Mathematics I (Calculus) 2065
Attempt all questions.
Group A
1. Verify Rolle’s theorem for the function 
on the interval [-3, 3].
2. Obtain the area between two curves y = sec2x and y = sin x from x = 0 to x = π/4.
3. Test the convergence of p – series 
for p > 1.
4. Find the eccentricity of the hyperbola 9x2 – 16y2 = 144.
5. Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).
6. Find the area enclosed by the curve r2 = 4cos2θ.
7. Obtain the values of 
and 
at the point (4, -5) if f(x,y) = x2+ 3xy + y -1.
8. Using partial derivatives 
, find if x2 + cos y – y2= 0.
9. Find the partial differential equation of the function (x – a)2 + (y – b)2 + z2= c2.
10. Solve the partial differential equation x2p + q = z2 .
Group B
11. State and prove the mean value theorem for a differential function.
12. Find the length of the Asteroid x = cos3t, y = sin3t for 0 ≤ t ≥ 2π.
13. Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.
14. What is meant by direction derivative in the plain? Obtain the derivative of the function f(x,y) = x2+ xy 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: x2 + y2 = 4 in the plane z = 0 and above by the paraboid z = 4 – x2 – y2.
16. Graph the function f(x) = -x3 + 12x + 5 for -3 ≤x ≤ 3.
Group C
17. Define Taylor’s polynomial of order n. Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = ex at x = 0.
18. Obtain the centroid and the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x2.
19. Find the maximum and the minimum values of f(x, y) = 2xy – 2y2– 5x2 + 4x – 4. Also find the saddle point if it exists.
OR
Evaluate the integral 
20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.
OR
Find the particular integral of the equation (D2 – D1)z =2y-x2 where 