The number of solutions of the equation sin^-1 ⁡(x^2+ 13)+ cos^-1 ⁡(x^2- 23)=x^2 for x ∈ [-1 ,1 ] and [x ]denotes the greatest integer less than or equal to x, is:

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Accepted Answer

Correct answer: B. (b) JEE Main 2021 Given equation sin^-1 ⁡(x^2+ 13)+ cos^-1 ⁡(x^2- 23)=x^2 Now, sin^-1 ⁡(x^2+ 13)is defined if -1 x^2+ 13<2 ⇒ -43 x^2< 53 ⇒0 x^2< 53 and cos^-1 ⁡(x^2- 23)is defined if -1 x^2- 23<2 ⇒ -13 x^2< 83 ⇒0 x^2< 83 So, form (1 ) and (2 ) we can conclude 0 x^2< 53 Case - 1 if 0 x^2< 23 sin^-1 ⁡ (0 )+ cos^-1 ⁡ (-1 )=x^2 ⇒x+ =x^2 ⇒x^2= but (0 , 23 ) ⇒No value of ' X^ '^ ' Case - II if 23 x^2< 53 sin^-1 ⁡ (1 )+ cos^-1 ⁡ (0 )=x^2 ⇒ 2+ 2=x^2 ⇒x^2= But ( 23 , 53 ) ⇒ No value of ' X^ ' So, number of solutions of the equation is zero.

$The number of solutions of the equation sin^{- 1} (x^{2} + \frac{1}{3}) + cos^{- 1} (x^{2} - \frac{2}{3}) = x^{2} for x \in [- 1, 1] and [x] denotes the greatest integer less than or equal to x, is:$

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