61.

If a,b, c are three vectors such that  and the angle between  is $\frac{\mathrm{\pi }}{2}$, then

• a² = b² + c²

• b² = c² + a²

• c² = a² + b²

• 2a² - b² = c²

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be the position vectors of the vertices A, B, C respectively of $∆$ABC. Thevector area of $∆$ABC is :

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}}\left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) + \overrightarrow{\mathrm{c}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}}\left(\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right) + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{a}}\right) + \overrightarrow{\mathrm{c}}\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)\right)$

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$, $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}}$ and $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \overrightarrow{\mathrm{c}} = \mathrm{\lambda }\overrightarrow{\mathrm{a}} + \mathrm{\mu }\overrightarrow{\mathrm{b}}$, then $\mathrm{\lambda } + \mathrm{\mu }$ is equal to :

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If $\mathrm{P}\left(\mathrm{A} \cup \mathrm{B}\right) = 0.8 \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = 0.3$, then $\mathrm{P}\left(\overline{\mathrm{A}}\right) + \mathrm{P}\left(\overline{\mathrm{B}}\right)$ is equal to :

• 0.3

• 0.5

• 0.8

• 0.9

A coin is tossed n times the probability of getting head at least once is greater than 0.8. Then, the least value of such n is :

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XOZ-plane divides the join of (2, 3, 1) and(6, 7, 1) in the ratio :

• 3 : 7

• 2 : 7

• - 3 : 7

• - 2 : 7

If the direction ratio of two lines are given by 3lm - 4ln + mn = 0 and l + 2m + 3n = 0, then the angle between the lines, is :

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

A plane makes intercepts 3 and 4 respectively on Z-axis and X-axis. If it is parallel to Y-axis, then its equation is

• 3x + 4z = 12

• 3z + 4x = 12

• 3y + 4z = 12

• 3z + 4y = 12

The equation of the plane passing through(1, 1, 1) and   (1, - 1, - 1) and perpendicular to 2x -y + z = 0 is :

• 2x +5y +z + 8 = 0

• x + y - z - 1 = 0

• 2x + 5y + z + 4 = 0

• x - y + z - 1 = 0

$\mathrm{A} \mathrm{minimum} \mathrm{value} \mathrm{of} {\int }_0^{\mathrm{x}}{\mathrm{te}}^{{\mathrm{t}}^2}\mathrm{dt} \mathrm{is}$

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