Question

Number of roots of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ which lie in the interval $[-\pi ,\pi ]$ is

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

4

$co{s}^{2}x+\frac{\sqrt{3}+1}{2}sinx-\frac{\sqrt{3}}{4}-1=0$
$\Rightarrow 1-si{n}^{2}x+\frac{\sqrt{3}+1}{2}sinx-\frac{\sqrt{3}}{4}-1=0$
$\Rightarrow si{n}^{2}x-\frac{\sqrt{3}+1}{2}sinx+\frac{\sqrt{3}}{4}=0$
$\Rightarrow (sinx-\frac{\sqrt{3}}{2})(sinx-\frac{1}{2})=0$
$\Rightarrow sinx=\frac{\sqrt{3}}{2},\frac{1}{2}$
Therefore, solutions in the given interval are,
$x=\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6}$
Hence, option 'B' is correct.