (d) Use the method of seperation of variable to solve $$ y^3 \dfrac {\partial z}{\partial x} + x^2 \dfrac {\partial z}{\partial y}=0 $$(6 marks)
5 (a) Choose the correct answer:
(i) The value of $$ \int^1_0 \int^{x^2}{0} e^{y/x}dydx \ is \ \_\_\_\_\_\_ $$ (a) 0
(b) 1
(c) 3
(d) 1/2
ii) The value of ? (1/2) is _______ $$ (a) \ 2\sqrt{\pi} \\ (b) \ \pi /2 \\ (c) \ \sqrt{\pi} \\ (d) \ \sqrt{2x} $$
(iii) The integral $$ \int^a_0 \int_y^a\dfrac {x}{x^2 + y^2}dxdy $$ after changing the order of integration is _______ $$ (a) \ \int^a_0 \int^x_0 dydx \\ (b) \ \int^a_0 \int^x_0 \dfrac{x}{x^2 + y^2}dxdy \$$c) \ \int^x_0 \int^x_0 \dfrac{x}{x^2 + y^2}dxdy \\ (d) \ \int^x_0 \int^a_0 \dfrac {x}{x^2 + y^2}dxdy $$
iv) The value of β (3, 1/2) is ________ $$ (a) \ \dfrac {15}{16} \$$b) \ \dfrac {16}{15} \\ (c) \ \dfrac {16}{5}\\ (d) \ \dfrac {16}{3} $$(4 marks)
5 (b) Change the order of integration in $$ \int^{4a}_0 \int^{\sqrt{az}}_{\frac {x^2}{4x}}dydx $$ and hence evaluate the same.(4 marks)
5 (c) $$ Evaluate \ \int^c_{-c} \int^{b}_{-b} \int^{a}_{-a}(x^2 + y^2 + z^2) dx \ dy \ dz $$(6 marks)
5 (d) $$ Prove \ that \ \int^1_0 \dfrac {x^2}{\sqrt{1-x^4}}dx^{-x} \int^1_0 \dfrac {1}{\sqrt{1+x}}dx = \dfrac {\pi}{4 \sqrt{2}} $$(6 marks)
6 (a) Let S be the closed boundary surface of a region of volume V then for a vector field of difined in V and S ∫s f.nds is ________
(a) ∫v curl of dv
(b) ∫v f dv
(c) ∫v grad dv
(d) None of these
$$ If \ \int_c f.dr \ where \ f=3xy\widehat{i}-y^2 \widehat{j}$$ and C is the part of the parabola y=2x2 from the region (0, 0) to the point (1, 2) is _______
(a) 7/6
(b) -7/6
(c) 3x+3y
(d) -35
iii) In the Green's theorem in the plane $$ \oint_c Mdx+Ndy = \_\_ \_\_ \_\_ \_\_ $$ $$ (a) \ \iint_R \left [ \dfrac {\partial M}{\partial y} + \dfrac {\partial N}{\partial x} \right ]dxdy \\ (b) \ \iint_R \left [ \dfrac {\partial M}{\partial y} - \dfrac {\partial N} { \partial x}\right ]dxdy \\ (c) \ \iint_R \left [ \dfrac {\partial N} { \partial x} - \dfrac {\partial M}{\partial y} \right ]dxdy \\ (d) \iint_R \left [ \dfrac {\partial N}{\partial x} + \dfrac {\partial M}{\partial y} \right ]dxdy $$
iv) A necessary and sufficient condition that the line integral $$ \int_c \overrightarrow{F}.d\overrightarrow{r} $$ for any closed curve C is _______ $$ (a) \ div \overrightarrow{F}=0 \$$b) \ div \overrightarrow{F} e 0 \$$c) \ curl \overrightarrow{F} =0 \$$d) \ grad \overrightarrow{F}=0 $$(4 marks)
6 (b) Using the divergence theorem, evaluate$$ \int_c f.nds \ where \ f=4xz\widehat{i}- y^2 \widehat{j}+ yz\widehat{k} $$ and S is the surface of the cube bounded by x=0, x=1, y=0, y=1, z=0, z=1.(4 marks)
