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Maths model%20 qp

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aryaskrishnan

This document is a 3 page model question paper for the B.Tech degree examination in Engineering Mathematics - I. It contains 5 parts with a total of 100 marks. Part A contains 5 questions worth 15 marks total. Part B contains 5 questions worth 25 marks total. Part C contains 2 modules with 2 questions each, worth 60 marks total. The questions cover topics like eigenvalues and eigenvectors, homogeneous functions, integration, differential equations, and Laplace transforms.

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(Pages : 3) B.TECH. DEGREE EXAMINATION, MAY 2011 MODEL QUESTION PAPER First and Second Semester ENGINEERING MATHEMATICS − I (Common to all branches) Time : Three Hours Maximum : 100 Marks PART A 1. Define eigen values and eigen vectors of a matrix. Find the sum of the eigen values of A= x3 y3 du du 2. State Euler’s theorem on homogeneous functions. If tan u , prove that x y sin 2u x y dx dy 1 x 3. Evaluate xy x y dxdy 0 x 4. Solve D 2 1 y cos(2 x 1) 5. State the first shifting property in Laplace transforms. Also find L e 3t t 3 . [5 x 3 marks = 15 marks] PART B 6. Find the rank of the matrix A = 2 2 2 2 u u u 1 u 7. If u f ( x, y) where x r cos and y r sin , prove that . x y r r2 44 x 8. Evaluate the integral dxdy by changing the order of integration. 0x x2 y2 d2y 9. Using the method of variation of parameters, solve 4y tan 2 x . dx 2 cos at cos bt 10. Find the Laplace transform of . t [5 x 5 marks = 25 marks] Turn over
2 PART C Module I 11. (a) Find values of a and b for which the equations x + ay + z = 3, x + 2y + 2z = b, x + 5y + 3z = 9 are consistent. (7 marks) (b) Show that the vectors (2, −2, 1), (1, 4, −1) and (4, 6, −3) are linearly independent. (5 marks) Or 12. Reduce the quadratic from 2 x1x2 2 x1x3 2 x2 x3 to a canonical form by orthogonal transformation. and specify the matrix of transformation. Also find the rank, index, signature and nature of the quadratic form. (12 marks) Module II (u , v, w) 13. (a) If u x 3y 2 z3, v 4x 2 yz , w 2z 2 xy , evaluate at (1, −1, 0). (5 marks) ( x, y , z ) (b) Expand x 2 y 3 y z in powers of (x – 1) and (y + 2) using Taylor’s theorem. (7 marks) Or x y u u 1 14. (a) If u sin 1 , prove that x y tan u (7 marks) x y x y 2 (b) In a plane triangle, find the maximum value of cosA cosB cosC. (5 marks) Module III 15. (a) Find, by triple integration, the volume of the sphere x 2 y2 z2 a2 (5 marks) 1 1 x2 1 dzdydx (b) Evaluate (7 marks) 0 0 x2 y 2 x2 y2 z2 Or
3 16. (a) Find the area between the circle x 2 y2 a 2 and the line x + y = a lying in the first quadrant, by double integration. (5 marks) (b) By transforming into cylindrical coordinates, evaluate the integral x2 y2 z 2 dx dy dz taken over the region of space defined by x 2 y 2 1 and 0 z 1 . (7 marks) Module IV 17. (a) Solve D 2 4 D 3 y sin 3x cos 2 x (7 marks) d2y y 1 (b) Solve x 2 x (5 marks) dx 2 x x2 Or 2 18. (a) Solve D 2 y 8 e2 x sin 2 x x 2 (7 marks) 2 2d y dy (b) Solve 1 x 1 x y 2sin log 1 x (5 marks) dx 2 dx Module V 2s 2 6s 5 19. (a) Find the inverse Laplace transform of (5 marks) s3 6s 2 11s 6 s (b) Using convolution theorem, find the inverse Laplace transform of (7 marks) 2 s2 a2 Or 20. Solve the following differential equation by the method of Laplace transforms y 3y 2y 4t e3t , when y (0) 1 and y (0) 1 (12 marks) [5 x 12 marks = 60 marks]

