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

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

• $\mathrm{ylog}\left({\mathrm{ab}}^2\right)$

• y²

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

Let f(x) = e^x, g(x) = ${\sin }^{-1}\left(\mathrm{x}\right)$ and h(x) = f[g(x)], then $\frac{\mathrm{h}\text{'}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{x}\right)}$ is equal to

• ${\mathrm{e}}^{{\sin }^{-1}\left(\mathrm{x}\right)}$

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

• ${\sin }^{-1}\left(\mathrm{x}\right)$

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

The maximum slope of the curve y = - x³ + 3x² + 2x - 27 is

• 5

• - 5

• $\frac{1}{5}$

• None of these

If $\mathrm{f}\left(\mathrm{x}\right) = \left|\log \left|\mathrm{x}\right|\right|$, then

• f(x) is continuous and differentiable for all x in its domain

• f(x) is continuous for all x in its domain but not differentiable at x = ± 1

• f(x) is neither continuous nor differentiable at x = ± 1

• None of the above

The derivative of ${\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\mathrm{x}}\right)$ with respect to ${\tan }^{-1}\left(\frac{2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}{1 - 2{\mathrm{x}}^2}\right)$ at x = 0 is

• $\frac{1}{8}$

• $\frac{1}{4}$

• $\frac{1}{2}$

• 1

Let f(x + y) = f(x) f(y) and f(x) = 1 + sin(2x) g(x) where g(x) is continuous. Then, f'(x) equals

• f(x) g(0)

• 2f(x) g(0)

• 2g(0)

• None of the above

For the curve y = xe, the point

• x = - 1 is a point of minimum

• x = 0 is a point of minimum

• x = - 1 is a point of maximum

• x = 0 is a point of maximum

78.

If ${\sin }^{-1}\left(\mathrm{x}\right) - {\cos }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{6}$, then x is

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• $\frac{- 1}{2}$

• None of these

If $\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{a}\right)}\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(\sin \left(\mathrm{x} - \mathrm{\alpha }\right)\right) + {\mathrm{c}}_1$, then the values of (A, B) is

• $\left(\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(\cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$

• $\left(- \sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(- \cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$

The value of ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\mathrm{x}}^3 + \mathrm{xcos}\left(\mathrm{x}\right) + {\tan }^5\left(\mathrm{x}\right) + 1\right)d\mathrm{x}$ is equal to

• 0

• 2

• $\mathrm{\pi }$

• None of these