31.

For any vector $\overrightarrow{\mathrm{a}}$, ${\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right|}^2$ = ?

• ${\left|\overrightarrow{\mathrm{a}}\right|}^2$

• $2{\left|\overrightarrow{\mathrm{a}}\right|}^2$

• $3{\left|\overrightarrow{\mathrm{a}}\right|}^2$

• $4{\left|\overrightarrow{\mathrm{a}}\right|}^2$

If the vectors $\mathrm{a}\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \mathrm{b}\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and}$ $\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}}$$$\begin{gathered} \left(\mathrm{a}, \mathrm{b}, \mathrm{c} \ne 1\right) \mathrm{are} \mathrm{coplanar}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \\ \frac{1}{1 - \mathrm{a}} + \frac{1}{1 - \mathrm{b}} + \frac{1}{1 - \mathrm{c}} = ? \end{gathered}$$

• 0

• 1

• a + b + c

• abc

$\mathrm{Let}, \hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{be} \mathrm{two} \mathrm{unit} \mathrm{vectors} \mathrm{and} \mathrm{\theta } \mathrm{be} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{them}$.

What is $\cos \left(\frac{\mathrm{\theta }}{2}\right)$ = ?

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{2}$

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{2}$

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{4}$

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{4}$

$\mathrm{Let}, \hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{be} \mathrm{two} \mathrm{unit} \mathrm{vectors} \mathrm{and} \mathrm{\theta } \mathrm{be} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{them}$.

What is $\sin \left(\frac{\mathrm{\theta }}{2}\right)$ = ?

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{2}$

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{2}$

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{4}$

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{4}$

What is a vector of unit length orthogonal to both the vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 3\hat{\mathrm{k}} ?$

• $\frac{ - 4\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{26}}$

• $\frac{ - 4\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{26}}$

• $\frac{ - 3\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{14}}$

• $\frac{ - 3\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{14}}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are the position vectors of the vertices of an equilateral triangle whose ortho centre is at the origin, then which one of the following is correct ?

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \mathrm{unit} \mathrm{vector}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{be} \mathrm{three} \mathrm{vectors} \mathrm{such} \mathrm{that} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = 0, \\ \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 10, \left|\overrightarrow{\mathrm{b}}\right| = 6 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 14. \\ \mathrm{What} \mathrm{is} \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} = ? \end{gathered}$$

•  - 332

•  - 166

• 0

• 166

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} + 4\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{d}}, \mathrm{where} \overrightarrow{\mathrm{c}} \mathrm{is} \mathrm{parallel} \mathrm{to} \\ \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{d}} \mathrm{is} \mathrm{perpendicular} \mathrm{to} \overrightarrow{\mathrm{a}}. What is \overrightarrow{c} = ? \end{gathered}$$

• $\frac{3\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)}{2}$

• $\frac{2\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)}{3}$

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

• $\frac{\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)}{3}$

$$\begin{gathered} \text{Let }\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} + 4\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{d}}, \mathrm{where} \overrightarrow{\mathrm{c}} \mathrm{is} \mathrm{parallel} \mathrm{to} \\ \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{d}} \mathrm{is} \mathrm{perpendicular} \mathrm{to} \overrightarrow{\mathrm{a}}. \mathrm{If} \overrightarrow{\mathrm{d}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}, \mathrm{then} \mathrm{which} \mathrm{of} \mathrm{the} \\ \mathrm{following} \mathrm{equations} \mathrm{is}/\mathrm{are} \mathrm{correct} ? \end{gathered}$$

1. y - x = 4
2. 2z - 3 = 0

Select the correct answer using the code given below :

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{be} \mathrm{three} \mathrm{vectors} \mathrm{such} \mathrm{that} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}, \mathrm{and} \left[\overrightarrow{\mathrm{a}}\right] = 10,\left[\overrightarrow{\mathrm{b}}\right] = 6 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 14. \\ \mathrm{What} \mathrm{is} \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} = ? \end{gathered}$$

• $30°$

• $45°$

• $60°$

• $75°$