If = _ 1 2 ^2 ^ -1 x 2 x^2-3 x+2 d x then the value of 7 ( 2 7 ) is ____________. (Here, the inverse trigonometric function ^ -1 x assumes values in (- 2 , 2 ).)

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Quantrex Academy · Accepted Answer

Correct answer: A. A

= _ 1 2 ^2 ^ -1 x 2 x^2-3 x+2 d x ....(i) aligned & Let x= 1 t \ & d x=- 1 t^2 d t \ & = _2^ 1 2 ^ -1 ( 1 t ) 2 t^2 - 3 t +2 ( -1 t^2 ) d t aligned = _ 1 2 ^2 ^ -1 t 2 t ^2-3 t +2 dt ....(ii) Now by (i) + (ii) aligned & 2 = _ 1 2 ^2 2 2 x^2-3 x+2 d x \ & = 8 _ 1 2 ^2 d x x^2- 3 x 2 +1 \ & = 8 _ 1 2 ^2 (x- 3 4 )^2+ 7 16 \ & = 8 7 4 [ ^ -1 ( x- 3 4 7 4 ) ]_ 1 2 ^2 aligned aligned & = 2 7 [ ^ -1 4 x-3 7 ]_ 1 2 ^2 \ & = 2 7 [ ^ -1 5 7 - ^ -1 (- 1 7 ) ] \ & = 2 7 ^ -1 ( 5 7 + 1 7 ) 1- 5 7 \ & = 2 7 ^ -1 (3 7 ) aligned Now 7 ( 2 7 ) aligned & 7 ( ^ -1 (3 7 ) ) \ & 7 3 7 \ & =21 aligned

JEE Advanced Mathematics Question (2025) — Solution

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