Question

If the vectors $a$ and $b$ are linearly independent satisfying $(\sqrt{3}\tan \theta +1)\overrightarrow{a}+(\sqrt{3}\sec \theta -2)\overrightarrow{b}=0$, then the most general values of $\theta$ are

• A. $n\pi -\frac{\pi }{6},n\in Z$
• B. $2n\pi \pm \frac{11\pi }{6},n\in Z$
• C. $n\pi \pm \frac{\pi }{6},n\in Z$
• D. $2n\pi +\frac{11\pi }{6},n\in Z$

Answer 1

The correct option is D $2n\pi +\frac{11\pi }{6},n\in Z$

Let $x\overrightarrow{a}+y\overrightarrow{b}=0$. and if $\overrightarrow{a}$ and $\overrightarrow{b}$ are linearly independent, then we have $x=y=0.$

So $\sqrt{3}\tan \theta +1=0$ and $\sqrt{3}\sec \theta -2=0$
$\Rightarrow \theta =\frac{11\pi }{6}$
$\Rightarrow \theta =2n\pi +\frac{11\pi }{6},n\in Z$