If A(- 2, 1), B(2, 3) and C( - 2, - 4) are three points. Then the angle between BA and BC is

• ${\tan }^{-1}\left(\frac{2}{3}\right)$

• ${\tan }^{-1}\left(\frac{3}{2}\right)$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

In a $∆\mathrm{ABC}$, $\frac{\left(\mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} + \mathrm{a} - \mathrm{b}\right)\left(\mathrm{a} + \mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2}$ equals

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

• ${\sin }^2\left(\mathrm{B}\right)$

83.

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l² + m² - n² = 0 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The plane through the point (- 1, - 1, - 1) and containing the line of intersection of the planes $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ = 0 and $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ = 0 is

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

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 5\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

If $\mathrm{\theta }$ is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1, 2, - 1), (2, 0, 3), (3, - 1, 2) respectively, then $\sqrt{462} \cos \left(\mathrm{\theta }\right)$ is equal to

• 20

• 10

• 30

• 40

Let the pairs $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{d}}$ each determine a plane. Then the planes are parallel, if :

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}\right) \mathrm{and} \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}\right) = \overrightarrow{0}$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}\right) . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}\right) = 0$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}\right) = \overrightarrow{0}$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) . \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}\right) = 0$

The area of a parallelogram with $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ as diagonal is :

• $\sqrt{72}$

• $\sqrt{73}$

• $\sqrt{74}$

• $\sqrt{75}$

In a $∆\mathrm{ABC}$, if $\left(\sqrt{3} - 1\right)$a = 2b, A = 3B, then C is

• $60°$

• 120$°$

• 30°

• 45°

The x - axis, y - axis and a line passing through the point A (6, 0) from a $∆$ ABC. If $\angle$A = 30°, then the area of the $∆$, in sq unit is

• $6\sqrt{3}$

• $12\sqrt{3}$

• $4\sqrt{3}$

• 8$\sqrt{3}$

The mid point of the line joining the points (- 10, 8) and (- 6, 12)divides the line joining the points ( 4, - 2) and (- 2, 4) in the ratio

• 1 : 2 internally

• 1 : 2 externally

• 2 : 1 internally

• 2 : 1 externally