Mathematics I (Calculus) 2068

Attempt all questions.

Group A (10×2=20)

1. Define one-to-one and onto functions with suitable examples.

2. Show by integral test that the series  converges if p>1.

3. Test the convergence of the series 

4. Find the focus and the directrix of the parabola y² = 10x.

5. Find the point where the line X = 8/3 + 2t, y = -2t, z = 1 + t intersects the plane 3x + 2y + 6z = 6.

6. Find a spherical coordinate equation for the sphere x² + y² + (z-1)² = 1.

7. Find the area of the region R bounded by y = x and y = x² in the first quadrant by using double integrals.

8. Define Jacobian determinant for X = g(u, v, w) ,y = h(u, v, w), z = k(u, v, w).

9. Find the extreme values of f(x,y) = x²+ y².

10.Define partial differential equations of the second order with suitable examples.

Group B (5×4=20)

11. State Rolle’s Theorem for a differential function. Support with examples that the hypothesis of theorem are essential to hold the theorem.

12. Test if the following series converges 

13. Obtain the polar equations 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 (5×8=40)

16. Find the area of the region enclosed by the parabola y = 2 – x² and the line y = -x.

OR

Evaluate the integrals

17. Define a curvature of a space curve. Find the curvature for the helix r(t) = (a cost)i + (a sint)j + btk(a,b ≥ 0, a² + b² ≠ 0).

18.Find the volume of the region D enclosed by the surfaces z = x²+ 3y² and z = 8 – x² – y².

19. Find the maximum and minimum values of the function f(x,y) = 3x + 4y on the circle x² +y² = 1.

OR

State the conditions of second derivative test for local extreme values. Find the local extreme values of the function f(x,y) = x² + xy + y² + 3x – 3y + 4.

20. Define one-dimensional wave equation and one-dimensional heat equations with initial conditions. Derive solution of any of them.