Tribhuvan University Institute of Science and Technology 2065

Transcription

1 1CSc. MTH Tribhuvan University Institute of Science and Technology 2065 Bachelor Level/First Year/ First Semester/ Science Full Marks: 80 Computer Science and Information Technology (MTH. 104) Pass Marks: 32 (Calculus and Analytic Geometry) Time: 3hours Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all questions: Group A (10 x 2 = 20) 1. Verify Rolle s theorem for the function ( ) on the interval [ ] 2. Obtain the area between two curves and from to. 3. Test the convergence of p-series for. 4. Find the eccentricity of the hyperbola. 5. Find a vector perpendicular to the plane of P (1, -1, 0), Q (2, 1, -1) and R (-1, 1, 2). 6. Find the area enclosed by the curve. 7. Obtain the values of and at the point (4,-5) if ( ). 8. Using partial derivatives, find if. 9. Find the partial differential equation of the function ( ) ( ). 10. Solve the partial differential equation. Group B (5 x 4 = 20) 11. State and prove the mean value theorem for a differentiable function. IOST, TU 1

2 1CSc. MTH Find the length of the Asteroid for. 13. Define a curvature of a curve. Prove that the curvature of a circle of radius is Ya. 14. What is meant by direction derivative in the plane? Obtain the derivative of the function ( ) at P(1, 2) in the direction of the unit vector ( ) ( ). 15. Find the center of mass of a solid of constant density δ, bounded below by the disk in the plane z = 0 and above by the paraboid. 16. Graph the function ( ) for. Group C (5 x 8 = 40) 17. Define Taylor s polynomial of order n. Obtain Taylor s polynomial and Taylor s series generated by the function ( ) at. 18. Obtain the centroid of the region in the first quadrant that is bounded above by the line and below by the parabola. 19. Find the maximum and minimum values of ( ). Also find the saddle point if it ermist. Evaluate the integral. 20. What do you mean by d Alembert s solution of the one dimensional wave equation? Derive it. Find the particular integral of the equation ( ) Where. IOST, TU 2

3 1CSc. MTH Tribhuvan University Institute of Science and Technology 2066 Bachelor Level/First Year/ First Semester/ Science Full Marks: 80 Computer Science and Information Technology (MTH. 104) Pass Marks: 32 (Calculus and Analytic Geometry) Time: 3hours Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all questions: 1. Find the length of the curve from to. Group A (10 x 2 = 20) 2. Find the critical points of the function ( ) ( ). 3. Does the following series converge? 4. Find the polar equation of the circle ( ). 5. Find the area of the parallelogram where vertices are A(0, 0), B(7, 3), C(9, 8) and D(2, 5). 6. Evaluate the integral ( ) 7. Evaluate the limit ( ) ( ) 8. Find ( ) if and. 9. Solve the partial differential equation. 10. Find the general integral of the linear partial differential equation ( ). IOST, TU 3

4 1CSc. MTH State and prove Rolle s Theorem. Group B (5 x 4 = 20) 12. Find the length of the cardioid. 13. Define the unit tangent vector of a differential curve. Find the unit tangent vector of the curve ( ) ( ) ( ). 14. What do you mean by critical point of the function f(x, y) in a region? Find local extreme values of the function ( ). 15. Find a particular integral of the equation 16. Graph the function. Group C (5 x 8 = 40) 17. What do you mean by Taylor s polynomial of order n? Obtain Taylor s polynomial and Taylor s series generated by the function ( ). 18. Find the volume of the region D enclosed by the surfaces and. 19. Obtain the maximum and minimum values of the function ( ) on the triangular plate in the first quadrant bounded by the lines. Evaluate the integral 20. Show the solution of the wave equation is ( ) [ ( ) ( )] ( ) and deduce the result if the velocity is zero. Find a particular integral of the equation ( ) ( ) where A, l, m are constants. IOST, TU 4

5 1CSc. MTH Tribhuvan University Institute of Science and Technology 2067 Bachelor Level/First Year/ First Semester/ Science Full Marks: 80 Computer Science and Information Technology (MTH. 104) Pass Marks: 32 (Calculus and Analytic Geometry) Time: 3hours Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all questions: Group A (10 x 2 = 20) 1. Define a relation and a function from the set into another set. Give suitable examples. 2. Show that the series converses by using integral test. 3. Investigate the convergence of the series. 4. Find the foci, vertices, center of the ellipse. 5. Find the equation of the plane through (-3, 0, 7) perpendicular to. 6. Define cylindrical coordinates (r,, z). Find an equation for the circular cylinder in cylindrical coordinates. 7. Calculate ( ) for ( ) 8. Define Jacobian determinant for ( ) ( ) ( ). 9. What do you mean by local extreme points of f(x, y)? Illustrate the concept by graphs. 10. Define partial differential equations of the first index with suitable examples. Group B (5 x 4 = 20) 11. State the mean value theorem for a differentiable function and verify if for the function ( ) on the interval [-1, 1]. IOST, TU 5

