If x in [0, 1],then the number of solution(s) | vedclass

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(B) Given the equation: $2[\cos^{-1}x] + 6[	ext{sgn}(\sin x)] = 3$ for $x \in [0, 1]$.Step $1$: Analyze the range of $\cos^{-1}x$ for $x \in [0, 1]$.

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(B) Given the equation: $2[\cos^{-1}x] + 6[	ext{sgn}(\sin x)] = 3$ for $x \in [0, 1]$.Step $1$: Analyze the range of $\cos^{-1}x$ for $x \in [0, 1]$.Since $x \in [0, 1]$,$\cos^{-1}x \in [0, \pi/2]$.Thus,$[\cos^{-1}x]$ can take values $0$ or $1$ (since $\pi/2 \approx 1.57$).Step $2$: Analyze the range of $	ext{sgn}(\sin x)$ for $x \in [0, 1]$.For $x \in (0, 1]$,$\sin x > 0$,so $	ext{sgn}(\sin x) = 1$.For $x = 0$,$\sin 0 = 0$,so $	ext{sgn}(0) = 0$.Step $3$: Test the cases.Case $1$: If $x = 0$,then $[\cos^{-1}0] = [\pi/2] = 1$ and $	ext{sgn}(\sin 0) = 0$.The equation becomes $2(1) + 6(0) = 2 
eq 3$.Case $2$: If $x \in (0, 1]$,then $	ext{sgn}(\sin x) = 1$,so $[	ext{sgn}(\sin x)] = 1$.The equation becomes $2[\cos^{-1}x] + 6(1) = 3$,which simplifies to $2[\cos^{-1}x] = -3$,or $[\cos^{-1}x] = -1.5$.Since the greatest integer function must result in an integer,there is no solution for this case.Conclusion: There are no values of $x$ that satisfy the equation. Therefore,the number of solutions is $0$.

List $I$	List $II$
$P$. $\left(\frac{1}{y^2}\left(\frac{\cos (\tan ^{-1} y)+y \sin (\tan ^{-1} y)}{\cot (\sin ^{-1} y)+\tan (\sin ^{-1} y)}\right)^2+y^4\right)^{1 / 2}$ takes value	$1$. $\frac{1}{2} \sqrt{\frac{5}{3}}$
$Q$. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is	$2$. $\sqrt{2}$
$R$. If $\cos (\frac{\pi}{4}-x) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+\cos (\frac{\pi}{4}+x) \cos 2 x$ then possible value of $\sec x$ is	$3$. $\frac{1}{2}$
$S$. If $\cot (\sin ^{-1} \sqrt{1-x^2})=\sin (\tan ^{-1}(x \sqrt{6})), x \neq 0$,then possible value of $x$ is	$4$. $1$

If $x \in [0, 1]$,then the number of solution$(s)$ of the equation $2[\cos^{-1}x] + 6[\text{sgn}(\sin x)] = 3$ is (where $[.]$ denotes the greatest integer function and $\text{sgn}(x)$ denotes the signum function of $x$)-

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