6 (c) Use the Green's theorem, evaluate $$ \iint_c (2x^2 - y^2)dx + (x^2 + y^2)dy $$ where C is the triangle formed by the lines x=0, y=0 and x+y=1.(6 marks)
6 (d) Verify the Stoke's theorem for $$ f=-y^3 \widehat{i}+ x^3 \widehat{j} $$ where S is the circle disc x2+y2 ≤ 1, z=0.(6 marks)
7 (a) Choose the correct answer:
$$ (i)\ L\{\sinh at \}= \_\_\_\_\_\_\_ \\ (a) \ \dfrac {s}{s^2 -a^2} \\ (b) \ \dfrac {s}{s}{s^2 + a^2} \\ (c) \ \dfrac {a}{s^2 -a^2} \\ (d) \ \dfrac {a}{s^2 +a^2}$$
ii) if L{f(t)}=F(s) then L{eatf(t)} is _______ (a) F(s+a)
(b) F(s-a)
(c) F(s)
(d) None of these
$$ iii) \ L\left \{ \dfrac {e^t \sin t}{t} \right \} \$$a) \ \dfrac {\pi}{2}+ \tan^{-1}(s-1) \$$b) \ \dfrac {\pi}{2}+ \tan^{-1}s \\ (c) \ \dfrac {\pi}{2}- \cot^{-1}s \$$d) \ \cot^{-1}(s-1) $$
iv) Transform of unit step function L{u(t-a)} is, ______ $$ (a)\ \dfrac {e^{as}}{s} \$$b) \ \dfrac {e^{-s}}{s} \$$c) \ \dfrac {e^{2s}}{s} \$$d) \ \dfrac {e^{-as}}{s} $$(4 marks)
7 (b) $$ Evaluate \ L\left \{ 3^t + \dfrac {\cos 2t - \cos 3t}{t}+ t \sin t \right \} $$(4 marks)
7 (c) Find the Laplace transform of the triangular wave, given by$$ f(t)= \left\{\begin{matrix}t &0 & and \ f(t+2c)= f(t) \\2C-1 &C<1<2C & \ \end{matrix}\right. $$(6 marks)
7 (d) $$ Express \ f(t) = \left\{\begin{matrix}\cos t &if 0<1<\pi \\\cos 2t & if \pi < t< 2\pi \\\cos 3t & if \ t> 2\pi \end{matrix}\right. $$ in terms of unit step function and hence find L{f(t)}(6 marks)
8 (a) Choose the correct answer:
$$ L^{-1} \left \{ \cot^{-1} \left ( \dfrac {s}{a} \right ) \right \} = \_\_\_\_\_\_\_\_ \$$a) \ \dfrac {\sin t}{t} \$$b) \ \dfrac {\sin at}{t} \$$c) \ \dfrac {\sinh at}{t} \$$d) \ \dfrac {\sinh t}{t} $$ $$ (ii) \ L^{-1} \left \{ \dfrac {1}{4s^2 -36} \right \}= \_\_\_\_\_\_\_\_ \$$a) \ \dfrac {\cosh 6t}{4} \$$b) \ \dfrac {\sin 3t}{12} \$$c) \ \dfrac{\sinh 3t}{12} \$$d) \ \dfrac{\cosh 3t}{6}\\ $$ $$ iii) \ L^{-1}\left \{ \dfrac {1}{s(s^2 +a^2)} \right \} = \_\_\_\_\_\_\_\_ \_\ \$$a) \ \dfrac {1-\cos at}{a^2} \$$b) \ \dfrac {1+ \cos at}{a^2} \$$c) \ \dfrac {1- \sin at}{a^2} \$$d) \ \dfrac {1+ \sin 3t}{6} $$ $$ (iv) \ L^{-1} \left \{ \dfrac {s^2 - 3s +4}{s^4} \right \} = \_\_\_\_\_\_\_\_ \_\ \$$a)\ 1-3t + 2t^3 \$$b) \ 1+ \dfrac {t^2}{3} \$$c) \ t+ \dfrac {3}{2}+ 1 \$$d) \ t-\dfrac {3}{2}t^2 + \dfrac {2}{3} t^3 $$(4 marks)
8 (b) $$ Find \ L^{-1} \left \{ \dfrac{3s+7}{s^2 - 2s -3} \right \} $$(4 marks)
8 (c) Using Convolution theorem evaluate $$ L^{-1} \left \{ \dfrac{1} {(s+1)(s^2+4)}\right \} $$(6 marks)
8 (d) $$ Solve \ \dfrac {d^2y}{dt} + 5 \dfrac {dy}{dt} + 6y = 5e^{2t} \ given \ that \ y(0)= 2, \ \dfrac {dy(0)}{dt}=1 $$ by using Laplace transform method.(6 marks)
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