If $\mathrm{y} = \sqrt{\sin \left(\mathrm{x}\right) + \sqrt{\sin \left(\mathrm{x}\right) + \sqrt{\sin \left(\mathrm{x}\right) + ... \infty }}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to :

• $\frac{\cos \left(\mathrm{x}\right)}{2\mathrm{y} - 1}$

• $\frac{- \cos \left(\mathrm{x}\right)}{2\mathrm{y} - 1}$

• $\frac{\sin \left(\mathrm{x}\right)}{1 - 2\mathrm{y}}$

• $\frac{- \sin \left(\mathrm{x}\right)}{1 - 2\mathrm{y}}$

Given f(0) = 0 and f(x) = $\frac{1}{\left(1 - {\mathrm{e}}^{- {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}}\right)}$ for $\mathrm{x} \ne 0$. Then only one of the following statements on f (x) is true. That is f(x), is :

• continuous at x = 0

• not continuous at x = 0

• both continuous and differentiable at x = 0

• not defined at x = 0

The value of f at x = 0 so that function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2^{\mathrm{x}} - 2^{- \mathrm{x}}}{\mathrm{x}}, \mathrm{x} \ne 0$.  is continuous at x = 0, is :

• 0

• log(2)

• 4

• log(4)

If y = a^x . b^{2x - 1}, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is :

• ${\mathrm{y}}^2 . \log \left({\mathrm{ab}}^2\right)$

• $\mathrm{y} . \log \left({\mathrm{ab}}^2\right)$

• y²

• $\mathrm{y} . {\left(\log \left({\mathrm{ab}}^2\right)\right)}^2$

785.

The value of $\frac{\mathrm{d}}{\mathrm{dx}}\left[{\tan }^{-1}\left(\frac{\sqrt{\mathrm{x}}\left(3 - \mathrm{x}\right)}{1 - 3\mathrm{x}}\right)\right]$ is

• $\frac{1}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{3}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{2}{\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{3}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

The derivative of y = (1 - x)(2 - x) ... (n - x) at x = 1 is equal to :

• 0

• (- 1)(n - 1)!

• n! - 1

• (- 1)^{n - 1}(n - 1)!

Let f(x + y) = f(x)f(y) and f(x) = 1 + sin(3x) g(x), where g(x) is continuous, then f'(x) is : 

• f(x)g(0)

• 3g(0)

• f(x)$\cos \left(3\mathrm{x}\right)$

• 3f(x)g(0)

Let f be continuous on [1, 5] and differentiable in (1, 5). If f (1) = - 3 and f'(x) $\ge$ 9 for all x $\in$ (1, 5), then

• $\mathrm{f}\left(5\right) \ge 33$

• $\mathrm{f}\left(5\right) \ge 36$

• $\mathrm{f}\left(5\right) \le 36$

• $\mathrm{f}\left(5\right) \ge 9$

Let f be twice differentiable function such that f"(x) = - f(x) and f'(x) = g(x), h(x) = {f(x)}² + {g(x)}². If h(5) = 11, then h(10) is equal to :

• 22

• 11

• 0

• 20

A differentiable function f(x) is defined for all x > 0 and satisfies f(x³) = 4x⁴ for all x > 0. The value of f'(8) is :

• $\frac{16}{3}$

• $\frac{32}{3}$

• $\frac{16\sqrt{2}}{3}$

• $\frac{32\sqrt{2}}{3}$