If the operation $\oplus$ is defined by a $\oplus$ b = a² + b² for all real numbers a and b, then (2 $\oplus$ 3) $\oplus$ 4 is equal to

• 120

• 185

• 175

• 129

If a, b, c are non-zero and different from 1, then the value of $\left|\begin{array}{ccc}{\log }_{\mathrm{a}}\left(1\right)& {\log }_{\mathrm{a}}\left(\mathrm{b}\right)& {\log }_{\mathrm{a}}\left(\mathrm{c}\right)\\ {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{b}}\right)& {\log }_{\mathrm{b}}\left(1\right)& {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{c}}\right)\\ {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{c}}\right)& {\log }_{\mathrm{a}}\left(\mathrm{c}\right)& {\log }_{\mathrm{c}}\left(1\right)\end{array}\right|$ is

• 0

• $1 + {\log }_{\mathrm{a}}\left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\hspace{0.17em}+ \mathrm{c}\right)$

• ${\log }_{\mathrm{a}}\left(\mathrm{ab} + \mathrm{bc} + \mathrm{ca}\right)$

• 1

The value of ${\sin }^{-1}\left(\frac{2\sqrt{2}}{3}\right) + {\sin }^{-1}\left(\frac{1}{3}\right)$ is equal to

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

If ab < 1 and ${\cos }^{-1}\left(\frac{1 - {\mathrm{a}}^2}{1 + {\mathrm{a}}^2}\right) + {\cos }^{-1}\left(\frac{1 - {\mathrm{b}}^2}{1 + {\mathrm{b}}^2}\right) = 2{\tan }^{-1}\left(\mathrm{x}\right)$, then x is equal to

• $\frac{\mathrm{a}}{1 + \mathrm{ab}}$

• $\frac{\mathrm{a}}{1 - \mathrm{ab}}$

• $\frac{\mathrm{a} +\hspace{0.17em}\mathrm{b}}{1 + \mathrm{ab}}$

• $\frac{\mathrm{a} +\hspace{0.17em}\mathrm{b}}{1 - \mathrm{ab}}$

${\cos }^{-1}\left[\cos \left(\frac{7\mathrm{\pi }}{5}\right)\right]$ is equal to

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{2\mathrm{\pi }}{5}$

• $\frac{- 7\mathrm{\pi }}{5}$

• $\frac{7\mathrm{\pi }}{5}$

The value of ${\sec }^2\left({\tan }^{-1}\left(3\right)\right) + \csc \left({\cot }^{-1}\left(2\right)\right)$ is equal to

• 5

• 13

• 15

• 23

If $\mathrm{a} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \left|\mathrm{b}\right| = 5$ and the angle between a and b is $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$, then the area of the triangle formed by these two vectors as two sides is

• $\frac{15}{4}$

• $\frac{15}{2}$

• 15

• $\frac{15\sqrt{3}}{2}$

If a - b = 0 and a + b makes an angle of 60° with a, then

• $\left|\mathrm{a}\right| = 2\left|\mathrm{b}\right|$

• $2\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$

• $\left|\mathrm{a}\right| = \sqrt{3}\left|\mathrm{b}\right|$

• $\sqrt{3}\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$

19.

If $\hat{\mathrm{i}} + \hat{\mathrm{j}}, \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{k}}$ are the position vectors of the vertices of a $∆$ABC taken in order, then $\angle$A is equal to

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

Let a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If b is a vector such that a · b = ${\left|\mathrm{b}\right|}^2$ and  $\left|\mathrm{a} - \mathrm{b}\right|$ = $\sqrt{7}$, then $\left|\mathrm{b}\right|$ is equal to

• $\sqrt{7}$

• $\sqrt{3}$

• 7

• 3