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$