21.

The value of f(0) so that $\frac{\left(- {\mathrm{e}}^{\mathrm{x}} + 2^{\mathrm{x}}\right)}{\mathrm{x}}$  may be continuous at x = 0 is

• $\log \left(\frac{1}{2}\right)$

• 0

• 4

• - 1 + $\log \left(2\right)$

Let [ ] denotes the greatest integer function and f(x) = [tan²(x)] Then,

• $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ does not exist

• f(x) is continuous at x = 0

• f(x) is not differentiable at x = 0

• f(x) = 1

A spherical balloon is expanding. If the radius is increasing at the rate of 2 cm/min, the rate at which the volume increases (in cubic centimetres per minute) when the radius is 5 cm, is

• $10\mathrm{\pi }$

• 100$\mathrm{\pi }$

• 200$\mathrm{\pi }$

• 50$\mathrm{\pi }$

Let y be the number of people in a village at time t. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increases in time. Then, the population of the village at any fixed time t is given by

• y = e^kt + c, for some constants $\mathrm{c} \le 0 \mathrm{and} \mathrm{k} \ge 0$

• y = ce^kt, for some constants $\mathrm{c} \ge 0 \mathrm{and} \mathrm{k} \le 0$

• y = e^ct + k,  for some constants  $\mathrm{c} \le 0 \mathrm{and} \mathrm{k} \ge 0$

• y = ke^ct , for some constants $\mathrm{c} \ge 0 \mathrm{and} \mathrm{k} \le 0$

Let S be a finite set containing n elements. Then the total number of commutative binary operation on S is

• ${\mathrm{n}}^{\left[\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right]}$

• ${\mathrm{n}}^{\left[\frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{2}\right]}$

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

• $2^{\left({\mathrm{n}}^2\right)}$

The system of equations

x + y + z = 0

2x + 3y + z = 0

and x + 2y = 0

has

• a unique solution; x = 0, y = 0, z = 0

• infinite solutions

• no solution

• finite number of non-zero solutions

${\left[\begin{array}{cc}0& \mathrm{a}\\ \mathrm{b}& 0\end{array}\right]}^4 = \mathrm{I}$, then

• a = 1 = 2b

• a = b

• a = b²

• ab = 1

If D = diag (d₁, d₂, ..., d_n), where d_i $\ne$ 0, for i = 1, 2, ... , n then D^-1 is equal to

• D^T

• D

• adj(D)

• $\mathrm{diag}\left({{\mathrm{d}}_1}^{-1}, {{\mathrm{d}}_2}^{-1}, ..., {{\mathrm{d}}_{\mathrm{n}}}^{-1}\right)$

If x, y, z are different from zero and $∆ = \left|\begin{array}{ccc}\mathrm{a}& \mathrm{b} - \mathrm{y}& \mathrm{c} - \mathrm{z}\\ \mathrm{a} - \mathrm{x}& \mathrm{b}& \mathrm{c} - \mathrm{z}\\ \mathrm{a} - \mathrm{x}& \mathrm{b} - \mathrm{y}& \mathrm{c}\end{array}\right|$ = 0, then the value of the expression $\frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}}$ is

• 0

• - 1

• 1

• 2

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be three unit vectors such that $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \frac{1}{2}\overrightarrow{\mathrm{b}}$, $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ being non-parallel. If ${\mathrm{\theta }}_1$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ and ${\mathrm{\theta }}_2$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{c}}$, then

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{6}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{6}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{2}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{2}$