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Maths model%20 qp

  • 1. (Pages : 3) B.TECH. DEGREE EXAMINATION, MAY 2011 MODEL QUESTION PAPER First and Second Semester ENGINEERING MATHEMATICS − I (Common to all branches) Time : Three Hours Maximum : 100 Marks PART A 1. Define eigen values and eigen vectors of a matrix. Find the sum of the eigen values of A= x3 y3 du du 2. State Euler’s theorem on homogeneous functions. If tan u , prove that x y sin 2u x y dx dy 1 x 3. Evaluate xy x y dxdy 0 x 4. Solve D 2 1 y cos(2 x 1) 5. State the first shifting property in Laplace transforms. Also find L e 3t t 3 . [5 x 3 marks = 15 marks] PART B 6. Find the rank of the matrix A = 2 2 2 2 u u u 1 u 7. If u f ( x, y) where x r cos and y r sin , prove that . x y r r2 44 x 8. Evaluate the integral dxdy by changing the order of integration. 0x x2 y2 d2y 9. Using the method of variation of parameters, solve 4y tan 2 x . dx 2 cos at cos bt 10. Find the Laplace transform of . t [5 x 5 marks = 25 marks] Turn over
  • 2. 2 PART C Module I 11. (a) Find values of a and b for which the equations x + ay + z = 3, x + 2y + 2z = b, x + 5y + 3z = 9 are consistent. (7 marks) (b) Show that the vectors (2, −2, 1), (1, 4, −1) and (4, 6, −3) are linearly independent. (5 marks) Or 12. Reduce the quadratic from 2 x1x2 2 x1x3 2 x2 x3 to a canonical form by orthogonal transformation. and specify the matrix of transformation. Also find the rank, index, signature and nature of the quadratic form. (12 marks) Module II (u , v, w) 13. (a) If u x 3y 2 z3, v 4x 2 yz , w 2z 2 xy , evaluate at (1, −1, 0). (5 marks) ( x, y , z ) (b) Expand x 2 y 3 y z in powers of (x – 1) and (y + 2) using Taylor’s theorem. (7 marks) Or x y u u 1 14. (a) If u sin 1 , prove that x y tan u (7 marks) x y x y 2 (b) In a plane triangle, find the maximum value of cosA cosB cosC. (5 marks) Module III 15. (a) Find, by triple integration, the volume of the sphere x 2 y2 z2 a2 (5 marks) 1 1 x2 1 dzdydx (b) Evaluate (7 marks) 0 0 x2 y 2 x2 y2 z2 Or
  • 3. 3 16. (a) Find the area between the circle x 2 y2 a 2 and the line x + y = a lying in the first quadrant, by double integration. (5 marks) (b) By transforming into cylindrical coordinates, evaluate the integral x2 y2 z 2 dx dy dz taken over the region of space defined by x 2 y 2 1 and 0 z 1 . (7 marks) Module IV 17. (a) Solve D 2 4 D 3 y sin 3x cos 2 x (7 marks) d2y y 1 (b) Solve x 2 x (5 marks) dx 2 x x2 Or 2 18. (a) Solve D 2 y 8 e2 x sin 2 x x 2 (7 marks) 2 2d y dy (b) Solve 1 x 1 x y 2sin log 1 x (5 marks) dx 2 dx Module V 2s 2 6s 5 19. (a) Find the inverse Laplace transform of (5 marks) s3 6s 2 11s 6 s (b) Using convolution theorem, find the inverse Laplace transform of (7 marks) 2 s2 a2 Or 20. Solve the following differential equation by the method of Laplace transforms y 3y 2y 4t e3t , when y (0) 1 and y (0) 1 (12 marks) [5 x 12 marks = 60 marks]