$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{b}} + 3\overrightarrow{\mathrm{c}} = 0 \mathrm{and} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} = \mathrm{\lambda }\left( \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right), \\ \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\lambda } ? \end{gathered}$$

• 2

• 3

• 4

• 6

$\mathrm{If} \mathrm{the} \mathrm{vectors} \overrightarrow{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{A}} \mathrm{are} \mathrm{parallel} \mathrm{to} \mathrm{each} \mathrm{other}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{k}\overrightarrow{\mathrm{k}} \mathrm{x} \overrightarrow{\mathrm{A}} \mathrm{equal} \mathrm{to} ?$

• ${\mathrm{k}}^2\overrightarrow{\mathrm{A}}$

• $\overrightarrow{0}$

• $- {\mathrm{k}}^2\overrightarrow{\mathrm{A}}$

• $\overrightarrow{\mathrm{A}}$

$$\begin{gathered} \mathrm{Which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} \mathrm{in} \mathrm{respect} \mathrm{of} \mathrm{the} \mathrm{function} \\ \mathrm{f} : \mathrm{ℝ} \to {\mathrm{ℝ}}^{+} \mathrm{defined} \mathrm{as} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{Ix} + 1\mathrm{I} ? \end{gathered}$$

• $\mathrm{f}\left({\mathrm{x}}^2\right) = {\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^2$

• $\mathrm{f}\left(\left|\mathrm{x}\right|\right) = \left|\mathrm{f}\left(\mathrm{x}\right)\right|$

• $\mathrm{f}\left(\mathrm{x} + \mathrm{y}\right) = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{y}\right)$

• None of the above

14.

$$\begin{gathered} \mathrm{Suppose} \mathrm{f} : \mathrm{ℝ} \to \mathrm{ℝ} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) =\frac{{\mathrm{x}}^2}{1 +{\mathrm{x}}^2} . \\ \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{the} \mathrm{function} ? \end{gathered}$$

• $\left(0, 1\right)$

• $\left[0, 1\right]$

• $\left(0, 1\right)$

• $\left(0, 1\right)$

$$\begin{gathered} \mathrm{Let} \left|\overrightarrow{\mathrm{a}}\right| \ne 0, \left|\overrightarrow{\mathrm{b}}\right| \ne 0 \\ \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) . \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) = {\left|\overrightarrow{\mathrm{a}}\right|}^2 + {\left|\overrightarrow{\mathrm{b}}\right|}^2 \\ \mathrm{holds} \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} \end{gathered}$$

• $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{perpendicular}$

• $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{parallel}$

• $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{inclined} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} 45°$

• $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{anti} \mathrm{parallel}$

$\mathrm{If} \overrightarrow{\mathrm{r}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}, \mathrm{then} \mathrm{what} \mathrm{is} \overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} +\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = ?$

• x

• x + y

•  - (x + y + z)

• (x + y + z)

$$\begin{gathered} \mathrm{A} \mathrm{unit} \mathrm{vector} \mathrm{perpendicular} \mathrm{to} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{vectors} \\ 2\hat{\mathrm{i}} -\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{is} \end{gathered}$$

• $\frac{1}{\sqrt{3}}\hat{\mathrm{i}} + \frac{1}{\sqrt{3}}\hat{\mathrm{j}} - \frac{1}{\sqrt{3}}\hat{\mathrm{k}}$

• $\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{2}\hat{\mathrm{j}} + \frac{1}{2}\hat{\mathrm{k}}$

• $\frac{1}{\sqrt{3}}\hat{\mathrm{i}} - \frac{1}{\sqrt{3}}\hat{\mathrm{j}} - \frac{1}{\sqrt{3}}\hat{\mathrm{k}}$

• $\frac{1}{\sqrt{3}}\hat{\mathrm{i}} + \frac{1}{\sqrt{3}}\hat{\mathrm{j}} +\frac{1}{\sqrt{3}}\hat{\mathrm{k}}$

$\mathrm{If} \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 4 \mathrm{and} \left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right| = 5, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| ?$

• 8

• 6

• $5\sqrt{2}$

• 5

$$\begin{gathered} \mathrm{Let} \mathrm{a}, \mathrm{b} \mathrm{and} \mathrm{c} \mathrm{be} \mathrm{three} \mathrm{mutually} \mathrm{perpendicular} \mathrm{vectors} \mathrm{each} \mathrm{of} \mathrm{unit} \mathrm{magnitude}. \mathrm{If} \overrightarrow{\mathrm{A}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}+ \overrightarrow{\mathrm{c}}, \\ \overrightarrow{\mathrm{B}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} +\overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{C}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} \end{gathered}$$

• $\left|\overrightarrow{\mathrm{A}}\right| > \left|\overrightarrow{\mathrm{B}}\right| > \left|\overrightarrow{\mathrm{C}}\right|$

• $\left|\overrightarrow{\mathrm{A}}\right| = \left|\overrightarrow{\mathrm{B}}\right| \ne \left|\overrightarrow{\mathrm{C}}\right|$

• $\left|\overrightarrow{\mathrm{A}}\right| = \left|\overrightarrow{\mathrm{B}}\right| = \left|\overrightarrow{\mathrm{C}}\right|$

• $\left|\overrightarrow{\mathrm{A}}\right| \ne \left|\overrightarrow{\mathrm{B}}\right| \ne \left|\overrightarrow{\mathrm{C}}\right|$

$\mathrm{What} \mathrm{is} \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) = ?$

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \mathrm{x} \overrightarrow{\mathrm{b}}$

• $2\left(\overrightarrow{\mathrm{a}} \mathrm{x} \overrightarrow{\mathrm{b}}\right)$

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