Download DBATU B.Tech 2017 May 1st Year Engineering Mathematics I Question Paper

DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE

SEMESTER EXAMINATION: MAY-2017

Mechanical/Electrical/ExTC/Chemical/Petrochemical/Computer/IT/Civil

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Subject: Engineering Mathematics-I (New) BSHIOI

Semester: 1

Time: 03 Hrs

Max. Marks: 60

INSTRUCTION: ATTEMPT ANY FIVE QUESTIONS.

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Q1.

(a) Find the rank of the matrix A = [1 2 3 2]
[2 3 5 1]
[1 3 5 4] by reducing it to normal form. [4 Marks]

(b) For what values of k is the following system of equations consistent, and hence solve for them: [4 Marks]

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(c) Find the eigen values and eigen vectors of the matrix A = [3 1 4]
[0 2 6]
[0 0 5] . [4 Marks]

Q2.

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(a) Find the nth derivative of tan^-1 (2x/1-x²) in terms of r and ?. [4 Marks]

(b) If y = (x² - 1)ⁿ, prove that (x² - 1)y_n+2 + 2xy_n+1 - n(n + 1)y_n = 0. [4 Marks]

(c) Expand f(x+h) = tan^-1(x + h) in powers of h and hence find the value of tan^-1(1.003) upto five places of decimal. [4 Marks]

Q3.

(a) If x²/a²+u + y²/b²+u + z²/c²+u = 1, prove that (?u/?x)² + (?u/?y)² + (?u/?z)² = 2 [x ?u/?x + y ?u/?y + z ?u/?z] [4 Marks]

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(b) If z is a homogeneous function of degree n in x, y then prove that x² ?²z/?x² + 2xy ?²z/?x?y + y² ?²z/?y² = n(n - 1)z. [4 Marks]

(c) If F = F(x, y, z) where x = u + v + w, y = uv + vw + wu, z = xyz, then show that u ?F/?u + v ?F/?v + w ?F/?w = x ?F/?x + 2y ?F/?y + 3z ?F/?z [4 Marks]

Q4.

(a) Expand f(x,y) = cos x sin y as far as the terms of third degree. [4 Marks]

(b) If the sides and angles of a plane triangle vary in such a way that its circum-radius remains constant, prove that da/cosA + db/cosB + dc/cosC = 0, where da, db, dc are smaller increments in the sides a, b, c respectively. [4 Marks]

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(c) Find the maximum and minimum distances from the origin to the curve 3x² + 4xy + 6y² = 140. [4 Marks]

Q5.

(a) Change to polar co-ordinates and evaluate I = ?0⁸ ?0⁸ e^-(x²+y²) dx dy. [4 Marks]

(b) Evaluate 1 = ?0¹ ?_vy/x^v4-x² dxdy by changing the order of integration. [4 Marks]

(c) Evaluate I = ?0¹ ?0^vz ?_vz/vx^v4z-x² dydxdz. [4 Marks]

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Q6.

(a) Find the interval of convergence of the series S n!/(nⁿ) *xⁿ. [4 Marks]

(b) Test the convergence of the series S (n+1)/n²+1 *xⁿ. [4 Marks]

(c) Test the convergence of the series S n!/(nn)² . [4 Marks]

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