PUC Science 2nd PUC Class 12 - Karnataka Board PUC Important Questions for Mathematics

Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?

Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x² + 2) − log 3 on [−1, 1] ?

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].

Evaluate : `int_0^3dx/(9+x^2)`

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Evaluate:

`int((x+3)e^x)/((x+5)^3)dx`

If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.

Evaluate :

`int_e^(e^2) dx/(xlogx)`

Evaluate the following integral:

Evaluate:

`∫ (1)/(sin^2 x cos^2 x) dx`

Evaluate the following integral:

Evaluate each of the following integral:

`int "dx"/(("x" - 8)("x" + 7))`=

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.