21.

The proposition $~ \left(\mathrm{p} \iff \mathrm{q}\right)$ is equivalent to

• $\left(\mathrm{p} \vee ~ \mathrm{q}\right) \wedge \left(\mathrm{q} \wedge ~ \mathrm{p}\right)$

• $\left(\mathrm{p} \wedge ~ \mathrm{q}\right) \wedge \left(\mathrm{q} \wedge ~ \mathrm{p}\right)$

• $\left(\mathrm{p} \wedge ~ \mathrm{q}\right) \wedge \left(\mathrm{q} \wedge ~ \mathrm{p}\right)$

• None of the above

If truth values of P be F and q be T. Then, truth value of $~ \left(~ \mathrm{p} \vee \mathrm{q}\right)$ is

• T

• F

• Either T or F

• Neither T or F

If the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, then the rank of the matrix $\left[\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& 1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& 1\\ {\mathrm{x}}_3& {\mathrm{y}}_3& 1\end{array}\right]$ will always be less than

• 3

• 2

• 1

• None of these

The value of the determinant

$\left|\begin{array}{ccc}1& \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)& \cos \left(\mathrm{\alpha }\right)\\ \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)& 1& \cos \left(\mathrm{\beta }\right)\\ \cos \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\beta }\right)& 1\end{array}\right|$ is

• ${\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2$

• ${\mathrm{\alpha }}^2 - {\mathrm{\beta }}^2$

• 1

• 0

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\left(1 + \left|\sin \left(\mathrm{x}\right)\right|\right)}^{\raisebox{1ex}{$\mathrm{a}$}\!\left/ \!\raisebox{-1ex}{$\left|\sin \left(\mathrm{x}\right)\right|$}\right.}, - \frac{\mathrm{\pi }}{6} < \mathrm{x} < 0 \\ \mathrm{b}, \mathrm{x} = 0 \\ {\mathrm{e}}^{\raisebox{1ex}{$\tan \left(2\mathrm{x}\right)$}\!\left/ \!\raisebox{-1ex}{$\tan \left(3\mathrm{x}\right)$}\right.}, 0 < \mathrm{x} < - \frac{\mathrm{\pi }}{6}\right\} \end{gathered}$$

then the value of a and b, if f is continuous at x = 0, are respectively

• $\frac{2}{3}, \frac{3}{2}$

• $\frac{2}{3}, {\mathrm{e}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $\frac{3}{2}, {\mathrm{e}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• None of these

If $\mathrm{f} : \mathrm{R} \to \mathrm{R}$ be such that f(1) = 3 and f'(1) = 6. Then $\underset{\mathrm{x}\to 0}{\lim }{\left\{\frac{\mathrm{f}\left(1 + \mathrm{x}\right)}{\mathrm{f}\left(1\right)}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$ equals to

• 1

• e^1/2

• e²

• e³

The domain of the function f(x) = $\frac{1}{{\log }_{10}\left(1 - \mathrm{x}\right)} + \sqrt{\mathrm{x} +\hspace{0.17em}2}$ is

• $\left[- 3, - 2.5 \left[\cap \right] - 2.5, - 2\right]$

• $\left[- 2, 0 \left[\cup \right] 0, 1\right]$

• [0, 1]

• None of the above

The relation R defined on set A = $\left\{\mathrm{x} : \left|\mathrm{x}\right| < 3, \mathrm{x} \in \mathrm{I}\right\}$ by $\mathrm{R} = \left\{\left(\mathrm{x}, \mathrm{y}\right) : \mathrm{y} = \left|\mathrm{x}\right|\right\}$ is

• {(- 2, 2), (- 1, 1), (0, 0), (1, 1), (2, 2)}

• {(- 2, - 2), (- 2, 2), (-1, 1), (0, 0), (1, - 2), (1, 2), (2, - 1), (2, - 2)}

• {(0, 0), (1, 1), (2, 2)}

• None of the above

If a, b and c are in AP, then determinnant

$\left|\begin{array}{ccc}\mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + 2\mathrm{a}\\ \mathrm{x} + 3& \mathrm{x} +\hspace{0.17em}4& \mathrm{x} +\hspace{0.17em}2\mathrm{b}\\ \mathrm{x} + 4& \mathrm{x} +\hspace{0.17em}5& \mathrm{x} + 2\mathrm{c}\end{array}\right|$ is

• 0

• 1

• x

• 2x

The value of $2{\tan }^{-1}\left(\csc \left({\tan }^{-1}\left(\mathrm{x}\right)\right) - \tan \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right)$ is

• ${\tan }^{-1}\left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$

• ${\csc }^{-1}\left(\mathrm{x}\right)$