Mathematics : Vector Algebra

$\mathrm{If} \mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}, \mathrm{b} = \mathrm{m}\hat{\mathrm{i}} + \mathrm{n}\hat{\mathrm{j}} + 12\hat{\mathrm{k}} \mathrm{and} \mathrm{a} \times \mathrm{b} = 0, \mathrm{then} \left(\mathrm{m}, \mathrm{n}\right) = ?$

• $\left(\frac{- 24}{5}, \frac{- 36}{5}\right)$

• $\left(\frac{ - 24}{5}, \frac{36}{5}\right)$

• $\left(\frac{24}{5}, \frac{- 36}{5}\right)$

• $\left(\frac{24}{5}, \frac{36}{5}\right)$

$\mathrm{If} \left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4 \mathrm{and} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} 120°, \mathrm{then} \left|4\mathrm{a} +\hspace{0.17em}3\mathrm{b}\right| = ?$

• 25

• 7

• 13

• 12

$$\begin{gathered} \mathrm{If} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{non}-\mathrm{zero} \mathrm{vectors} \mathrm{such} \mathrm{that} \left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c} = \frac{1}{3}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a}, \mathrm{c} ┴ \mathrm{a} \mathrm{and} \mathrm{\theta } \mathrm{is} \mathrm{the} \\ \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{vectors} \mathrm{b} \mathrm{and} \mathrm{c}, \mathrm{then} \sin \left(\mathrm{\theta }\right) = ? \end{gathered}$$

• $\frac{2\sqrt{2}}{3}$

• $\frac{1}{3}$

• $\frac{\sqrt{2}}{3}$

• $\frac{2}{3}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }\left(\mathrm{\alpha } \times \mathrm{\beta }\right) + \mathrm{b}\left(\mathrm{\beta } \times \mathrm{\gamma }\right) + \mathrm{c}\left(\mathrm{\gamma } \times \mathrm{\alpha }\right) = 0 \mathrm{and} \mathrm{atleast} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{scalars} \mathrm{a},\mathrm{b}, \mathrm{c} \mathrm{is} \mathrm{non}-\mathrm{zero}, \\ \mathrm{then} \mathrm{the} \mathrm{vectors} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \end{gathered}$$

• parallel

• non coplanar

• coplanar

• mutually perpendicular

If a non-zero vector a is parallel to the line of intersection of the plane determined by the vectors $\hat{\mathrm{j}} - \hat{\mathrm{k}}, 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ the plane determined by the vectors $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, \widehat{\mathrm{i} }- 3\hat{\mathrm{j}}$ then the angle between the vectors a and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$  is

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

• ${\cos }^{-1}\left(\pm \frac{2}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\sqrt{3}\right)$

• ${\cos }^{-1}\left( \pm \frac{1}{\sqrt{3}}\right)$

If three numbers are drawn at random successively without replacement from a set S = {1, 2, ... 10}, then the probability that the minimum of the chosen numbers is 3 or their maximum is 7

• $\frac{11}{40}$

• $\frac{5}{40}$

• $\frac{3}{40}$

• $\frac{1}{40}$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{points} \mathrm{having} \mathrm{the} \mathrm{position} \mathrm{vectors} \\ 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, 2\mathrm{i} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, - \hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \\ 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{are} \mathrm{coplanar}, \mathrm{then} \mathrm{\lambda } = ? \end{gathered}$$

• $- \frac{146}{17}$

• 8

•  - 8

• $\frac{146}{17}$

If a, b and c are non-zero vectors such that a and b are not perpendicular to each other, then the vector r which is perpendicular to a and satisfying r x b = c x b  is

• $\frac{\left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c}}{\mathrm{c} \mathrm{a}}$

• $\frac{\mathrm{b} \times \left(\mathrm{a} \times \mathrm{c}\right)}{\mathrm{b} \mathrm{c}}$

• $\frac{\left(\mathrm{b} \times \mathrm{c}\right) \times \mathrm{a}}{\mathrm{a} \mathrm{b}}$

• $\frac{\left(\mathrm{c} \times \mathrm{b}\right) \times \mathrm{a}}{\mathrm{ac}}$

$$\begin{gathered} \mathrm{The} \mathrm{triad} (\mathrm{x}, \mathrm{y}, \mathrm{z}) \mathrm{of} \mathrm{real} \mathrm{number} \mathrm{such} \mathrm{that} \\ \left(3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) = \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)\mathrm{x} + \left(\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)\mathrm{y} + \left( - 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)\mathrm{z} \mathrm{is} \end{gathered}$$

• (- 2, 5, 3)

• (2, - 5, 3)

• (2, 5, 3)

• (2, 5, - 3)

If the volume of the tetrahedron formed by the coterminous edges a, b and c is 4, then the volume of the parallelopiped formed by the coterminous edges a x b, b x c and c x a is

• 576

• 48

• 16

• 144