Download PTU B.Tech 2020 March ME 4th Sem Mathematics III Question Paper

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Roll No. [T Total No. of Pages : 02

Total No. of Questions : 09

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B.Tech. (ME) (Sem.-4)

MATHEMATICS-III

Subject Code : AM-201

M.Code : 54035

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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1. Write briefly :
   1. Give Dirichlet’s conditions for the Fourier series expansion of f'(x).
   2. Find the value of a0, in the Fourier series expansion of f(x) = x, -p < x < p.
   3. Find Laplace transform of f(t) = t
   4. Write the definition of unit step function.
   5. Write the Laplace transform of periodic function f(t) with period T.

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   6. Find the complementary function of PDE : (2D² + 5DD’ + 2D’²) z = 0.
   7. Form the partial differential equation from z = ax + a² y² + b.
   8. Give definition of singular point.
   9. Give definition of conformal mapping.
   10. Evaluate ?[(x² + 2y) dx + (3x — y)dy] along the curve x = 2t, y = t² + 3 between (0, 3), (2, 4).

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SECTION-B

2. Find the Fourier series expansion of f(x) =
   -1, 0 < x < p
   2, p < x < 2p

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3. 1) Find L{t sin t} 1) L[ (s²-3s+4) / s ]
4. Prove the recurrence relation xJ'?(x) = xnJ?1(x) for Bessel function.
5. Solve the linear partial differential equation (mz — ny) p + (nx — lz)q = ly — mx.
6. Check if the function f(z) = 2xy + i (x² - y²) is analytic ?

SECTION-C

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7. a) Find the half range Fourier cosine series expansion of f(x) = x, 0 < x < p.
   b) Find L?¹{ 1 / [(s+2)(s-1)] } using convolution theorem.
8. a) Solve the differential equation y'' + 2y' = e?³?, y(0) = 1 using Laplace transform.
   b) Solve (D² + 4DD’ —5D’²) z = sin (2x + 3y).
9. a) Find the analytic function, whose real part is u = sin2x / (cosh2y - cos2x)
   

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   b) Find the Taylor’s series expansion of f(z) = 1 / [(z+1)(z+3)] for the region | z | < 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.

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