If $\mathrm{y} = {\log }_{\mathrm{a}}\left(\mathrm{x}\right) + {\log }_{\mathrm{x}}\left(\mathrm{a}\right) + {\log }_{\mathrm{x}}\left(\mathrm{x}\right) + {\log }_{\mathrm{a}}\left(\mathrm{a}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{\mathrm{x}} + \mathrm{xlog}\left(\mathrm{a}\right)$

• $\frac{\log \left(\mathrm{a}\right)}{\mathrm{x}} + \frac{\mathrm{x}}{\log \left(\mathrm{a}\right)}$

• $\frac{1}{\mathrm{xlog}\left(\mathrm{a}\right)} + \mathrm{xlog}\left(\mathrm{a}\right)$

• None of the above

If $\mathrm{y} = {\sin }^{-1}\left(3\mathrm{x} - 4{\mathrm{x}}^3\right) + {\cos }^{-1}\left(4{\mathrm{x}}^3 - 3\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{E}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 5

• 0

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

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

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x} - 1}{4} + \frac{{\left(\mathrm{x} - 1\right)}^3}{12} + \frac{{\left(\mathrm{x} - 1\right)}^5}{20} + \frac{{\left(\mathrm{x} - 1\right)}^7}{28} + ...$, where 0 < x < 2, then f'(x) is equal to

• $\frac{1}{4\mathrm{x}\left(2 - \mathrm{x}\right)}$

• $\frac{1}{4{\left(\mathrm{x} - 2\right)}^2}$

• $\frac{1}{2 - \mathrm{x}}$

• $\frac{1}{2 + \mathrm{x}}$

If f(x) = sin(x), the derivative of f(log(x)) w.r.t. x is

• cos(x)

• f'(log(x))

• cos(log(x))

• $\frac{\cos \left(\log \left(\mathrm{x}\right)\right)}{\mathrm{x}}$

If f(x + y) = 2 f(x) f(y), f'(5) = 1024log(2) and f(2) = 8,  then the value of f'(3) is

• 64(log(2))

• 128(log(2))

• 256

• 256log(2)

The number of discontinuities of the greatest integer function f (x) = [x], $\mathrm{x} \in \left(- \frac{7}{2}, 100\right)$ is equal to

• 104

• 100

• 102

• 103

If f(x) = $$\begin{gathered} \left\{\frac{3\sin \left(\mathrm{\pi x}\right)}{5\mathrm{x}}, \mathrm{x} \ne 0 \\ 2\mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k is equal to

• $\frac{3\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{2}$

If $\mathrm{y} = \sin \left[{\cos }^{-1}\left\{\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right)\right\}\right], \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x} = \frac{1}{2}$ is equal to

• 0

• - 1

• $\frac{2}{\sqrt{3}}$

• 1

If x² + y² = $\mathrm{t} - \frac{1}{\mathrm{t}}$ and x⁴ + y⁴ = ${\mathrm{t}}^2 + \frac{1}{{\mathrm{t}}^2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\frac{1}{{\mathrm{x}}^2{\mathrm{y}}^3}$

• $\frac{1}{{\mathrm{xy}}^3}$

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

• $\frac{1}{{\mathrm{x}}^3\mathrm{y}}$

If $\mathrm{y} = {\sec }^{-1}\left[\csc \left(\mathrm{x}\right)\right] + {\csc }^{-1}\left[\sec \left(\mathrm{x}\right)\right] + {\sin }^{-1}\left[\cos \left(\mathrm{x}\right)\right] + {\cos }^{-1}\left[\sin \left(\mathrm{x}\right)\right]$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 0

• 2

• - 2

• - 4