1. Tardigrade
  2. Question
  3. Mathematics
  4. 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° cot 405°+ tan 765° cot 675° is equal to Q 1 C The number N =( log 5 250/ log 50 5)-( log 5 10/ log 1250 5), when simplified reduces to a natural number, then N equals R 2 D The value of log 10((1/1+ x m - n + x p - n )+(1/1+ x n - p + x m - p )+(1/1+ x p - m + x n - m )) is equal to S 3 T 4

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Q.
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

Continuity and Differentiability

Solution:

(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)$N =\frac{\log _5 250}{\log _{50} 5}-\frac{\log _5 10}{\log _{1250} 5}, \\
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) $
$\left. N =\left(\log _5 2\right)^2+5 \log _5 2+6-\left[\left(\log _5 2\right)^2+5 \log _5 2+4\right)\right]$
$N =6-4=2$