If ( cot -1 x)2-7( cot -1 x)+100 , then the range of x will be

Question

If ( cot -1 x)2-7( cot -1 x)+10>0 , then the range of x will be (A) (.-∈ fty, cot 2.) (B) (- ∈ fty , cot 5) (C) (cot 2 , cot ⁡ 5) (D) (cot 2 , ∈ fty)

Tardigrade Admin · Accepted Answer

((cot)- 1 x - 5)((cot)- 1 ⁡ x - 2)>0 ⇒ (cot)- 1 x ∈ (- ∈ fty , 2)∪(5 , ∈ fty) ⇒ (cot)- 1 x∈ (0 , 2) (Taking intersection with range of cot- 1 x ) ⇒ n∈ (.cot 2 , ∈ fty .)

Q. If $(cot^{- 1} x)^{2} - 7 (cot^{- 1} x) + 10 > 0$ , then the range of $x$ will be

$(- \in f t y, co t 2)$

$(- \in f t y, co t 5)$

$(co t 2, co t 5)$

$(co t 2, \in f t y)$

$((co t)^{- 1} x - 5) ((co t)^{- 1} x - 2) > 0$
$\Rightarrow (co t)^{- 1} x \in (- \in f t y, 2) \cup (5, \in f t y)$
$\Rightarrow (co t)^{- 1} x \in (0, 2)$ (Taking intersection with range of $co t^{- 1} x$ )
$\Rightarrow n \in (co t 2, \in f t y)$

Q. If (cot−1x)2−7(cot−1x)+10>0 , then the range of x will be

(−∈fty,cot2)

(−∈fty,cot5)

(cot2,cot⁡5)

(cot2,∈fty)