Question

Column I		Column II
A	If $x_1$ and $x_2$ satisfy the equation $x^{{\log }_{10}x}=100x$ then the value of $x_1{x}_2$ equals	P	irrational
B	Sum of the squares of the roots of the equation ${\log }_2\left(9-2^x\right)=3-x$ is	Q	rational
C	If ${\log }_{1/8}\left({\log }_{1/4}\left({\log }_{1/2}x\right)\right)=\frac{1}{3}$ then $x$ is	R	prime
		S	composite

• A. (A) Q, S; (B) Q, S; (C) P
• B. (A) Q, S; (B) Q, S; (C) Q
• C. (A) Q, S; (B) Q, S; (C)R
• D. (A) Q, S; (B) Q, S; (C) S

Answer 1

Answer: (A)

(A)${\log }_{10}^{2}x=\log 100x=2+{\log }_{10}x$
$\text{ put }\log x=t$
$t^2=2+t\Rightarrow t^2-t-2=0\Rightarrow (t-2)(t+1)=0$
$\Rightarrow t=2\text{ or }t=-1$
${\log }_{10}x=2\text{ or }{\log }_{10}x=-1$
$x=100\text{ or }x=1/10$
(B)${\log }_2\left(9-2^x\right)=3-x$
$2^{(3-x)}=9-2^x$
$\frac{8}{2^x}=9-2^x$
$\text{ put }{2}^x=t$
$8=9t-t^2\Rightarrow t^2-9t+8=0\Rightarrow (t-8)(t-1)=0$
$t=8\text{ or }t=1$
$2^x=8\text{ or }{2}^x=2^3\Rightarrow x=3$
$2^x=1\text{ or }{2}^x=2^0\Rightarrow x=0$
(C)
${\log }_{1/8}\left({\log }_{1/4}\left({\log }_{1/2}x\right)\right)=\frac{1}{3}$
${\left(\frac{1}{8}\right)}^{1/3}={\log }_{1/4}\left({\log }_{1/2}x\right);\frac{1}{2}={\log }_{1/4}\left({\log }_{1/2}x\right);{\left(\frac{1}{4}\right)}^{1/2}={\log }_{1/2}x;\frac{1}{2}={\log }_{1/2}x$
${\left(\frac{1}{2}\right)}^{1/2}=x;\therefore x=\frac{1}{\sqrt{2}}.$