PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?

Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?

Differentiate  \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?

Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?

Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?

Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?

 Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?

Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?

Differentiate 

\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?

Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] with respect to x.

If  \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that  \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?

If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that  \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?

Differentiate the following with respect to x: 

\[\cos^{- 1} \left( \sin x \right)\]

If  \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\],  show that \[\frac{dy}{dx}\] is independent of x. ? 

If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ? 

If  \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?

If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?

If  \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?

If the derivative of tan⁻¹ (a + bx) takes the value 1 at x = 0, prove that 1 + a² = b ?