411.

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

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = - \hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{and} \overrightarrow{\mathrm{c}} = - 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \\ \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} 2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{Suppose} \overrightarrow{\mathrm{a}} = \mathrm{\lambda }\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \mathrm{\lambda }\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\mathrm{\lambda }\hat{\mathrm{k}}. \mathrm{If} \\ \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} \mathrm{greater} \mathrm{than} 90°, \mathrm{then} \mathrm{A} \mathrm{satisfies} \mathrm{the} \mathrm{inequality} \end{gathered}$$

• $- 7 < \mathrm{\lambda } < 1$

• $\mathrm{\lambda } > 1$

• $1 < \mathrm{\lambda } < 7$

• $- 5 < \mathrm{\lambda } < 1$

$$\begin{gathered} \mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{tetrahedron} \mathrm{having} \mathrm{the} \mathrm{edge} \\ \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} - \hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{as} \mathrm{coterminous}, \mathrm{is} \frac{2}{3}\mathrm{cubic} \mathrm{unit}. \mathrm{Then} \mathrm{\lambda } = ? \end{gathered}$$

• 1

• 2

• 3

• 4

The image of the point (3, 2, 1) in the plane 2x - y + 3z = 7 is

• (1, 2, 3)

• (2, 3, 1)

• (3, 2, 1)

• (2, 1, 3)

A point moves in the xy-plane such that the sum of its distance from two mutually perpendicular lines is always equal to 5 units. The area(in sq units) enclosed by the locus of the point,is

• $\frac{25}{4}$

• 25

• 50

• 100

If the pair of lines given by $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right){\cos }^2\left(\mathrm{\theta }\right) = {\left(\mathrm{xcos}\left(\mathrm{\theta }\right) + \mathrm{ysin}\left(\mathrm{\theta }\right)\right)}^2$ are perpendicular to each other, then $\mathrm{\theta }$ is equal to

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{3\mathrm{\pi }}{4}$

If the foot of the perpendicular from (0, 0, 0) to a plane is (1, 2, 3), then the equation of the plane is

• 2x + y + 3z = 14

• x + 2y + 3z = 14

• x + 2y + 3z + 14 = 0

• x + 2y - 3z = 14

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, {\mathrm{r}}_1{\mathrm{r}}_2 + {\mathrm{r}}_2{\mathrm{r}}_3 + {\mathrm{r}}_3{\mathrm{r}}_1 = ?$

• $\frac{{∆}^2}{{\mathrm{r}}^2}$

• $\frac{∆}{\mathrm{r}}$

• $\frac{2∆}{\mathrm{r}}$

• ${∆}^2$

$$\begin{gathered} \mathrm{If} \mathrm{in} \mathrm{a} ∆\mathrm{ABC}, \frac{1}{\mathrm{a} +\hspace{0.17em}\mathrm{c}} + \frac{1}{\mathrm{b} + \mathrm{c}} = \frac{3}{\mathrm{a} + \mathrm{b} + \mathrm{c}}, \mathrm{then} \\ \angle \mathrm{C} = ? \end{gathered}$$

• $30°$

• $45°$

• $60°$

• $90°$