6 1CSc. MTH Find the Taylor series and Taylor polynomials generated by the function ( ) at x = Find the length of the cardioid 14. Define the partial derivative of f(x, y) at a point (x 0, y 0 ) with respect to all variables. Find the derivative of ( ) ( ) at the point (2, 0) in the direction of. 15. Find a general solution of the differential equation ( ) Group C (5 x 8 = 40) 16. Find the area of the region in the first quadrant that is bounded above and below by the x-axis and the line. Investigate the convergence of the integrals a. b. ( ) 17. Calculate the curvature and torsion for the helix ( ) ( ) ( ) 18. Find the volume of the region D enclosed by the surfaces and 19. Find the absolute maximum and minimum values of ( ) on the triangular plate in the first quadrant bounded by the lines x = 0, y = 0 and x + y = 9. Find the points on the curve defined. nearest to the origin. How are the Lagrange multipliers 20. Define D Alembert s solution satisfying the initial conditions of the one dimensional wave equation. IOST, TU 6

7 1CSc. MTH Tribhuvan University Institute of Science and Technology 2068 Bachelor Level/First Year/ First Semester/ Science Full Marks: 80 Computer Science and Information Technology (MTH. 104) Pass Marks: 32 (Calculus and Analytic Geometry) Time: 3hours Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all questions: 1. Define one-to-one and onto function with suitable examples. 2. Show by integral test that the series Group A (10 x 2 = 20) converges if p > Test the convergence of the series ( ) 4. Find the focus and directrix of the parabola. 5. Find the point where the line intersects the plane. 6. Find the spherical coordinate equation for the sphere ( ). 7. Find the area of the region R bounded by and in the first quadrant by using double integrals. 8. Define Jacobian determinant for ( ) ( ) ( ) 9. Find the extreme values of ( ). 10. Define partial differential equations of the second order with suitable examples. IOST, TU 7

8 1CSc. MTH Group B (5 x 4 = 20) 11. State Rolle s theorem for a differential function. Support with examples that the hypothesis of the theorem are essential to hold the theorem. 12. Test if the following series converges a. b. 13. Obtain the polar equation for circles through the origin centered on the x- and y-axis and radius a. 14. Show that the function ( ) { ( ) ( ) ( ) is continuous at every point except the origin. 15. Find the solution of the equation Group C (5 x 8 = 40) 16. Find the area of the region enclosed by the parabola and the line. Evaluate the integrals a) ( ) b) 17. Define a curvature of a space curve. Find the curvature of the helix ( ) ( ) ( ) ( ) 18. Find the volume of the region D enclosed by the surfaces and. 19. Find the maximum and minimum values of the function ( ) on the circle. State the conditions of second derivative test for local extreme values. Find the local extreme values of the function ( ). 20. Define one dimensional wave equation and one dimensional heat equation with initial conditions. Derive solution of any of them. IOST, TU 8

9 CSc. MTH Tribhuvan University Institute of Science and Technology 2069 Bachelor Level/First Year/ First Semester/ Science Full Marks: 80 Computer Science and Information Technology (MTH. 104) Pass Marks: 32 (Calculus and Analytic Geometry) Time: 3 hours Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all questions: Downloaded from: Group A (10 x 2 = 20) 1. Verify the mean value theorem for the function ( ) ( ) in the interval of [0, 1]. 2. Find the length of the curve for 3. Test the convergence of the series by comparison test. 4. Obtain the semi-major axis, semi-minor axis, foci, vertices 5. Find the angel between the vectors 2i + j + k and -4i + 3j +k. 6. Obtain the area of the region R bounded by y = x, and y = x 2 in the first quadrant. 7. Show that the function ( ) { ( ) ( ) ( ) is continuous at every point in the plane except the origin. 8. Using partial derivatives, find if 2xy + tan y 4y 2 = Verify that the partial differential equation is satisfied by ( ) ( ) 10. Find the general solution of the equation ( ) IOST, TU 9

10 CSc. MTH State and prove mean value theorem for definite integral. Group B (5 x 4 = 20) 12. Find the area of the region that lies in the plane enclosed by the cardioid r = 2(1 + cosθ). 13. What do you mean by principal unit normal vector? Find unit tangent vector and principal unit normal vector for the circular motion ( ) ( ) ( ) 14. Define partial derivative of a function f(x, y) with respect to x at the point (x 0, y 0 ). State Euler s theorem, verify it for the function ( ) ( ) 15. Find a particular integral of the equation Group C (5 x 8 = 40) 16. Graph the function 17. What is mean by Maclaurin series? Obtain the Maclaurin series for the function ( ). 18. Evaluate the double integral by applying the transformation and integrating over an appropriate region in the uv plane. 19. Define maximum and minimum of a function at a point. Find the local maximum and local minimum of the function ( ) Find the volume of the region D enclosed by the surface z = x 2 + 2y 2 and z = 8 x 2 y Find the solution of the equation Find the particular integral of the equation (D 2 D ' ) z = 2y x 2 Where. IOST, TU 10