Question
The complete solution set of $${\left[ {{{\cot }^{ - 1}}x} \right]^2} - 6\left[ {{{\cot }^{ - 1}}x} \right] + 9 \leqslant 0,$$ where [.] denotes the greatest integer function, is
A.
$$\left( { - \infty ,\cot 3} \right]$$
B.
$$\left[ {\cot 3,\cot 2} \right)$$
C.
$$\left[ {\cot 3,\infty} \right)$$
D.
None of these
Answer :
$$\left( { - \infty ,\cot 3} \right]$$
Solution :
$$\eqalign{
& {\left[ {{{\cot }^{ - 1}}x} \right]^2} - 6\left[ {{{\cot }^{ - 1}}x} \right] + 9 \leqslant 0 \cr
& \Rightarrow {\left( {\left[ {{{\cot }^{ - 1}}x} \right] - 3} \right)^2} \leqslant 0 \cr
& \Rightarrow \left[ {{{\cot }^{ - 1}}x} \right] = 3 \cr
& \Rightarrow 3 \leqslant {\cot ^{ - 1}}x < 4 \cr
& \Rightarrow x \in \left( { - \infty ,\cot 3} \right] \cr} $$