Download PTU B-Tech AR-Automation-And-Robotics 2020 Dec 3rd Sem 76502 Mathematics Iii Question Paper

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Total No. of Pages : 02
Total No. of Questions : 18

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B.Tech. (Automation & Robotics) (2018 Batch) (Sem.-3)
MATHEMATICS-III
Subject Code : BTAR-303-18
M.Code : 76502
Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

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Write briefly :

1. Find the Fourier series of the function f(x) = | x | over the interval [-2, 2].
2. Find Laplace transform of t sin²t.
3. State and prove Second Shifting Property for Laplace transform.
4. Find inverse Laplace transform of ^2s2+5/_s⁴+4s²+13.

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5. Express sum of Legendre polynomials 8P₄(x) + 2P₃(x) + P₀(x) in terms of powers of x.
6. For Legendre polynomial P_n(x), show that P_n(-x) = (-1)ⁿ P_n(x)
7. Form a partial differential equation by eliminating arbitrary function f from the relation z = y² + 2f(x+logy).
8. Solve z(xp - yq) = y² - x².
9. Show that the function u (x, y) = 2x³ + y³ - 3x²y is harmonic.

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10. State Cauchy Integral Theorem.

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SECTION-B

1. Find the Fourier series expansion of the function f(x) = { 0, for -p < x < 0 1, for 0 < x < p }
2. State and prove Convolution Theorem for Laplace transform.

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3. For Legendre polynomial P_n(x), show that : ?¹_-1 P_m(x) P_n(x) dx = { 0, form ? n 2/(2n+1), form = n }
4. Solve by Charpit’s method z = p² + q².
5. Evaluate ? (z²+5)/(z² + 2z) dz, C:|z|=1.

SECTION-C

1. a) Using Laplace transform, solve y” — 6y’ + 9y = e^3t, y (0) = 2, y' (0) = 6.
   

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   b) Find inverse Laplace transform of (2s²+1)/((s+2)²(s-1)).
2. a) Solve Legendre differential equation (1- x²)y" - 2xy' + n (n+1) y = 0.
   b) Using the method of separation of variables, solve ?u/?t = k(?²u/?x²), U (x, 0)=x², u (0, t)=u (2p, t) = 0.
3. a) Find all Taylor and Laurent series expansions of f(z) = 1/(z²+1) about the point z = i.
   b) Compute the residues at the singular points z=1, -2 of f(z) = (1 +z+z²)/((z-1)² (z+2)).

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NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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