Question

Column I		Column II
A	The expression $\sqrt{{\log }_{0.5}^{2}8}$ has the value equal to	P	1
B	The value of expression $(\log 2{)}^3+(\log 8)(\log 5)+(\log 5{)}^3+3$, is	Q	2
C	Let $N={\log }_{2}5\cdot {\log }_{1/6}2\cdot {\log }_3\frac{1}{6}$, then $3^N$ is equal to	R	3
D	$\frac{{\log }_{3}243}{{\log }_2\sqrt{32}}$ is equal to	S	4
		T	5

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

Answer 1

Answer: (B)

(A) $\sqrt{{\left({\log }_{0.5}^{2}8\right)}^2}=\left|{\log }_{\frac{1}{2}}8\right|=|-3|=3$
(B)$(\log 2{)}^3+(\log 8\cdot \log 5)+(\log 5{)}^3+3$
$(\log 2{)}^3+(\log 5{)}^3+\log 8\cdot \log 5+3$
$(\log 2+\log 5)\left((\log 2{)}^2+(\log 5{)}^2-\log 2\cdot \log 5\right)+3\log 2\cdot \log 5+3$
$\log 10\left((\log 2{)}^2+(\log 5{)}^2-\log 2\log 5\right)+3\log 2\cdot \log 5+3$
$(\log 2{)}^2+(\log 5{)}^2+2\log 2\cdot \log 3+3$
$(\log 2+\log 5{)}^2+3$
$(\log 10{)}^2+3=1+3=4$
(C)$N={\log }_{2}5\cdot {\log }_{\frac{1}{6}}2\cdot {\log }_3\frac{1}{6}$
$={\log }_{2}5\cdot {\log }_{6}2\cdot {\log }_{3}6={\log }_{3}5$
$3^N=3^{{\log }_{3}5}=5$
(D) $\frac{{\log }_{3}243}{{\log }_2\sqrt{32}}=\frac{5{\log }_{3}3}{\left(\frac{5}{2}\right)\cdot {\log }_{2}2}=\frac{2}{5}\times 5=2$