DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
End — Semester Examination (Supplementary): November 2018
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Branch: B. Tech (Common to all) Semester: II
Subject with code: Engineering Mathematics — II (MATH 201)
Date: 27/11/2018 Max Marks: 60 Duration: 03 Hrs.
INSTRUCTION: Attempt any FIVE of the following questions. All questions carry equal marks.
Q.1
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(a) Prove that cos8? — sin8? = 1—16 (cos6? + 15cos2?). [6 Marks]
(b) If tan(A + iB) = x + iy, prove that
(i) tan2A = 2x / (1-x2-y2) (ii) tanh2B = 2y / (1+x2+y2) [6 Marks]
Q.2
(a) Solve (1 + ex/y) dx + ex/y(1- x/y) dy = 0. [6 Marks]
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(b) Solve x dy/dx + y = logx/x. [6 Marks]
Q.3 Solve any TWO:
(a) Solve d2y/dx2 + 4dy/dx + 13y = 18e-2x. [6 Marks]
(b) Solve (D2+5D +4)y = x2+7x+9. [6 Marks]
(c) Solve by the method of variation of parameters d2y/dx2 — 3dy/dx +y = cosecx . [6 Marks]
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Q.4
(a) Find the Fourier series of f(x) = x2 in the interval (0, 2p) and hence deduce that p2/12 = 1 - 1/22 + 1/32 - 1/42 + ... [6 Marks]
(b) Expand the function f(x) = px — x2 in a half — range sine series in the interval(0, p). [6 Marks]
Q.5
(a) The necessary and sufficient condition for vector F(t) to have constant magnitude is F(t) . dF(t)/dt = 0. [6 Marks]
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(b) A point moves in a plane so that its tangential and normal components of acceleration are equal and the angular velocity of the tangent is constant and equal to ?. Show that the path is equiangular spiral r = Ae?t + B, where A & B are constants. [6 Marks]
Q.6 Solve any TWO:
(a) Find curl F, where F = V (x3 +y3 + z3 — 3xyz) . [6 Marks]
(b) If r is a position vector with r = |r| , show that V.(rnr) = (n+ 3)rn. [6 Marks]
(c) Show that ?s F . n ds = ?v (? . F) dv. [6 Marks]
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