21.

If the angle between the curves y = 2^x and y = 3^x is $\mathrm{\alpha }$, then the value of tan($\mathrm{\alpha }$) is equal to

• $\frac{\log \left({\displaystyle \frac{3}{2}}\right)}{1 + \log \left(2\right)\log \left(3\right)}$

• $\frac{6}{7}$

• $\frac{1}{7}$

• $\frac{\log \left({\displaystyle 6}\right)}{1 + \log \left(2\right)\log \left(3\right)}$

The function f(x) = 3x³ - 36x + 99 is increasing for

• $- \infty < \mathrm{x} < 2$

• $- 2 < \mathrm{x} < \infty$

• - 2 < x < 2

• x < - 2 or x > 2

Let f(x) = x³ - x + p ($0 \le \mathrm{x} \le 2$) where p is a constant. The value c of mean value theorem is

• $\frac{\sqrt{3}}{2}$

• $\frac{\sqrt{6}}{3}$

• $\frac{\sqrt{3}}{3}$

• $\frac{2\sqrt{3}}{3}$

The minimum value of the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{1}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}$ in the interval $\left[0, \frac{\mathrm{\pi }}{2}\right]$ is

• $\frac{\sqrt{2}}{2}$

• $- \frac{\sqrt{2}}{2}$

• $\frac{2}{\sqrt{3} + 1}$

• $-\frac{2}{\sqrt{3} + 1}$

If n(A) = 5 and n(B) = 7, then the number of relations on A x B is

• 2³⁵

• 2⁴⁹

• 2²⁵

• $2^{35 \times 35}$

Let $\mathrm{\varphi }\left(\mathrm{x}\right) = \frac{\mathrm{b}\left(\mathrm{x} - \mathrm{a}\right)}{\mathrm{b} - \mathrm{a}} + \frac{\mathrm{a}\left(\mathrm{x} - \mathrm{b}\right)}{\mathrm{a} - \mathrm{b}}$, where x $\in$ R and a and b are fixed real numbers with $\mathrm{a} \ne \mathrm{b}$. Then, $\mathrm{\varphi }\left(\mathrm{a} + \mathrm{b}\right)$ is equal to

• $\mathrm{\varphi }\left(\mathrm{ab}\right)$

• $\mathrm{\varphi }\left(- \mathrm{ab}\right)$

• $\mathrm{\varphi }\left(\mathrm{a}\right) + \mathrm{\varphi }\left(\mathrm{b}\right)$

• $\mathrm{\varphi }\left(\mathrm{a} - \mathrm{b}\right)$

The range of the function f(x) = $\frac{{\mathrm{x}}^2 + 8}{{\mathrm{x}}^2 + 4}, \mathrm{x} \in \mathrm{R}$ is

• $\left[- 1, \frac{3}{2}\right]$

• (1, 2]

• (1, 2)

• [1, 2]

If n(A) = 1000, n(B) = 500 and if n(A $\cap$ B) $\ge$1 and  $\mathrm{n}\left(\mathrm{A} \cup \mathrm{B}\right) = \mathrm{p}$, then

• $500 \le \mathrm{p} \le 1000$

• $1001 \le \mathrm{p} \le 1498$

• $1000 \le \mathrm{p} \le 1498$

• $1000 \le \mathrm{p} \le 1499$

The domain of the function $\mathrm{f}\left(\mathrm{x}\right) = {\sin }^{-1}\left(\frac{\mathrm{x} + 5}{2}\right)$ is

• [- 1, 1]

• [2, 3]

• [3, 7]

• [- 7, - 3]

If f(x) = x + 1 and g(x) = 2x, then f{g(x)} is equal to

• 2(x + 1)

• 2x(x + 1)

• x

• 2x + 1