Sub Code: KAS103
B.Tech. (SEM-I) THEORY EXAMINATION 2018-19
MATHEMATICS-I
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Time: 3 Hours
Total Marks: 100
Note: Attempt all Sections. If require any missing data; then choose suitably.
SECTION A
1. Attempt all questions.
| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | Find the rank of the matrix
| 2 | 1 |
| b. | Find the stationary point of f(x, y) = x² + y² + 3axy, a > 0 | 2 | 1 |
| c. | If x = rcos?, y = rsin?, z = z then find ?(x, y, z) / ?(r, ?, z) | 2 | 2 |
| d. | Define del operator and gradient. | 2 | 2 |
| e. | If f = 3x²y - y³z², find grad f | 2 | 3 |
| f. | Evaluate ?01 ?0x ex² dydx | 2 | 4 |
| g. | If the eigen values of matrix A are 1, 1, -1, then find the eigen values of A² + 2A + 3I. | 2 | 1 |
| h. | Define Rolle's Theorem | 2 | 3 |
| i. | If u = xy² sin-1 (y/x), then find x ?u/?x + y ?u/?y | 2 | 1 |
| j. | State the Taylor's Theorem for two variables | 2 | 3 |
| k. | In R = E/I, if possible error in E and I are 2% and 10% respectively, then find the error in R. | 2 | 3 |
SECTION B
2. Attempt any three of the following:
| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | Using Cayley-Hamilton theorem find the inverse of the matrix A= Also express the polynomial B= A8 - 11A7 - 4A6 + A5 + A4 - 11A3 - 3A2 + 2A + I as a quadratic polynomial in A and hence find B. | 8 | 1 |
| b. | If y = sin(m sin-1x), prove that : (1 - x²) yn+2 - (2n+1)x yn+1 + (m² - n²)yn = 0 and find yn at x = 0. | 8 | 2 |
| c. | If u, v, w are the roots of the equation (x-a)3 + (x-b)3 + (x-c)3 = 0, then find ?(u, v, w) / ?(a, b, c) | 8 | 2 |
| d. | Evaluate ?-88 ?-88 e-(x² + y²) dxdy by changing to polar coordinates. Hence show that ?08 e-x² dx = v(p/2) | 8 | 2 |
| e. | Verify the divergence theorem for F = (x³ - y)i + (y³ - zx)j + (z³ - xy)k, taken over the cube be planes x = 0, y = 0, z = 0, x = 1, y = 1, z = 1. | 8 | 3 |
SECTION C
3. Attempt any one part of the following:
| Qno. | Question | Marks | CO |
|---|---|---|---|
| a. | Find inverse employing elementary transformation A =
| 10 | 1 |
| b. | Reduce the matrix A to its normal form when A = Hence find the rank of A. | 10 | 1 |
4. Attempt any one part of the following:
| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | If sin y = 2 log(x + 1) show that (x + 1)²yn+2 + (2n+1)(x+1)yn+1 + (n²+4)yn = 0 | 10 | 2 |
| b. | Verify Lagrange's Mean value Theorem for the function f(x) = x(x-1)(x-2) in [-2, 2] | 10 | 2 |
5. Attempt any one part of the following:
| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | Find the maximum or minimum distance of the point (1, 2, -1) from the sphere x²+y²+z² = 24. | 10 | 3 |
| b. | If u = cos-1 ( (x + y) / (vx + vy) ) then show that x ?u/?x + y ?u/?y = 1/2 cot u | 10 | 3 |
6. Attempt any one part of the following:
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| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | Change the order of integration and then evaluate: ?02 ?x2-x xy dy dx | 10 | 4 |
| b. | Calculate the volume of the solid bounded by the surface x+y+z=1 & z=0. | 10 | 4 |
7. Attempt any one part of the following:
| Q no. | Question | Marks | CO |
|---|---|---|---|
| a. | Prove that F = (y²z - 2xz +3y)i + (3xz+2xy)j + (3xy - 2xz+2z)k is Solenoidal and Irrotational. | 10 | 3 |
| b. | Find the directional derivative of f = 5x²y - 5yz + z²x at P(1, 1, 1) in the direction of the line (x-1)/2 = (y-1)/-2 = (z-1)/1 | 10 | 3 |
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