(i) [ − 1, 0)

(ii) ( − ∞, cot 3) U (cot 2, ∞ )

(iii) – sin1 < x ≤ 1

(iv) cos 2 < x ≤ 1

(v) no solution

Sol. (i) cos − 1 x > cos − 1 x 2 .

Defined for

As function decreases

x < x 2 ⇒ x 2 – x > 0

x(x – 1) > 0 ⇒

[ − 1, 0)

(ii) Let arc cotx = p ⇒ p 2 – 5p + 6 > 0 ⇒ (p – 2)(P – 3) > 0

cot –1 x is a decreasing function ,

( − ∞, cot 3) U (cot 2, ∞ )

(iii) sin –1 x > – 1

x ∈ (– sin 1, 1]

(iv) cos –1 x < 2 ⇒ x ∈ (cos 2, 1]

(v) cot –1 x < –

cot –1 x > 0  x ∈ R ⇒ no solution

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