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+4 x^2+4\right)-4 x^2=\left(x^2+2\right)^2-(2 x)^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