A) \[\frac{\sqrt{23}}{6}\]
B) \[\frac{23}{36}\]
C) \[\frac{2}{3}\]
D) \[\frac{3}{2}\]
E) \[\frac{1}{6}\]
Correct Answer: A
Solution :
\[\because \] \[\left( \frac{1}{2},\frac{1}{3},n \right)\] are the direction cosines of a line, then\[{{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{3} \right)}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \] \[\frac{1}{4}+\frac{1}{9}+{{n}^{2}}=1\] \[\Rightarrow \] \[{{n}^{2}}=1-\frac{1}{4}-\frac{1}{9}=\frac{23}{36}\] \[\Rightarrow \] \[n=\frac{\sqrt{23}}{6}\]
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