3 out of 6 vertices of a regular hexagon are chosen at a time at random. The probability that the triangle formed with these three vertices is an equilateral triangle, is

• $\frac{1}{10}$

• $\frac{1}{20}$

$$\begin{gathered} \mathrm{If} \left|{\mathrm{z}}_1\right| = 1, \left|{\mathrm{z}}_2\right| = 2, \left|{\mathrm{z}}_3\right| = 3 \mathrm{and} \left|9{\mathrm{z}}_1{\mathrm{z}}_2 + 4{\mathrm{z}}_1{\mathrm{z}}_3 + {\mathrm{z}}_2{\mathrm{z}}_3\right| = 12, \mathrm{then} \\ \mathrm{the} \mathrm{value} \mathrm{of} \left|{\mathrm{z}}_1 +{\mathrm{z}}_2 + {\mathrm{z}}_3\right| \mathrm{is} \end{gathered}$$

• 3

• 4

• 8

• 2

The cartesian equation of the plane whose vector equation is $\mathrm{\gamma } = \left(1 + \mathrm{\lambda } - \mathrm{\mu }\right)\hat{\mathrm{i}} + \left(2 - \mathrm{\lambda }\right)\hat{\mathrm{j}} + \left(3 - 2\mathrm{\lambda } + 2\mathrm{\mu }\right)\hat{\mathrm{k}}, \mathrm{where} \mathrm{\lambda }, \mathrm{\mu } \mathrm{are} \mathrm{scalars}, \mathrm{is}$ 

• 2x + y = 5

• 2x - y = 5

• 2x - z = 5

• 2x + z = 5

$\mathrm{For} \mathrm{three} \mathrm{vectors} \mathrm{p}, \mathrm{q} \mathrm{and} \mathrm{r}, \mathrm{if} \mathrm{r} = 3\mathrm{p} +\hspace{0.17em}4\mathrm{q} \mathrm{and} 2\mathrm{r} = \mathrm{p} - 3\mathrm{q}, \mathrm{then}$

• $\left|\mathrm{r}\right| < 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{the} \mathrm{same} \mathrm{direction}$

• $\left|\mathrm{r}\right| > 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{opposite} \mathrm{directions}$

• $\left|\mathrm{r}\right| < 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{opposite} \mathrm{directions}$

• $\left|\mathrm{r}\right| > 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{same} \mathrm{directions}$

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

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

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

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

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

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

• 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

In a city, 10 accidents take place in a span of 50 days. Assuming that the number of accidents follow the Poisson distribution, the probability that three or more accidents occur in a day, is

• $\sum _{\mathrm{k} = 3}^{\infty } \frac{{\mathrm{e}}^{ - \mathrm{\lambda }}{\mathrm{\lambda }}^{\mathrm{k}}}{\mathrm{k}!}, \mathrm{\lambda } = 0.2$

• $\sum _{\mathrm{k} = 3}^{\infty } \frac{{\mathrm{e}}^{\mathrm{\lambda }}{\mathrm{\lambda }}^{\mathrm{k}}}{\mathrm{k}!}, \mathrm{\lambda } = 0.2$

• $1 - \sum _{\mathrm{k} = 0}^3 \frac{{\mathrm{e}}^{ - \mathrm{\lambda }}{\mathrm{\lambda }}^{\mathrm{k}}}{\mathrm{k}!}, \mathrm{\lambda } = 0.2$

• $\sum _{\mathrm{k} = 0}^3 \frac{{\mathrm{e}}^{ - \mathrm{\lambda }}{\mathrm{\lambda }}^{\mathrm{k}}}{\mathrm{k}!}, \mathrm{\lambda } = 0.2$

70.

If the plane 56x + 4y + 9z = 2016 meets the coordinate axes in A, B, C, then the centroid of the $∆$ABC is

• $\left(12, 168, \frac{224}{3}\right)$

• (12, 168, 224)

• (12, 168, 112)

• $\left(12, -168, \frac{224}{3}\right)$