If A is a non-singular matrix of order 3, then adj (adj A) is equal to

• A

• A^{- 1}

• $\frac{1}{\left|\mathrm{A}\right|}\mathrm{A}$

• $\left|\mathrm{A}\right|\mathrm{A}$

If $\left[\begin{array}{c}\mathrm{x} - \mathrm{y} - \mathrm{z}\\ - \mathrm{y} +\hspace{0.17em}\mathrm{z}\\ \mathrm{z}\end{array}\right] = \left[\begin{array}{c}0\\ 5\\ 3\end{array}\right]$, then the value of x, y and z are respectively

• 5, 2, 2

• 1, - 2, 3

• 0, - 3, 3

• 11, 8, 3

Which one of the following is true always for any two non-singular matrices A and B of same order ?

• AB = BA

• (AB)^t = A^tB^t

• (A + B)(A - B) = A² - B²

• (AB)^{- 1} = B^{- 1}A^{- 1}

If ${\tan }^{-1}\left(\mathrm{x} + 2\right) + {\tan }^{-1}\left(\mathrm{x} - 2\right) - {\tan }^{-1}\left(\frac{1}{2}\right) = 0$, then one of the values of x is equal to

• - 1

• 5

• $\frac{1}{2}$

• 1

25.

The value of ${\sin }^{-1}\left(\frac{4}{5}\right) + 2{\tan }^{-1}\left(\frac{1}{3}\right)$ is equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

The value of ${\sin }^{-1}\left(\cos \left(4095°\right)\right)$ is equal to

• $- \frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $- \frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

If $\left|\overrightarrow{\mathrm{a}}\right| = 5, \left|\overrightarrow{\mathrm{b}}\right| = 6 \mathrm{and} \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} = - 25, \mathrm{then} \left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right| \mathrm{is} \mathrm{equal} \mathrm{to}$

• 25

• $6\sqrt{11}$

• 11$\sqrt{5}$

• $5\sqrt{11}$

If $\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}} \mathrm{and} \overrightarrow{\mathrm{r}}$ are perpendicular to $\overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}},$ $\overrightarrow{\mathrm{r}} + \overrightarrow{\mathrm{p}},$ $\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}}$ respectively and if $\left|\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}}\right| = 6,$ $\left|\overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}}\right|= 4\sqrt{3}$ and $\left|\overrightarrow{\mathrm{r}} + \overrightarrow{\mathrm{p}}\right| = 4,$  then $\left|\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}}\right|$ is

• 5$\sqrt{2}$

• 10

• 15

• 5

The vectors of magnitude a, 2a, 3a meet at a point and their directions are along the diagonals of three adjacent faces of a cube. Then, the magnitude of their resultant is

• 5a

• 6a

• 10a

• 9a

Which one of the following vectors is of magnitude 6 and perpendicular to both $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ $\mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ ?

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$

• $2\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $3\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

• $2\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$