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 $x y z=1000 $ $\log _{10} x \log _{10} y+\log _{10} x y \log _{10} z=1$ 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)
$a b+b c+c a=1 $ .....(2)
$a b c=3$ ....(3)
Now, $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
$a ^3+ b ^3+ c ^3=3 \cdot(3 \cdot 2-3.1)+3.3 $
$=18+9=27$
$\therefore \sqrt[3]{ a ^3+ b ^3+ c ^3}=3$