All x satisfying the inequality
Question:

All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>0$, lie in the interval:

  1. (1) $(-\infty, \cot 5) \cup(\cot 4, \cot 2)$

  2. (2) $(\cot 2, \infty)$

  3. (3) $(-\infty, \cot 5) \cup(\cot 2, \infty)$

  4. (4) $(\cot 5, \cot 4)$


Correct Option: , 2

Solution:

$\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>0$

$\left(\cot ^{-1} x-5\right)\left(\cot ^{-1}-2\right)>0$

$\cot ^{-1} x \in(-\infty, 2) \cup(5, \infty)$ …(1)

But $\cot ^{-1} x$ lies in $(0, \pi)$

Now, from equation (1)

$\cot ^{-1} x \in(0,2)$

Now, it is clear from the graph

$x \in(\cot 2, \infty)$

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