If f(x) = $$\begin{gathered} \left\{\frac{\sin \left(5\mathrm{x}\right)}{{\mathrm{x}}^2 + 2\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k} + \frac{1}{2}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k is

• 1

• - 2

• 2

• 1/2

$\mathrm{y} = {\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - \sqrt{1 - {\mathrm{x}}^2}}{\sqrt{1 + {\mathrm{x}}^2} + \sqrt{1 - {\mathrm{x}}^2}}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{{\mathrm{x}}^2}{\sqrt{1 - {\mathrm{x}}^4}}$

• $\frac{{\mathrm{x}}^2}{\sqrt{1 + {\mathrm{x}}^4}}$

• $\frac{\mathrm{x}}{\sqrt{1 + {\mathrm{x}}^4}}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^4}}$

Which one of the following is not true always?

• If f(x) is not continuous at x = a, then it is not differentiable at x = a

• If f(x) is continuous at x = a, then it is differentiable at x = a

• If f(x) and g(x) are differentiable at x = a, then f(x) + g(x) is also differentiable at x = a

• If a function f(x) is continuous at x = a, then $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ f(x) exists

If y = $1 + \frac{1}{\mathrm{x}} + \frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{x}}^3} + ... \infty \mathrm{with} \left|\mathrm{x}\right| > 1, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ :

• $\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}$

• x²y²

• $\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}$

• $\frac{- {\mathrm{y}}^2}{{\mathrm{x}}^2}$

275.

If f(x) and g(x) are two functions with g(x) = $\mathrm{x} - \frac{1}{\mathrm{x}}$and fog(x) = ${\mathrm{x}}^3 - \frac{1}{{\mathrm{x}}^3}$, then f' (x) is :

• 3x² + 3

• ${\mathrm{x}}^2 - \frac{1}{{\mathrm{x}}^2}$

• $1 +\hspace{0.17em}\frac{1}{{\mathrm{x}}^2}$

• $3{\mathrm{x}}^2 + \frac{3}{{\mathrm{x}}^4}$

If sin(x + y) + cos(x + y) = log(x + y), then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is :

• $\frac{- \mathrm{y}}{\mathrm{x}}$

• 0

• - 1

• 1

If f(x) is a function such that f''(x) + f(x) = 0 and g(x) = [f(x)]² + [f'(x)]² and (3) = 3 then g(8) is equal to :

• 5

• 0

• 3

• 8

If the function f(x) = $$\begin{gathered} \left\{\frac{1 - \cos \left(\mathrm{x}\right)}{{\mathrm{x}}^2}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k is

• 1

• 0

• $\frac{1}{2}$

• - 1

If ${\sec }^{-1}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{y}}\right) = \mathrm{a}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{\mathrm{y} - 1}{\mathrm{x} + 1}$

• $\frac{\mathrm{y} + 1}{\mathrm{x} - 1}$

• $\frac{\mathrm{x} - 1}{\mathrm{y} - 1}$

• $\frac{\mathrm{x} - 1}{\mathrm{y} + 1}$

If y = ${\cos }^2\left(\frac{3\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{3\mathrm{x}}{2}\right), \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• $- 3\sqrt{1 - {\mathrm{y}}^2}$

• 9y

• - 9y

• $3\sqrt{1 - {\mathrm{y}}^2}$