Question

Find the number of solutions of the equation $(1-2cos\theta {)}^2+(tan\theta +\sqrt{3}{)}^2=0$ in interval $[0,2\pi ]$

• A. $1$
• B. $2$
• C. $0$
• D. None of these

Answer 1

Answer: (A)

$1$

If $a^2+b^2=0$

then $a=0$ and $b=0$.

Thus in equation ${(1-2\cos \theta )}^2+(\tan \theta +\sqrt{3})=0$,

${(1-2\cos \theta )}^2=0$ and

$(\tan \theta +\sqrt{3})=0$

For ${(1-2\cos \theta )}^2=0$,

$\theta =\frac{\pi }{3}\&\frac{5\pi }{3}$

For $(\tan \theta +\sqrt{3})=0$,

$\theta =\frac{2\pi }{3}\&\frac{5\pi }{3}$

$\therefore \theta =\frac{5\pi }{3}or{360}^o$

Hence the number of solutions to this equation is 1 and answer is A