11.

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\hspace{0.17em}+ \mathrm{y}\right) = \mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right) \mathrm{and} \mathrm{f}\left(\mathrm{x}\right) = 1 + \mathrm{xg}\left(\mathrm{x}\right)\mathrm{\varphi }\left(\mathrm{x}\right), \mathrm{where} \\ \underset{\mathrm{x}\to 0}{\lim }\mathrm{g}\left(\mathrm{x}\right) = \mathrm{a} \mathrm{and} \underset{\mathrm{x}\to 0}{\lim }\mathrm{\varphi }\left(\mathrm{x}\right) = \mathrm{b}. \mathrm{Then} \mathrm{what} \mathrm{is} \mathrm{f}\text{'}\left(\mathrm{x}\right) = ? \end{gathered}$$

• 1 + abf(x)

• 1 +ab

• ab

• abf(x)

$\mathrm{What} \mathrm{is} \underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{\mathrm{x}} - 1 - \mathrm{x}}{{\mathrm{x}}^2} = ?$

• 0

• $\frac{1}{2}$

• 1

• 2

$\mathrm{What} \mathrm{is} {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{d\mathrm{\theta }}{1 + \cos \left(\mathrm{\theta }\right)} = ?$

• $\frac{1}{2}$

• 1

• $\sqrt{3}$

• None of the above

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sqrt{9 - {\mathrm{x}}^2}, \mathrm{then} \mathrm{what} \mathrm{is} \underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(1\right)}{\mathrm{x} - 1} = ?$

• $- \frac{1}{4\sqrt{2}}$

• $\frac{1}{8}$

• $- \frac{1}{2\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

Consider the following statements : 

1. If $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ and $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{g}\left(\mathrm{x}\right)$ both exist, then $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\left\{\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\right\} \mathrm{exists}$
2. If $\mathrm{If} \underset{\mathrm{x}\to \mathrm{a}}{\lim }\left\{\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\right\}, \mathrm{then} \mathrm{both} \underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{f}\left(\mathrm{x}\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{g}\left(\mathrm{x}\right)$ must exist.

Which of the above statements is/are correct?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

Consider the following statements:

1. Derivative of f(x) may not exist at some point.
2. Derivative of f(a) may exist finitely at some point
3. Derivative of f(x) may be infinite (geometrically) at some point.

Which of the above statements are correct ?

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

If $\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}} = \mathrm{l} \mathrm{and} \underset{\mathrm{x}\to \infty }{\lim }\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}} = \mathrm{m}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• l = 1, m = 1

• $\mathrm{l} = \frac{2}{\mathrm{\pi }}, \mathrm{m} = \infty$

• $\mathrm{l} = \frac{2}{\mathrm{\pi }}, \mathrm{m} = 0$

• l = 1, m = $\infty$

The left-hand derivative of f(X) = [x] $\sin \left(\mathrm{\pi x}\right)$ at x = k

where k is an integer and [x] is the greatest integer function, is

• ${\left( - 1\right)}^{\mathrm{k}}\left(\mathrm{k} - 1\right)\mathrm{\pi }$

• ${\left( - 1\right)}^{\mathrm{k} - 1}\left(\mathrm{k} - 1\right)\mathrm{\pi }$

• ${\left( - 1\right)}^{\mathrm{k}}\mathrm{k\pi }$

• ${\left( - 1\right)}^{\mathrm{k} - 1}\mathrm{k}\mathrm{\pi }$

Consider the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{a}}^{\left[\mathrm{x}\right] + \mathrm{x}} - 1}{\left[\mathrm{x}\right] + \mathrm{x}} \\ \mathrm{where} \left[ . \right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function} \\ \mathrm{What} \mathrm{is} \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right) = ? \end{gathered}$$

• 1

• ln(a)

• 1 - a ^{- 1}

• Limit does not exist

Consider the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{a}}^{\left[\mathrm{x}\right] + \mathrm{x}} - 1}{\left[\mathrm{x}\right] + \mathrm{x}} \\ \mathrm{where} \left[ . \right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function} \\ \mathrm{What} \mathrm{is} \underset{\mathrm{x}\to 0^{-}}{\lim }\mathrm{f}\left(\mathrm{x}\right) = ? \end{gathered}$$

• 0

• ln(a)

• $1 - {\mathrm{a}}^{ - 1}$

• Limit does not exist