Question Paper: Applied Mathematics 2 : Question Paper May 2015 - First Year Engineering (Semester 2) | Mumbai University (MU)
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Applied Mathematics 2 - May 2015

First Year Engineering (Semester 2)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) $$Evaluate \ \int^\infty_0 \dfrac {x^4}{4^x} dx$$(3 marks) 1 (b) Find P.I. of (D2-4D+4) y=ex+cos 2x(3 marks) 1 (c) Show that ∇=1-E-1.(3 marks) 1 (d) $$Evaluate \ \int^1_0 \int^{\sqrt{1+x^2}}_0 \dfrac {dydx}{1+x^2+y^2}$$(3 marks) 1 (e) $$Solve \ \left ( 1+e^{x/y} \right )dx+e^{x/y} \left ( 1- \dfrac {x}{y} \right )dy = 0$$(4 marks) 1 (f) Evaluate $$\int^\infty_0 \int^\infty_0 e^{-(x^2+y^2)}dxdy$$ by changing to polar co-ordinates(4 marks) 2 (a) Solve $$y^4 dx = \left ( x^{-3/4} - y^{3}x \right )dy$$(6 marks) 2 (b) Change the order of integration and evaluate $$\int^1_0 \int^{1/x}_0 \dfrac {y}{(1+xy)^2 (1+y^2)}dydx$$(6 marks) 2 (c) (i) $$P.T. \ \int^\infty_0 \dfrac {x^{n=1}}{(a+bx)^{m-n}}dx = \dfrac {1}{a^n b^m} \beta (m,n)$$(4 marks) 2 (c) (ii) $$P.T. \ \int^\infty_0 \dfrac{\log (1+ax^2)}{x^2} dx = \pi \sqrt{a}, \ where \ a>0$$(4 marks) 3 (a) Evaluate $$\int^{\log 2}_0 \int^x_0 \int^{x+\log y}_0 e^{x+y+z}dzdydx$$(6 marks) 3 (b) Find the area bounded between the paraboala
x2=4ay and x2=-4a(y-2a)
(6 marks)
3 (c) Solve by the method of variation of parameters $$\dfrac {d^2 y}{dx^2} + y = \sec x \tan x$$(8 marks) 4 (a) Find the length of the cardioid r=a(1-cos ?) lying outside the circle r=a cos ?(6 marks) 4 (b) Solve $$\dfrac {d^2y}{dx^2} - 4 \dfrac {dy}{dx}+3y = 2xe^{3x}+3e^x \cos 2x$$(6 marks) 4 (c) Using R.K. Method of fourth order, solve, $$\dfrac {dy}{dx} = \dfrac {y^2 - x^2}{y^2+x^2} \ given \ y(0)=1 \ at \ x=0.2, 0.4$$(8 marks) 5 (a) Solve x sin x dy+(xycos x- ysinx -2)dx=0(6 marks) 5 (b) Solve $$\dfrac {dy}{dx} =2 + \sqrt{xy} \ with \ x_0=1.2, y_0 =1.6403$$ by moditied Euler's method, for x=1.4 correct to 4-decimal places, (taking h=0.2)(6 marks) 5 (c) Evaluate $$\int^6_0 x f(x) dx \ by$$ i) Trapezoidal rule
ii) simpson's 1/3rd rule
using the following table

 x 0 1 2 3 4 5 6 f(x) 0.146 0.161 0.176 0.19 0.204 0.217 0.23
(8 marks) 6 (a) The charge Q on the plane of a condensor of Capacity C charged through a resistance R by a steady voltage V satisfies the differential equation $$R\dfrac {dQ}{dt}+ \dfrac {Q}{c} =V, \ if \ Q=0 \ at \ t=0 , \ Show \ that \ i= \dfrac {V}{R}e^{-1/ac} \because \ i=\dfrac {dQ}{dt}$$(6 marks) 6 (b) Evaluate $$\iint_A x^2 dxdy$$ where A is the region in the first qadrant bounded by the hyperbola xy=16 and the lines y=x, y=0 and x=8.(6 marks) 6 (c) Find the volume of the tetrahedron bounded by the planes, x=0, y=0, z=0 and x+y+z=a.(8 marks)