Download DBATU B-Tech 1st Year 2018 Dec Engineering Mathematics I 2 Question Paper

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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE

End — Semester Examination (Supplementary): November 2018

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Branch: B. Tech (Common to all) Semester: I

Subject with code: Engineering Mathematics — I (MATH 101)

Date: 26/11/2018 Marks: 60 Duration: 03 Hrs.

INSTRUCTION: Attempt any FIVE of the following questions. All questions carry equal marks.

Q.1 (a) Find the rank of the matrix A = 1 -1 2 3 2 1 0 2 0 1 0 2 by reducing it to normal form. [6 Marks]

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

Q.2 (a) If y = e^{m sin-1 x}, prove that (1 — x²)y_n+2 — (2n + 1)xy_n+1 — (n² + a²)y_n = 0. [6 Marks]

(b) Using Taylor’s theorem, express the polynomial f(x) = 2x³ + 7x² + x — 6 in powers of (x — 1). [6 Marks]

Q.3 Solve any TWO:

(a) If y = log(x² + y² + z²) , prove that (x² + y² + z²) (x ?z/?x + y ?z/?y + z ?z/?z) = 2. [6 Marks]

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(b) If z is a homogeneous function of degree n in x,y , prove that x² ?²z/?x² + 2xy ?²z/?x?y + y² ?²z/?y² = n(n—1)z. [6 Marks]

(c) If z= f(x,y) where x =e^u +e^v & y = e^u —e^v, then show that ?z/?u - ?z/?v = x ?z/?x - y ?z/?y [6 Marks]

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Q.4 (a) If u= x/y, v =y/z and w =z/x, show that ?(u, v, w)/?(x, y, z) = 0 [4 Marks]

(b) The focal length of a mirror is found from the formula 1/f = 1/u - 1/v. Find the percentage error in f if u & v are both in error by 2% each. [4 Marks]

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(c) Find the maximum value of x^myⁿz^p, when x +y+z=c. [4 Marks]

Q.5 (a) Evaluate the integral ?₀¹ ?_x^2-x e^x+y dydx. [6 Marks]

(b) Change to polar co-ordinates to evaluate I = ?₀⁸ ?₀⁸ e^-(x2+y2) dy dx . [6 Marks]

(c) Evaluate the integral I = ?₀¹ ?_vy¹ ?₀^1-x x dz dx dy . [6 Marks]

Q.6 (a) State D’ Alembert’s ratio test, and hence check the convergence of the series: S (n/2ⁿ) [6 Marks]

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(b) State Cauchy’s root test, and hence check the convergence of the series: S (1 + 1/n)^-n2 [6 Marks]

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