Sardar Vallabhbhai National Institute of Technology 2012 B.Tech Mathematics-II - Question Paper
This course is common to all SVNIT 1st year students
S. V. National Institute of Technology, Surat-7.
End Semester Examination, April/May- 2012 B. Tech I (2ml Sem), All Branches Engg. Mathematics -II (ASM102)
Date-30/04/2012
Time: 2 hours (1.30 pm-3.30pm) M .M. - 50
Instruction :(i) Attempt all Questions
(ii)Figures in right indicates Marks
(iii)Draw figures if necessary
I 1. (aState and Prove Eulers Theorem. Deduce x2 uxx + 2xyuxy + y2 uyy = n(n - l)u,where u is a homogenous -'function of two variables x,y of degree n. (5)
Or
(a) Define Jacobian for a function of three variables. If u = , v = , w = , Find .
y 7 xyz d(u,v,w)
1 (b) Solve any two of the followings. (4x2)
*{1) Find the minimum value of x2 + y2 + z2 subject to the conditions x + y + z = 3a.
1 (ii) Find the equation of tangent plane and Normal line to the surface z2 = 4(1 + x2 + y2) at the point (2,2,6).
J (iii) The focal length f of a convex lens measured by observing the image distance and the object distance on an
ill . . optical bench and using formula - + - = - .if u and v can be measured to an accuracy of 1% and the image is
u v f
I a real one.show that the maximum possible error in f is 1 %.
((Expand tan-1 0) about the point (1,1). (4)
2. (a) Obtain the differential equation for Detection of Diabetes Model. Find its solution with physical interpretations.
ir (5)
(a) An E.M.F E Sinpt is applied at t=0 to a circuit containing a condenser C and Inductance L in series. The
current satisfies the equation idt = E sin pt , where i = 7. If p2 = 77 and initially the current
ut c - dt Lc
/ and the charge 17 are zero, find the current in the circuit at any time t.
( 0
(b) Splve any two of the followings w (4x2) (i) (D2 -2D + l)y = xexsinx
(ii) (D2 + 2D)y = 1 + x4
(iii) (xzD3 + 3xD2 + D)y = x2logx
1 t
(c) Solve by using variation of parameter if y 3y + 2y = W
3. (a) Find the series solution about x=0 of the equation y" + xy' + y = 0 ,y(0) = 3 ,y (0) = 7 . (5) Or
(a) Define Regular singular point. Find the Frobenius series solution of 2 x2y" xy + (1 x2)y = 0.
(b) Solve any Two of the followings (4x2)
\fS 0) If f(t) is a piecewise continuous periodic function with period T, Then show that
um) = T:r,!Ze-mdt ' n
(*) Apply convolution theorem to evaluate L_1 (XQ
(iii) Find Laplace Transform of cosatcosbtj
...... -, d *
(c) Solve y"' + 2y" y 2y = 0 , where y = and when y = l,y = 2, y = 2 at t = 0 by using Laplace
v' Transformation. (3)
Or cA-~
(c) Solve y 2 J0fy(a) sin(t - a) da = 0, when y(0) = -1 by using Laplace Transform.
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