Download PTU B.Tech 2021 Jan CE 3rd Sem 76373 Mathematics Iii Transform And Discrete Mathematics Question Paper

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Roll No. Total No. of Pages : 02

Total No. of Questions: 18 B.Tech.(CE) (2018 Batch) (Sem.-3)

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MATHEMATICS-III (TRANSFORM & DISCRETE MATHEMATICS)

Subject Code : BTAM-301-18

M.Code : 76373

Time : 3 Hrs. Max. Marks : 60

INSTRUCTIONS TO CANDIDATES :

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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

Write briefly :

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1. Define gradient of a scalar point function.
2. If F = (x + y + 1)i + (j + (x + y)k. Show that F. curl F = 0
3. Define Laplace transform
4. Write the relation between Laplace and Fourier transform.
5. Represent f(t) = sin 2t, 2p < t < p and 0 otherwise, in terms of unit step function.

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6. Define Solenoidal and irrotational fields.
7. State convolution theorem of Fourier transform.
8. State Stokes theorem.
9. Write Euler's formula of Fourier series.
10. Write Gibbs phenomenon.

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SECTION-B

11. Find the values of a and b such that the surfaces ax² - byz = (a + 2) x and 4x²y + z³ = 4 cut orthogonally at (1, -1, 2).
12. Apply Convolution theorem to evaluate the inverse Laplace transform of:

    s² / (s²+a²) (s²+b²)

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13. Find the Fourier sine transform of e^-mx. Hence show that

    ?₀⁸ x sin mx / (1+x²) dx = pe^-m / 2 , m > 0

14. Apply Green's theorem to evaluate ?_c [(2x² - y²)dx + (x² + y²)dy], where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle x²+ y²= a².
15. If A and B are irrotational, prove that A × B is solenoidal.

SECTION-C

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16. Verify Gauss divergence theorem for F = (x²-yz)i + (y²-zx)j + (z²-xy)k taken over the parallelepiped 0 = x = a, 0 = y = b, 0 = z = c.
17. Find the Fourier cosine series of the function f(x) = p - x in 0 < x < p. Hence show that

    ?_r=0⁸ 4 / (2r+1)² = p²/8

18. a) Use Laplace transform method to solve :

    d²x/dt² - 2dx/dt + x = e^t

    With x=2, dx/dt =-1 at t = 0.

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    b) Find the directional derivative of f = x² - y² + 2z² at the point P (1, 2, 3) in the direction of the line PQ where Q is the point (5, 0, 4). Also calculate the magnitude of the maximum directional derivatives.

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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