Question

Number of solutions of the equation ${\log }_{10}\left(\sqrt{5{\cos }^{-1}x-1}\right)+\frac{1}{2}{\log }_{10}\left(2{\cos }^{-1}x+3\right)+{\log }_{10}\sqrt{5}=1$ is

• A. 0
• B. 1
• C. more than one but finite
• D. infinite

Answer 1

Answer: (B)

${\cos }^{-1}x=t\Rightarrow x\in [-1,1]\text{ and }t\in [0,\pi ]$
${\log }_{10}\sqrt{5t-1}+\frac{1}{2}{\log }_{10}(2t+3)+\frac{1}{2}{\log }_{10}5=1;\left(t>\frac{1}{5}\text{ andt }=-\frac{3}{2}\right)$
${\log }_{10}((5t-1)(2t+3)\cdot 5)=2$
$(5t-1)(2t+3)\cdot 5=100$
$(5t-1)(2t+3)=20$
$10t^2+13t-3=20$
$10t^2+23t-10t-23=0$
$t(10t+23)-(10t+23)=0$
$(t-1)(10t+23)=0\Rightarrow t=1\text{ or }t=-\frac{23}{10}\text{ (rejected) }$
${\cos }^{-1}x=1\Rightarrow x=\cos 1$