MCQ
The value of $\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} } \right)$ is
- ✓$\frac{21}{19}$
- B$\frac{19}{21}$
- C$\frac{22}{23}$
- D$\frac{23}{22}$
$ = \cot \left[ {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}\left( {1 + {n^2} + n} \right)} } \right]$
$ = \cot \left[ {\sum\limits_{n = 1}^{19} {{{\tan }^{ - 1}}\left( {\frac{1}{{1 + {n^2} + n}}} \right)} } \right]$
$ = \cot \left[ {\sum\limits_{n = 1}^{19} {{{\tan }^{ - 1}}\left( {n + 1} \right) - {{\tan }^{ - 1}}1} } \right]$
$ = \cot \left[ {{{\tan }^{ - 1}}20 - {{\tan }^{ - 1}}1} \right]$
$ = \cot \left( {{{\tan }^{ - 1}}\frac{{19}}{{21}}} \right)$
$ \Rightarrow \frac{{21}}{{19}}$
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| $\text{X}$ | $2$ | $3$ | $4$ | $5$ |
| $\text{P}(\text{X})$ | $\frac{5}{\text{k}}$ | $\frac{7}{\text{k}}$ | $\frac{9}{\text{k}}$ | $\frac{11}{\text{k}}$ |