Question

Following question contains statements given in two columns, which have to be matched. The statements in List-1 are labelled as A, B, C, and D while the statements in List-2 are labelled as p, q, r, and s. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column-II.
1. Match values of $x$ in List 2 which satisfy the equation in List 1

Answer 1

A) $2{\log }_{10}(x-3)={\log }_{10}(x^2-21)$
$\Rightarrow {(x-3)}^2=x^2-21\Rightarrow 6x=30\Rightarrow x=5$
B) $x^{{\log }_{2}x+4}=32\Rightarrow ({\log }_{2}x+4){\log }_{2}x={\log }_{2}32$
Let ${\log }_{2}x=y\Rightarrow y^2+4y-5=0\Rightarrow (y+5)(y-1)=0$
${\log }_{2}x=-5$ or ${\log }_{2}x=1\Rightarrow x=\frac{1}{32}$ or $x=2$
C) $5\log {{}_{10}}^x+\frac{5\log {{}_{10}}^x}{5}=3.3\log {{}_{10}}^x+\frac{3\log {{}_{10}}^x}{3}$
$\Rightarrow \left(\frac{6}{5}\right)5\log {{}_{10}}^x=\left(\frac{10}{3}\right)3\log {{}_{10}}^x$
$\Rightarrow {\left(\frac{5}{3}\right)}^{\log {{}_{10}}^x}={\left(\frac{5}{3}\right)}^2\Rightarrow \log {{}_{10}}^x=2\Rightarrow x=100$
D) $9.9^{\log x}-{3.3}^{\log x}-210=0\Rightarrow 9x^2-3x-210=0$
$\Rightarrow 3x^2-x-70=0\Rightarrow 3x^2-15x+14x-70=0$
$\Rightarrow x=5$ or $x=\frac{-14}{3}$