JEE Main 2021 (Online) 26th February Morning ShiftQuestion 1
If $\sqrt{3}\left(\cos^{2}x\right)=(\sqrt{3}-1)\cos x+1$, the number of solutions of the given equation when $x\in\left[0,\frac{\pi}{2}\right]$ is $\_\_\_\_\_\_$ .
Trigonometric Ratio & Identities
Answer$1$
Solution
Concept. Solve the quadratic in $c=\cos x$; only $c\ge0$ is allowed on $\left[0,\frac\pi2\right]$. Formulas used. (1) Quadratic formula; $(\sqrt3-1)^2+4\sqrt3=(\sqrt3+1)^2$Step 1 - Solve. $\sqrt3c^2-(\sqrt3-1)c-1=0$. By (1), $$c=\frac{(\sqrt3-1)\pm(\sqrt3+1)}{2\sqrt3}=1\ \text{or}\ -\frac{1}{\sqrt3}.$$ Step 2 - Count. On $\left[0,\frac\pi2\right]$, $\cos x\ge0$, so $\cos x=1$ and $x=0$: $1$ solution. Answer: 1