Question

Column I		Column II
A	The value of $2^{{\log }_{3}5}-5^{{\log }_{3}2}+3^{{\log }_{2}7}-7^{{\log }_{2}3}+6^{{\log }_{6}11}-4^{{\log }_{2}3}$ is equal to	P	0
B	The value of $\tan 225^{\circ }\cot 405^{\circ }+\tan 765^{\circ }\cot 675^{\circ }$ is equal to	Q	1
C	The number $N=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}$, when simplified reduces to a natural number, then $N$ equals	R	2
D	The value of ${\log }_{10}\left(\frac{1}{1+x^{m-n}+x^{p-n}}+\frac{1}{1+x^{n-p}+x^{m-p}}+\frac{1}{1+x^{p-m}+x^{n-m}}\right)$ is equal to	S	3
		T	4

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

Answer 1

Answer: (C)

(A) $5^{{\log }_{3}2}-5^{{\log }_{3}2}+3^{{\log }_{2}7}-3^{{\log }_{2}7}+11-2^{2{\log }_{2}3}=11-2^{{\log }_2{3}^2}=11-9=2$
(B) $\tan \left(180^{\circ }+45^{\circ }\right)\cot \left(360^{\circ }+45^{\circ }\right)+\tan \left(720^{\circ }+45^{\circ }\right)\cot \left(720^{\circ }{-}^{\circ }\right)$ $=\tan 45^{\circ }\cot 45^{\circ }+\tan 45^{\circ }\left(\cot 45^{\circ }\right)=0$ Ans.
(C)$$\begin{gathered} N=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}, \\ <br/>N={\left(3+{\log }_{5}2\right)}^2\cdot \left(2+{\log }_{5}2\right)-\left(1+{\log }_{5}2\right)\left(4+{\log }_{5}2\right) \end{gathered}$$
$N={\left({\log }_{5}2\right)}^2+5{\log }_{5}2+6-\left[{\left({\log }_{5}2\right)}^2+5{\log }_{5}2+4\right)]$
$N=6-4=2$