MCQ
The angle between the lines whose direction cosines are connected by the relations $l + m + n = 0$ and $2lm + 2nl - mn = 0$, is
- A$\frac{\pi }{3}$
- ✓$\frac{{2\pi }}{3}$
- C$\pi $
- DNone of these
When $2l + m = 0,$ then $\frac{l}{1} = \frac{m}{{ - 2}} = \frac{n}{1}$
When $l - m = 0,$ then $\frac{l}{1} = \frac{m}{1} = \frac{n}{{ - 2}}$
$\therefore $ Direction ratios are $1, -2, 1$ and $1, 1, -2.$
$\cos \theta = \frac{{\sum {a_1}{a_2}}}{{\sqrt {(\sum a_1^2)\,} .\sqrt {(\sum a_2^2)\,} }} = - \frac{1}{2}\,$
$ \Rightarrow \,\,\theta = {120^o} = \frac{{2\pi }}{3}.$
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| $X$ | 0 | 1 | 2 | 3 |
| $P(X)$ | $\frac{4}{k}$ | $\frac{6}{k}$ | $\frac{4}{k}$ | $\frac{2}{k}$ |