Question

Column I		Column II
A	If $x_1$ and $x_2$ satisfy the equation $x^{\log _{10} x}=100 x$ 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)$\log _{10}^2 x=\log 100 x=2+\log _{10} x$
$\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=9 t - t ^2 \Rightarrow t ^2-9 t +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}} . $