Verified Solution • 2025–26 Session moderate

QJEEM21Maeda5: JEE Main 2021 Mathematics - Class 12 JEEM21Maeda5 Solution: The number of solutions of the equation , for x [ 1, 1], and…

The number of solutions of the equation

sin⁡−1[x2+13]+cos⁡−1[x2−23]=x2{\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2} , for x ∈\in [ −- 1, 1], and [x] denotes the greatest integer less than or equal to x, is :

Multiple Choice Options

  • A 0
  • B Infinite
  • C 2
  • D 4

Step-by-Step Verified Solution

There are three cases possible for x∈[−1,1]x \in [ - 1,1]

Case I : x∈[−1,−23)x \in \left[ { - 1, - \sqrt {{2 \over 3}} } \right)

∴\therefore sin⁡−1(1)+cos⁡−1(0)=x2{\sin ^{ - 1}}(1) + {\cos ^{ - 1}}(0) = {x^2}

⇒x2=π2+π2=π\Rightarrow {x^2} = {\pi \over 2} + {\pi \over 2} = \pi

⇒x=±π\Rightarrow x = \pm \sqrt \pi →\to (Reject)

Case II : x∈(−23,23)x \in \left( { - \sqrt {{2 \over 3}} ,\sqrt {{2 \over 3}} } \right)

∴\therefore sin⁡−1(0)+cos⁡−1(−1)=x2{\sin ^{ - 1}}(0) + {\cos ^{ - 1}}( - 1) = {x^2}

⇒0+π=x2\Rightarrow 0 + \pi = {x^2}

⇒x=±x\Rightarrow x = \pm \sqrt x →\to (Reject)

Case III : x∈(23,1)x \in \left( {\sqrt {{2 \over 3}} ,1} \right)

∴\therefore sin⁡−1(0)+cos⁡−1(0)=x2{\sin ^{ - 1}}(0) + {\cos ^{ - 1}}(0) = {x^2}

⇒x2−π⇒x−±x\Rightarrow {x^2} - \pi \Rightarrow x - \pm \sqrt x (Reject)

∴\therefore No solution. There, the correct answer is (1).

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