AKTU B-Tech Last 10 Years 2010-2020 Previous Question Papers || Dr. A.P.J. Abdul Kalam Technical University
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Sub Code: KAS203
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Roll No.
Printed Pages: 02
Paper Id: 199243
B. TECH.
(SEM II) THEORY EXAMINATION 2018-19
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MATHEMATICS-II
Time: 3 Hours
Total Marks: 100
Note: Attempt all Sections. If require any missing data; then choose suitably.
SECTION A
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- Attempt all questions in brief. 2 x 10 = 20
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Find the P.I of d²y/dx² + 4y = sin2x | 2 | 1 |
| b. | Solve simultaneous equations dx/dt = 3y dy/dt = 3x | 2 | 1 |
| c. | Find the volume of solid generated by revolving the circle x² + y² = 25 about y-axis. | 2 | 2 |
| d. | Evaluate G(5/2) where G is gamma function | 2 | 2 |
| e. | Find the Fourier constant a1 of f(x) = x², -p = x = p | 2 | 3 |
| f. | Discuss the convergence of sequence an = 2n/n²+1 | 2 | 3 |
| g. | Show that complex function f(z) = z³ is analytic. | 2 | 4 |
| h. | Define Conformal mapping. | 2 | 4 |
| i. | Evaluate ?1+i (x² – iy)dz along the path y = x | 2 | 5 |
| j. | Find residue of f(z) = cos z/z(z+5) at z = 0 | 2 | 5 |
SECTION B
- Attempt any three of the following:
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Use Frobenius method to solve 9x(1-x) d²y/dx² -12 dy/dx + 4y = 0 | 10 | 1 |
| b. | Apply Dirichlet integral to find the volume of an octant of the sphere x² + y² + z² = 25. | 10 | 2 |
| c. | Find half range sine series of f(x) = 0<x<2 4-x 2<x<4 | 10 | 3 |
| d. | Show that u = x4 – 6x2y2 + y4 is harmonic function. Find complex function f(z) whose u is a real part. | 10 | 4 |
| e. | Expand f(z) = 1/(z-1)(z-2) in regions (i) 1 < |z| < 2 (ii) 2 < |z| | 10 | 5 |
SECTION C
- Attempt any one part of the following:
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| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Solve d²y/dx² + y = tanx by method of variation of parameter. | 10 | 1 |
| b. | Solve x² d²y/dx² + 2(x² + x) dy/dx + (x² + 2x + 2)y = 0 by Normal Form. | 10 | 1 |
- Attempt any one part of the following:
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Prove that ß(m,n)= GmGn/G(m+n) where G is gamma function | 10 | 2 |
| b. | Use Beta and Gamma function to solve ?0p 1/1+x4 dx | 10 | 2 |
- Attempt any one part of the following:
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Find the Fourier series of f(x) = xsinx, -p = x = p | 10 | 3 |
| b. | State D' Alembert's test. Test the series 1+ x/2 + x²/5 + x³/10 + ... + xn/n²+1 + ... | 10 | 3 |
- Attempt any one part of the following:
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Let f(z) = x²y5(x+iy)/x4+y10 when z ? 0, f(z) = 0 when z = 0. Prove that Cauchy Riemann satisfies at z = 0 but function is not differentiable at z = 0. | 10 | 4 |
| b. | Find Mobius transformation that maps points z = 0, -i, 2i into the points w = 5i,8, -i/3 | 10 | 4 |
- Attempt any one part of the following:
| QNo. | Question | Marks | CO |
|---|---|---|---|
| a. | Using Cauchy Integral formula evaluate ?c sin z/z²+25 dz where c is circle |z| = 8 | 10 | 5 |
| b. | Apply residue theorem to evaluate ?-88 x²/(x²+1)(x²+4) dx | 10 | 5 |
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