Download PTU B-Tech ME 2020 Dec 5th Sem 70601 Mathematics Iii Question Paper

FirstRanker.com

Firstranker's choice

Roll No. Total No. of Pages : 02

--- Content provided by‌ FirstRanker.com ---

Total No. of Questions : 18

B.Tech. (ME) (2012 Onwards) (Sem. - 5)

MATHEMATICS-III

Subject Code : BTAM-500

M.Code : 70601

--- Content provided by FirstRanker.com ---

Time : 3 Hrs. Max. Marks : 60

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.

--- Content provided by​ FirstRanker.com ---

SECTION-A

Write briefly :

1. Expand f(x) = | sin x | in Fourier series.
2. Find Laplace transform of sin t cos t.
3. Find Laplace transform of t^-at t^-bt

--- Content provided by​ FirstRanker.com ---

4. Find inverse Laplace transform of 1/(s-3)³
5. Express x⁴+2x³-6x²+5x-3 in terms of Legendre polynomials.
6. For Legendre polynomial P_n(x), show that P_n(1) = n(n+1)/2
7. Form a partial differential equation by eliminating arbitrary functions from the relation z = yf(x) + xg(y).
8. Solve xp + yq = 3z.

--- Content provided by‌ FirstRanker.com ---

9. Show that the function f(z) = | z |² satisfies the Cauchy-Riemann equations only at origin.
10. State Cauchy Integral Theorem.

FirstRanker.com

FirstRanker.com

SECTION-B

--- Content provided by‌ FirstRanker.com ---

1. Find the Fourier series expansion of the function f(x) = x², -p < x < p. Deduce that S 1/n² = 1 + 1/2² + 1/3² + 1/4² + ... = p²/6
2. State and prove Convolution theorem for Laplace transform.
3. For Bessel’s function J_n(x), show that J'₀ = 2(J₂ + J₄ + J₆ + ...)=1
4. Solve by Charpit’s method q + xp = p²
5. Evaluate ? dz / ((z²+4)²) where |z|=2

--- Content provided by FirstRanker.com ---

SECTION-C

1. a) Using Laplace transform, solve y'' + 2y' = 1 - H (t - 1), y (0) = 2, where H (t) is Heaviside’s unit step function.
2. b) Find inverse Laplace transform of 1/(s²(s+1))
3. a) Using Frobenius method, find two linearly independent solutions of the equation 2x²y'' +xy' — (x² +1)y=0.
4. b) A rod of length l with insulated side is initially at a uniform temperature u. Its ends are suddenly cooled at 0°C and kept at that temperature. Find the temperature function u (x, t).

--- Content provided by‍ FirstRanker.com ---

5. a) Find all Taylor and Laurent series expansions of f(z) = 1/(z(z-1)) about the point z=0.
6. b) Compute the residues at all the singular points of f(z) = z²/(z⁴+1)

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.

FirstRanker.com

--- Content provided by⁠ FirstRanker.com ---