The number of solutions in [0, 2pi] of the eq | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given equation: $16^{\sin^2 x} + 16^{\cos^2 x} = 10$ Since $\cos^2 x = 1 - \sin^2 x$,we have: $16^{\sin^2 x} + 16^{1 - \sin^2 x} = 10$ $16^{\sin^2

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(D) Given equation: $16^{\sin^2 x} + 16^{\cos^2 x} = 10$ Since $\cos^2 x = 1 - \sin^2 x$,we have: $16^{\sin^2 x} + 16^{1 - \sin^2 x} = 10$ $16^{\sin^2 x} + \frac{16}{16^{\sin^2 x}} = 10$ Let $t = 16^{\sin^2 x}$. Then $t + \frac{16}{t} = 10$,which implies $t^2 - 10t + 16 = 0$. Solving for $t$: $(t - 8)(t - 2) = 0$,so $t = 8$ or $t = 2$. Case $1$: $16^{\sin^2 x} = 2$ $\Rightarrow 2^{4\sin^2 x} = 2^1$ $\Rightarrow 4\sin^2 x = 1$ $\Rightarrow \sin^2 x = \frac{1}{4}$ $\Rightarrow \sin x = \pm \frac{1}{2}$. In $[0, 2\pi]$,$\sin x = \pm \frac{1}{2}$ gives $x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$ ($4$ solutions). Case $2$: $16^{\sin^2 x} = 8$ $\Rightarrow 2^{4\sin^2 x} = 2^3$ $\Rightarrow 4\sin^2 x = 3$ $\Rightarrow \sin^2 x = \frac{3}{4}$ $\Rightarrow \sin x = \pm \frac{\sqrt{3}}{2}$. In $[0, 2\pi]$,$\sin x = \pm \frac{\sqrt{3}}{2}$ gives $x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$ ($4$ solutions). Total number of solutions = $4 + 4 = 8$.

The number of solutions in $[0, 2\pi]$ of the equation $16^{\sin^2 x} + 16^{\cos^2 x} = 10$ is

Similar Questions

The number of solutions of the trigonometric equation $1+\cos x \cdot \cos 5 x=\sin ^2 x$ in $[0, 2 \pi]$ is

If $\cos \theta + \cos 2\theta + \cos 3\theta = 0$,then the general value of $\theta$ is

The most general value of $\theta$ which satisfies both the equations $\tan \theta = -1$ and $\cos \theta = \frac{1}{\sqrt{2}}$ is

The number of values of $x$ in the interval $[0, 3\pi]$ satisfying the equation $2 \sin^2 x + 5 \sin x - 3 = 0$ is

If $5\cos 2\theta + 2\cos^2\frac{\theta}{2} + 1 = 0$ and $-\pi < \theta < \pi$,then $\theta = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module