Using Rolle's theorem, prove that there is no real a for which the equation        x² - 3x + a = 0 two different roots in [-1,1].

Find c of Lagrange's mean value theorem for the following function          x³ + x in [1,2].

Solve (1 + y²)dx = (tan^-1y - x)dy

Identify the nature of singular points of the differential equation                          
x²(x - 2)y" + (x - 1)y' + 2xy = 0 

For any two real numbers a and b         (a < b) prove that a²+ ab+ b² = 3c²,for some c ∈ (a,b) using Lagrange's mean value theorem.

Solve equation                                                       
y= 3x + log_e p

Using Maclaurin's series expand tan x up to the term containing x⁵

Find the volume of sphere                    x² + y² + z² =  a² using triple integrals.

Find the volume bounded by xy- plane, the cylinder x²+y²= 1 and the plane       
x+y+z = 3

Find the volume in the positive octant bounded by the plane x+2y+3z =4 and the coordinate planes.

A circular hole of radius b is made centrally through a sphere of radius a . Find the volume of remaining sphere.