Verified Solution • 2025–26 Session moderate

QJEEM21Mabec9: JEE Main 2021 Mathematics - Class 12 JEEM21Mabec9 Solution: If the domain of the function is the interval ( , ], then + is…

If the domain of the function f(x)=cos⁡−1x2−x+1sin⁡−1(2x−12)f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }} is the interval ( α\alpha , β\beta ], then α\alpha + β\beta is equal to :

Multiple Choice Options

  • A 32{3 \over 2}
  • B 2
  • C 12{1 \over 2}
  • D 1

Step-by-Step Verified Solution

O≤x2−x+1≤1O \le {x^2} - x + 1 \le 1

⇒x2−x≤0\Rightarrow {x^2} - x \le 0

⇒x∈[0,1]\Rightarrow x \in [0,1]

Also, 0 < {\sin ^{ - 1}}\left( {{{2x - 1} \over 2}} \right) \le {\pi \over 2}

\Rightarrow 0 < {{2x - 1} \over 2} \le 1

\Rightarrow 0 < 2x - 1 \le 2

1 < 2x \le 3

{1 \over 2} < x \le {3 \over 2}

Taking intersection

x∈(12,1]x \in \left( {{1 \over 2},1} \right]

⇒α=12,β=1\Rightarrow \alpha = {1 \over 2},\beta = 1

⇒α+β=32\Rightarrow \alpha + \beta = {3 \over 2}

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