Inverse Trigonometric FunctionHard
Question
The number of real solutions of 
is
isOptions
A.zero
B.two
C.one
D.infinite
Solution
Given function is
Function is defined, if
(i) x(x + 1) ≥ 0 ∵ Domain of square root function.
(ii) x2 + x + 1 ≥ 0 ∵ Domain of square root function.
(iii)
Domain of sin-1 function
From (ii) and (iii)
0 ≤ x2 + x + 1 ≤
x2 + x ≥ 0
⇒ 0 ≤ x2 + x + 1 ≤ x2 + x + 1 ≥ 1
⇒ x2 + x + 1 = 1
⇒ x2 + x = 0
⇒ x(x + 1) = 0
⇒ x = 0, x = - 1
Function is defined, if
(i) x(x + 1) ≥ 0 ∵ Domain of square root function.
(ii) x2 + x + 1 ≥ 0 ∵ Domain of square root function.
(iii)
Domain of sin-1 function From (ii) and (iii)
0 ≤ x2 + x + 1 ≤
x2 + x ≥ 0 ⇒ 0 ≤ x2 + x + 1 ≤
x2 + x + 1 ≥ 1⇒ x2 + x + 1 = 1
⇒ x2 + x = 0
⇒ x(x + 1) = 0
⇒ x = 0, x = - 1
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