If $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}\left(1 + \mathrm{acos}\left(\mathrm{x}\right)\right) - \mathrm{bsin}\left(\mathrm{x}\right)}{{\mathrm{x}}^3} = 1$, then

• $\mathrm{a} = - \frac{5}{2}, \mathrm{b} = - \frac{1}{2}$

• $\mathrm{a} = - \frac{3}{2}, \mathrm{b} = - \frac{1}{2}$

• $\mathrm{a} = - \frac{3}{2}, \mathrm{b} = - \frac{5}{2}$

• $\mathrm{a} = - \frac{5}{2}, \mathrm{b} = - \frac{3}{2}$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the rooots of ax² + bx + c = 0, then $\underset{\mathrm{x}\to \mathrm{\alpha }}{\lim }\frac{1 - \cos \left({\mathrm{ax}}^2 + \mathrm{bx} + \mathrm{c}\right)}{{\left(\mathrm{x} - \mathrm{\alpha }\right)}^2}$ is equal to

• $\frac{{\mathrm{a}}^2}{2}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

• $- \frac{{\mathrm{a}}^2}{2}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

• 0

• 1

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

• - 1

• 1

• 2

• - 2

If $\mathrm{x} = {\mathrm{acos}}^3\left(\mathrm{t}\right) \mathrm{and} \mathrm{y} = {\mathrm{asin}}^3\left(\mathrm{t}\right)$, then ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\mathrm{t} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ is equal to

• 1

• - 1

• 0

• $\infty$

The anti-derivative F of f defined by f(x) = 4x³ - 6x² + 2x + 5, F(0) = 5, is

• x⁴ - 2x³ + x² + 5x

• 12x² - 12x + 2

• 16x⁴ - 18x³ + 4x² + 5x

• x⁴ - 2x³ + x² + 5x + 5

$\underset{\mathrm{x}\to 0}{\lim }\mathrm{x}{\left(- 1\right)}^{\left[\frac{1}{\mathrm{x}}\right]}$

• 0

• 1

• - 1

• Does not exist

If f(x) = $$\begin{gathered} \left\{\frac{\sin \left[\mathrm{x}\right]}{\left[\mathrm{x}\right]} \mathrm{for} \left[\mathrm{x}\right] \ne 0 \\ 0 \mathrm{for} \left[\mathrm{x}\right] = 0\right\} \end{gathered}$$

where [x] denotes the greatest integer less than or eual to x, then $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ is equal to

• 1

• - 1

• 0

• Does not exist

78.

If $\underset{\mathrm{n}\to \infty }{\lim }\sum \frac{\log \left(\mathrm{n} + \mathrm{r}\right) - \log \left(\mathrm{n}\right)}{\mathrm{n}} = 2\left(\log \left(2\right) - \frac{1}{2}\right)$, then $\underset{\mathrm{n}\to \infty }{\lim }\frac{1}{{\mathrm{n}}^{\mathrm{\lambda }}}{\left[{\left(\mathrm{n} + 1\right)}^{\mathrm{\lambda }}{\left(\mathrm{n} + 2\right)}^{\mathrm{\lambda }} ... {\left(\mathrm{n} + \mathrm{n}\right)}^{\mathrm{\lambda }}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.}$ is equal to

• $\frac{4\mathrm{\lambda }}{\mathrm{e}}$

• ${\left(\frac{4}{\mathrm{e}}\right)}^{\mathrm{\lambda }}$

• ${\left(\frac{4}{\mathrm{e}}\right)}^{\frac{1}{\mathrm{\lambda }}}$

• ${\left(\frac{\mathrm{e}}{4}\right)}^{\mathrm{\lambda }}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(\sin \left(\mathrm{x}\right)\right) - \sin \left(\mathrm{x}\right)}{{\mathrm{ax}}^3 + {\mathrm{bx}}^5 + \mathrm{c}} = \frac{- 1}{12}$, then

• $\mathrm{a} = 2, \mathrm{b} \in \mathrm{R}, \mathrm{c} = 0$

• $\mathrm{a} = - 2, \mathrm{b} \in \mathrm{R}, \mathrm{c} = 0$

• $\mathrm{a} = 1, \mathrm{b} \in \mathrm{R}, \mathrm{c} = 0$

• $\mathrm{a} = - 1, \mathrm{b} \in \mathrm{R}, \mathrm{c} = 0$

The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left({\mathrm{x}}^4\right) - {\mathrm{x}}^4\cos \left({\mathrm{x}}^4\right) + {\mathrm{x}}^{20}}{{\mathrm{x}}^4\left({\mathrm{e}}^{2{\mathrm{x}}^4} - 1 - 2{\mathrm{x}}^4\right)}$ is equal to

• 0

• $- \frac{1}{6}$

• $\frac{1}{6}$

• Does not exist