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

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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE

SEMESTER EXAMINATION: A 2.1

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

Subject : Engineering Mathematics-I (New) BS ]"} 1 O \ Semester 51

Time : 03 Hrs Max. Marks : 60

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2 MAY 2017

INSTRUCTION: ATTEMPT ANY FIVE QUESTIONS.

Q1. (a) Find the rank of the matrix A = 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]

x+y+z=1, 4x+2y+4z=k, x+4y+10z = k².

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

Q2. (a) Find the n^th derivative of tan^-1(x/a) in terms of n and a. [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 x 2 a 2 + u + y 2 b 2 + u + z 2 c 2 + u = 1 , prove that ? u ? x + ? u ? y + ? u ? z = 2 [4 Marks]

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(b) If z is a homogeneous function of degree n in x, y , then prove that x ?z/?x + y ?z/?y = nz. [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]

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Q4. (a) Expand f(x,y) = cosx siny , 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 ?₀⁸ ?₀⁸ e^-(x²+y²) dx dy. [4 Marks]

(b) Evaluate ?₀^a ?_y^v(a²-y²) x/(v(x²+y²)) dx dy by changing the order of integration. [4 Marks]

(c) Evaluate ?₀¹ ?₀^v(1-x²) ?₀^{v(a²-x²-y²)} dydxdz. [4 Marks]

Q6. (a) Find the interval of convergence of the series S xⁿ/(n+1). [4 Marks]

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

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

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