Question

Column I		Column II
A	The value of $K$, where $\log (\log 4)+\log (\log 25)=\log K+\log (\log 2)+\log (\log 5)$ is	P	1
B	Number of values of $x\in N$, for which $x^4+4$ is prime, is	Q	2
C	If $b$ is a positive real number different from 1 , let ${\log }_{b}x$ denotes the base $b$ logarithm of $x$. Let $n$ be the number of solutions of $x$ to the equation, ${\log }_{b}x={\log }_{x}b$ where $x$ is a positive real different from 1 . Then $n$ equals	R	3
D	The expression $\sqrt{{\log }_{0.5}{}^{2}8}$ has the value equal to	S	4

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

Answer 1

Answer: (A)

(A)$\log (\log 4)+\log (\log 25)=\log K+\log (\log 2)+\log (\log 5)$
$\log (2\log 2)+\log (2\log 5)=\log k+\log (\log 2)+\log (\log 5)$
$\log 2+\log 2=\log K$
$\log 4=\log K\Rightarrow K=4$
(B) $x^4+4=\left(x^4+4x^2+4\right)-4x^2={\left(x^2+2\right)}^2-(2x{)}^2;x=1$
(C)${\log }_{b}x=\frac{1}{{\log }_{b}x}$
${\left({\log }_{b}x\right)}^2=1\Rightarrow {\log }_{b}x=1\text{ or }-1$
$x=b\text{ or }\frac{1}{b}\Rightarrow n=2$
(D) Note $\sqrt{x^2}=-x$, if $x$ is negative