$\mathrm{If} \mathrm{x} + {\log }_{10}\left(1 + 2^{\mathrm{x}}\right) = {\mathrm{xlog}}_{10}\left(5\right) + {\log }_{10}\left(6\right) \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 2, - 3

• 2 only

• 1

• 3

A function is defined as follows : $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{- \frac{\mathrm{x}}{\sqrt{{\mathrm{x}}^2}}, \mathrm{x}\ne 0 \\ 0, \mathrm{x} = 0\right\} \end{gathered}$$

Which one of the following is correct in respect of the above function ?

• f(x) is continuous at x = 0 but not differentiable at x = 0

• f(x) is continuous as well as differentiable at x = 0

• f(x) is discontinuous at x = 0

• None of thee above

Consider the following :

1. x + x² is continuous at x = 0
2.  $\mathrm{x} +\hspace{0.17em}\cos \left(\frac{1}{\mathrm{x}}\right)$ is discontinuous at x= 0
3. ${\mathrm{x}}^2 + \cos \left(\frac{1}{\mathrm{x}}\right)$ is continuous at x = 0

Which of the above are correct ?

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3 

Consider the following statements :

1. $\frac{\mathrm{dy}}{\mathrm{dx}}$ at a point on the curve gives slope of the tangent at that point.

2. If a(t) denotes acceleration of a particle, then $\int \mathrm{a}\left(\mathrm{t}\right)\mathrm{dt} + \mathrm{c}$ gives velocity of the particle.

3. If s(t) gives displacement of a particle at time t,

then $\frac{\mathrm{ds}}{\mathrm{dt}}$ gives its acceleration at that instant.

Which of the above statements is/are correct ?

• 1 and 2 only

• 2 only

• 1 only

• 1, 2 and 3

$\mathrm{If} \mathrm{y} = {\sec }^{-1}\left(\frac{\mathrm{x} + 1}{\mathrm{x} - 1}\right) + {\sin }^{-1}\left(\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• 0

• 1

• $\frac{\mathrm{x} - 1}{\mathrm{x} \hspace{0.17em}+ 1}$

• $\frac{x + 1}{x - 1}$

If f(x) = $\frac{\mathrm{x}}{2}$ - 1, then on the interval [0, $\mathrm{\pi }$] which one of the following is correct ?

• tan[f(x)], where [.] is the greatest integer function, and $\frac{1}{\mathrm{f}\left(\mathrm{x}\right)}$ are both continuous

• tan[f(x)], where [.] is the greatest integer function, and f ^{- 1}(x) are both continuous

• tan[f(x)], where [.] is the greatest integer function, and $\frac{1}{\mathrm{f}\left(\mathrm{x}\right)}$ are both discontinuous

• tan[f(x)], where [.] is the greatest integer function, is discontinuous but $\frac{1}{\mathrm{f}\left(\mathrm{x}\right)}$ is continuous

The set of all points, where the function $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{1 - {\mathrm{e}}^{ - {\mathrm{x}}^2}} \mathrm{is} \mathrm{differentiable}, \mathrm{is} :$

• $\left(0, \infty \right)$

• $\left( - \infty , \infty \right)$

• $\left( - \infty , 0\right) \cup \left(0, \infty \right)$

• $\left( - 1, \infty \right)$

Match List-I with List-II and select the correct answer using the code given below the lists :

List-I	List-II
(Function)	(Maximum value)
a) sin(x) + cos(x)	1. $\sqrt{10}$
b) 3sin(x) + 4cos(x)	2.  $\sqrt{2}$
c) 2sin(x) + cos(x)	3. 5
d) sin(x) + 3cos(x)	4. $\sqrt{5}$

19.

If $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}\left(\sqrt{\mathrm{x}} - \sqrt{\mathrm{x} + 1}\right), \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} :$

• continuous but not differentiable at x = 0

• differentiable at x = 0

• not continuous at x = 0

• None of the above

Let g be the greatest integer function. Then the function f (x) = (g(x))²  - g(x²) is discontinuous at :

• all integers

• All integers except 0 and 1

• all integers except 0

• all integers except 1