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