Download SGBAU B-Tech 4th Sem Applied Mathematics III Question Paper

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Time : Three Hours

B.Tech. Fourth Semester (Chem. / Poly / Food / Pulp / Oil / Petro) (Old)

Applied Mathematics — 111 : 4 SCE 1

Notes:

1. Answer three questions from Section A and three questions from Section B.

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2. Due credit will be given to neatness and adequate dimensions.
3. Assume suitable data wherever necessary.
4. Illustrate your answer necessary with the help of neat sketches.
5. Use of calculator is permitted.
6. Use of pen Blue/Black ink/refill only for writing the answer book.

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SECTION -A

1. a) Solve d²y/dx² +2dy/dx+y =x²cosx
   b) Solve (D² +3D+2)y=e^-x by using variation of parameter.
   OR
2. a) Solve (D³ +1)y =sin3x —cos² x
   

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   b) Solve x² d²y/dx² - xdy/dx +4y = cos(log x) + x sin (logx)
3. a) Evaluate Laplace transform of ?₀^t e^t/t dt
   b) Prove that L^-1 {log(1 +1/s²)} = ?₀⁸ (1-cosu)/u du
   c) Express f(t) in terms unit step function and hence find its Laplace transform
   f(t)=t, 0<t<1
   

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   =4t, t>1
   OR
4. a) Find the Laplace transform of
   f(t)=asin(pt) , 0<t<p/p
   =0 , p/p<t<2p/p
   

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   where f(t+2p/p)=f(t)

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5. a) Use convolution theorem to find L^-1{1/((s+1)(s²+1))}
   b) Solve the differential equation using Laplace transform
   d²y/dt² =2t if y(0)=1, y'(0)=1/2

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6. a) Find the Fourier sine transform of e^-ax/x
   OR
   b) Using Fourier integral show that
   ?₀⁸ (sin(tx)/t) dt = p/2, 0<x<p
   =0, x>p

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7. Use Laplace transform to solve the differential equation dy/dt +2y + ?₀^t ydt =sint when y (0) = 1.

SECTION - B

8. Solve the following difference equation -
   i) y_x+2 +4y_x+1+y_x = x²
   ii) y_p+2 -4y_p =n²+n-1

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9. Find the inverse z-transform of (4z²-2z)/(z³+5z²+4z)
10. Solve the difference equation y_p+2 -2cosay_p+1 +y_p =0 with y(0) =1, y(1) = cosa using method of z-transform.
11. Find the z-transform of
    i) 1/(n+1)
    ii) (cos?+isin?)ⁿ

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12. Find the tangential & normal component of acceleration at any time t of a particle whose position (x, y) at any time t is given by x = log (t² +1), y=t -2tan^-1t.

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13. a) Find the directional derivative of f = e^2x cosyz at the origin in the direction of the tangent to the curve X =asint, y=acost, z=at at t=p/2
    OR
    b) If pF = ?P, where p, P and F are point functions, prove that F. curlF =0.

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14. a) Prove that a x ?(B.?(1/r)) = 3(a.r)(B x r)/r⁵ - (B.r)(a x r)/r⁵
    b) A vector field is given by F=siny i+ x(1+cos y)j . Evaluate the line integral over the circular path given by x²+ y² = a², z=0.
15. a) Apply Stokes theorem to evaluate ?_C (x+y)dx +(2x—2z)dy + (y+z)dz where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), (0, 0, 6).
    OR
    b) Use Divergence theorem to evaluate ?_S(y²z³i + z²x³j + x²y³k).dS where s is the upper part of the sphere x² +y² +z² =1.

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16. Prove that F = (x² —yz)i+(y² —zx) j+ (z² —xy)k is irrotational and find f if F=?f.