Question

Number of solution of the equations$co{s}^{3}x-3cosxsi{n}^{2}x+cos3x$which lie in the interval (0,1) is

• A. 1
• B. 2
• C. more than three
• D. 3

Answer 1

The correct option is A 1

${\cos }^{3}x-3\cos x{\sin }^{3}x+\cos 3x=0$

$\Rightarrow {\cos }^{3}x-3\cos x(1-{\cos }^{2}x)+\cos 3x=0$

$\Rightarrow {\cos }^{3}x-3\cos x+3{\cos }^{3}x+\cos 3x=0$

$\Rightarrow \underset{}{4{\cos }^{3}x-3\cos x}+\cos 3x=0$

$\cos 3x$

$\Rightarrow 2\cos 3x=0$

$\Rightarrow \cos 3x=0$

$[\cos \theta =0;GeneralSolution:\theta =n\pi +\frac{\pi }{2}]$

So $3x=n\pi +\frac{\pi }{2}$ where $N=0,1,2$

$x=\frac{n\pi }{3}+\frac{\pi }{6}$ where $n=0,1,2$

For $n=0x=\frac{\pi }{6}$

$n=1x=\frac{\pi }{3}+\frac{\pi }{6}=\frac{\pi }{2}$

Since only $\frac{\pi }{6}ϵ(0,1)$ is

$\therefore$ There exist only one solution of the equation in the interval $(0,1)$