MCQ
If $A + B = 225^\circ ,$ then $\frac{{\cot A}}{{1 + \cot A}}.\frac{{\cot B}}{{1 + \cot B}} = $
- A$1$
- B$-1$
- C$0$
- ✓$1/2$
$= \frac{1}{{(1 + \tan A)\,(1 + \tan B)}}$ $ = \frac{1}{{\tan A + \tan B + 1 + \tan A\tan B}}$ $[\,\because \tan (A + B) = \tan {225^o}]$
$ \Rightarrow \,\tan \,A + \tan \,B = 1 - \tan \,A\,\tan B$
$ = \frac{1}{{1 - \tan A\,\tan B + 1 + \tan A\tan B}} $
$= \frac{1}{2}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$f_n(x)=\sum_{j=1}^n \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)$
(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ )
Then, which of the following statement(s) is (are) TRUE?
$(A)$ $\sum_{ j =1}^5 \tan ^2\left( f _{ j }(0)\right)=55$
$(B)$ $\sum_{ j =1}^{10}\left(1+ f _{ j }^{\prime}(0)\right) \sec ^2\left( f _{ j }(0)\right)=10$
$(C)$ For any fixed positive integer $n$, $\lim _{x \rightarrow \infty} \tan \left(f_n(x)\right)=\frac{1}{n}$
$(D)$ For any fixed positive integer $n, \lim _{x \rightarrow \infty} \sec ^2\left(f_n(x)\right)=1$
| Column | Maximum of $z$ |
| $A$ | $300$ |
| $B$ | $325$ |