Question

Column I		Column II
A	Let $x$ be a real number satisfying $x^2+\frac{1}{x^2}=23$ and ${\log }_5\left(x^3+\frac{1}{x^3}\right)$ lies between two consecutive natural numbers $a\&b$ then $a+b$ is equal to	P	3
B	If ${\log }_{e}x+{\log }_{x}e=3$ then $\left({\ln }^{2}x+{\ln }_x^{2}e\right)$ is	Q	5
C	A rational number which is 50 times its own logarithm to the base 10 , is	R	7
D	If the positive numbers $x,y\&z$ satisfy $xyz=1000$ ${\log }_{10}x{\log }_{10}y+{\log }_{10}xy{\log }_{10}z=1$ and ${\log }_{x}10+{\log }_{y}10+{\log }_{z}10=\frac{1}{3}$ then the value of $\sqrt[3]{{\left({\log }_{10}x\right)}^3+{\left({\log }_{10}y\right)}^3+{\left({\log }_{10}z\right)}^3}$, is	S	50
		T	100

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

Answer 1

Answer: (C)

(A)${\left(x+\frac{1}{x}\right)}^2=x^2+\frac{1}{x^2}+2=25$
$\left(x+\frac{1}{x}\right)=\pm 5$
Now $x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)\left(x^2-1+\frac{1}{x^2}\right)=(\pm 5)(23-1)=\pm 110$
Taking positive value ${\log }_{5}110$ lies between $2\&3$ hence $2+3=a+b=5$
(B) $\ln x+\frac{1}{\ln x}=3$
${\left(\ln x+\frac{1}{\ln x}\right)}^2=9$
${\ln }^{2}x+\frac{1}{{\ln }^{2}x}=7$
(C) $N=50{\log }_{10}N\Rightarrow {\log }_{10}N=\frac{N}{50};N=(10{)}^{\frac{N}{50}};N^{\frac{1}{N}}=(10{)}^{\frac{1}{50}}=(10{)}^{\frac{2}{100}}=(100{)}^{\frac{1}{100}}$ $\Rightarrow N=100$ Ans.
(D) Let ${\log }_{10}x=a{\log }_{10}y=b{\log }_{10}z=c$
$a+b+c=3$ .....(1)
$ab+bc+ca=1$ .....(2)
$abc=3$ ....(3)
Now, $a^3+b^3+c^3-3abc=(a+b+c)\left(a^2+b^2+c^2-ab-bc-ca\right)$
$a^3+b^3+c^3=3\cdot (3\cdot 2-3.1)+3.3$
$=18+9=27$
$\therefore \sqrt[3]{a^3+b^3+c^3}=3$