Subject Code: 3110015
GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER- I & II (NEW) EXAMINATION — WINTER 2019
Subject Name: Mathematics —2
Date: 01/01/2020
Time: 10:30 AM TO 01:30 PM
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q1 (a) Find the length of curve of the portion of the circular helix r(t) = cost i + sint j + t k from t=0 to t=p. [03]
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(b) ? (xy2 + y3)dx + (x2y + 3x2)dy is independent of path joining the points (1, 2) and (3,4). Hence, evaluate the integral. [04]
(c) Verify tangential form of Green’s theorem for F = (x — sin y)i + (cos y) j, where C is the boundary of the region bounded by the lines y=0, x= p/2 and y=x. [07]
Q2 (a) Find the Laplace transform of f(t) defined as f(t) = 0, 0<t<k = 1, t>k [03]
(b) Find the inverse Laplace transform of s / ((s2+a2)(s2+b2)) [04]
(c) (i) Calculate the curl of the vector xyzi +3x2y j +(xz2 - yz2)k [07]
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(ii) The temperature at any point in space is given by T =xy+yz+zx. Determine the derivative of T in the direction of the vector 3i —4k at the point (1, 1, 1).
OR
(c) Let F=xi+yj+zk, r=|r|, and a is a constant vector. Find the value of div(a x F / rn) [07]
Q3 (a) Find constants a, b and c such that V= (x+2y+az)i+(bx+3y-z)j+(4x+cy+2z)k is irrotational. [03]
(b) Using Fourier cosine integral representation show that ?08 (cos wx / (k2 + w2)) dw = (p e-kx) / (2k) [04]
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(c) Solve the following differential equations: (i) cos(x+y)dy =dx (ii) sec2y dy/dx +xtany =x3 [07]
OR
(c) Find the Laplace transform of f(t) = t2 e-3t cos(2t) [07]
Q4 (a) Using Convolution theorem obtain L-1 {1 / (s2(s +a ))} [03]
(b) Find the power series solution of d2y/dx2 +xy=0 [04]
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(c) Find the Laplace transform of the waveform f(t) = t, 0<t<3 = 0, t>3 [07]
OR
(c) Using the Laplace transforms, find the solution of the initial value problem y"+25y=10cos5t y(0)=2, y'(0)=0 [07]
Q.5 (a) Using variation of parameter method solve (D2 + 1) y=xsinx [03]
(b) Solve (d2y/dx2) -3(dy/dx) +(dy/dx) -y=4t [04]
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(c) Solve (D2 + 4)y =cos2x [07]
OR
(c) Solve (i) y exdx+(2y+ex)dy=0 (ii) dy/dx +2ytanx=sinx [07]
Q.5 (a) Solve dy/dx = (x+y) / (ex+y) [03]
(b) If y1 = x is one of solution of x2y"+xy' —y =0 find the second solution. [04]
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(c) Using Frobenius method solve x2y" +4xy' +(x2 + 2) y=0 [07]
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