Question

Column I		Column II
A	Let $p =\log _5\left(\log _5 3\right)$. If $3^{ C +5^{- p }}=45$, then the value of $C$ is	P	1
B	If $x$ and $y$ satisfy simultaneously the equations,$(2 x)^{\log 2}=(3 y)^{\log 3} \text { and } 3^{\log x}=2^{\log y} \text {, }$then the value of $\left(x^{-1}+y^{-1}\right)$, is	Q	2
C	The value of $x$ satisfying the equation $3^{\log _{\sqrt{3}} \sqrt{x}}+9^{\log _3 x}+27^{\log _3 2 x^2}=3$ is coprime with	R	3
		S	5

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

Answer 1

Answer: (D)

(A)$\text { Consider } 5^{- p }=5^{-\log _5\left(\log _5 3\right)}=\frac{1}{5^{\log _5\left(\log _5 3\right)}}=\frac{1}{\log _5 3}\left( a ^{\log _{ a } N }= N \right)$
$\therefore 5- p =\frac{1}{\log _5 3}=\log _3 5$
Now, $ 3^{ C +\log _3 5}=45$
$\Rightarrow 3^C \cdot 3^{\log _3 5}=45 \Rightarrow 3^C \cdot 5=45 \Rightarrow 3^C=9=3^2 \Rightarrow C=2 \text {. Ans. }$
(B) We have $(2 x )^{\log 2}=(3 y )^{\log 3}$
taking log on both side, we get
$\Rightarrow (\log 2)(\log 2+\log x)=\log 3(\log 3+\log y) $
$\Rightarrow \log ^2 2-\log ^2 3=\log 3 \cdot \log y-\log 2 \cdot \log x$ ....(1)
$\text { Again, } 3^{\log x}=2^{\log y}$
taking log on both the sides, we get
$\Rightarrow \log x \cdot \log 3=\log y \cdot \log 2 \Rightarrow \frac{\log x }{\log 2}=\frac{\log y }{\log 3}= k \text { (let) } $
$\Rightarrow \log x = k \cdot \log 2 \text { and } \log y = k \cdot \log 3(\text { put in (1)) } $
$\therefore \text { we will get } k =-1 \Rightarrow x =1 / 2, y =1 / 3$
Hence $\left(x^{-1}+y^{-1}\right)=(2+3)=5$. Ans
(C)$3^{\log _3 x}+3^{2 \log _3 x}+3^{3 \log _3 x}=3$
$\Rightarrow x+x^2+x^3=3 \Rightarrow (x-1)+\left(x^2-1\right)+\left(x^3-1\right)=0 $
$\Rightarrow (x-1)\left(1+x+1+x^2+x+1\right)=0 $
$\Rightarrow x=1\left(\because x^2+2 x+3 \neq 0\right)$