Download PTU B.Tech 2020 March PE 3rd Sem Engineering Mathematics III Question Paper

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Roll No. ‘ ‘ ‘ ‘ ‘ ‘ | ‘ ‘ ‘ ‘ Total No. of Pages : 02

Total No. of Questions : 09

B.Tech.(Petroleum Refinary Engineering) (2013 Onwards) (Sem.-3)

ENGINEERING MATHEMATICS-III

Subject Code : BTAM-201

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M.Code : 72189

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.

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3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

1. Write briefly :
   1. Define Fourier series expansion for an even function.
   2. Find Fourier sine series of the function f(x)=1, 0 < x < 2.
   3. Find the inverse Laplace transform of (5m/2) / (s + 16).

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   4. Find the Laplace transform of f(t) = t sin z.
   5. Obtain a partial differential equation by eliminating the arbitrary constants c and w from z = ce^wt cos (wx)
   6. State and prove first shifting property of Laplace transforms.
   7. Find singular points of the differential equation (1 — x²)y" — 2xy' + n (n+ 1)y = 0
   8. State Cauchy’s integral formula.

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   9. Show that the function u (x, y) = 2x + y² — 3x²y is harmonic.
   10. Is f(z)=|z|² analytic function? Justify your answer.

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SECTION-B

2. Find the solution of the given homogeneous partial differential equation [D² - 3DD' + 3D(D')² + (D')³]z=0.
3. Show that the function : f(z)= (x³(1+i)-y³(1-i)) / (x² +y²) , z!=0 0, z=0 satisfies the Cauchy Riemann equations at z = 0 but f'(0) does not exist.

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4. Evaluate the integral ?F(z)dz, C:|z|=2 / (z(z+1))
5. Find inverse Laplace transform of 1 / (s²+9)
6. Express the Bessel’s function J₄(x) in terms of J₀(x) and J₁(x).

SECTION-C

7. Find series solution about x = 0, of the differential equation x(1+x)y"+3x'+y=0

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8. Find all possible Taylor and Laurent series expansions of the function f(z) = 1 / ((z+1)(z+2))
9. Find the Fourier series expansion of the function : f(x) = { 1, -p < x < 0 x, 0 < x < p and hence show that 1+1/2²+1/3²+1/4²+1/5²...=p²/6

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