MCQ
If ${\sin ^{ - 1}}\frac{1}{2} = {\tan ^{ - 1}}x,$ then $ x =$
- A$\sqrt 3 $
- ✓$\frac{1}{{\sqrt 3 }}$
- C$\frac{1}{{\sqrt 2 }}$
- DNone of these
$ \Rightarrow \,\,{\tan ^{ - 1}}x = \frac{\pi }{6}$
$ \Rightarrow \,\,x = \tan {30^o} = \frac{1}{{\sqrt 3 }}$.
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$L _1: \overrightarrow{ r }=\lambda \hat{ i }, \lambda \in R ,$
$L _2: \overrightarrow{ r }=\hat{ k }+\mu \hat{ j }, \mu \in R \text { and }$
$L _3: \overrightarrow{ r }=\hat{ i }+\hat{ j }+ vk , v \in R$
are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?
$(1)$ $\hat{k}+\hat{j}$ $(2)$ $\hat{ k }$ $(3)$ $\hat{ k }+\frac{1}{2} \hat{ j }$ $(4)$ $\hat{k}-\frac{1}{2} \hat{j}$
$2 x+y+z=5$
$x-y+z=3$
$x+y+a z=b$
has no solution, then :