$\left(\overrightarrow{\mathrm{a}} + \overrightarrow{2\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) . \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) = ?$

• $- \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $3\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

$\mathrm{If} \overrightarrow{\mathrm{u}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{v}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}, \left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 2, \mathrm{then} \left|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}\right| = ?$

• $2\sqrt{16 - {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2}$

• $2\sqrt{16 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 - {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

443.

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{angle} \mathrm{\theta } \mathrm{between} \mathrm{the} \mathrm{vectors} \overrightarrow{\mathrm{a}} = 2{\mathrm{x}}^2\hat{\mathrm{i}} + 4\mathrm{x}\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 7\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\mathrm{x}\hat{\mathrm{k}} \\ \mathrm{is} \mathrm{such} \mathrm{that} 90° <\mathrm{\theta } < 180°, \mathrm{then} \mathrm{x} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval} \end{gathered}$$

• $\left(0, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, 1\right)$

• $\left(1, \frac{3}{2}\right)$

• $\left(\frac{1}{2}, \frac{3}{2}\right)$

Let OA, OB, QC be the co-terminal edges of a rectangular parallelopiped of volume V and let P be the vertex opposite to O. Then, $\left[\overrightarrow{\mathrm{AP}} \overrightarrow{\mathrm{BP}} \overrightarrow{\mathrm{CP}}\right]$ is equal to

• 2V

• 12V

• $3\sqrt{3}\mathrm{V}$

• 0

If the vectors ii - 2xj - 3yk and i + 3xj + 2yk are orthogonal to each other, then the locus of the point (x, y) is

• a circle

• an ellipse

• a parabola

• a straight line

The magnitude of the projection of the vector a = 4i - 3j + 2k on the line which makes equal angles with the coordinate axes is

• $\sqrt{2}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

$$\begin{gathered} \mathrm{For} \mathrm{any} \mathrm{vector} \mathrm{r} \\ \mathrm{i} \times \left(\mathrm{r} \times \mathrm{i}\right) + \mathrm{j} \times \left(\mathrm{r} \times \mathrm{j}\right) + \mathrm{k} \times \left(\mathrm{r} \times \mathrm{k}\right) = \end{gathered}$$

• 0

• 2r

• 3r

• 4r

If the vectors AB = - 3i + 4k and AC = 5i - 2j + 4k are the sides of a $∆$ABC, then the length of the median through A

• $\sqrt{14}$

• $\sqrt{18}$

• $\sqrt{25}$

• $\sqrt{29}$

$\mathrm{If} \left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2 \mathrm{and} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} 120°, \mathrm{then} {\left\{\left(\mathrm{a} +\hspace{0.17em}3\mathrm{b}\right) \times \left(3\mathrm{a} - \mathrm{b}\right)\right\}}^2 \mathrm{is} \mathrm{equal} \mathrm{to}$

• 425

• 375

• 325

• 300

$$\begin{gathered} \mathrm{Let} \mathrm{v} = 2\mathrm{i} + \mathrm{j} - \mathrm{k} \mathrm{and} \mathrm{w} = \mathrm{i} + 3\mathrm{k}. \mathrm{If} \mathrm{u} \mathrm{is} \mathrm{any} \mathrm{unit} \mathrm{vector}, \\ \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{scalar} \mathrm{triple} \mathrm{product} \left[\begin{array}{ccc}\mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right] \mathrm{is} \end{gathered}$$

• 1

• $\sqrt{10} + \sqrt{6}$

• $\sqrt{59}$

• $\sqrt{60}$