JEE Advanced Mathematics Question (2026) — Solution

Step-by-step solution

Let the given expression be . First, we simplify the angles in the product. ( 27 11 ) = (2 + 5 11 ) = ( 5 11 ) ( 81 11 ) = (7 + 4 11 ) = - ( 4 11 ) = ( - 4 11 ) = ( 7 11 ) Thus, the angles are 11 , 3 11 , 5 11 , 7 11 , 9 11 , which can be written as _k = (2k-1) 11 for k = 1, 2, 3, 4, 5. The expression becomes: = _ k=1 ^ 5 (1 - 2 _k ) Using the trigonometric identity 1 - 2 = 1 - 2(2 ^2( /2) - 1) = 3 - 4 ^2( /2). Multiplying and dividing by ( /2), we get: 1 - 2 = 3 ( /2) - 4 ^3( /2) ( /2) = - (3 /2) ( /2) Applying this identity to the product: = _ k=1 ^ 5 ( - (3 _k/2) ( _k/2) ) = (-1)^5 _ k=1 ^ 5 (3 _k/2) _ k=1 ^ 5 ( _k/2) The terms in the denominator are ( 22 ), ( 3 22 ), ( 5 22 ), ( 7 22 ), ( 9 22 ). The terms in the numerator are ( 3 22 ), ( 9 22 ), ( 15 22 ), ( 21 22 ), ( 27 22 ). We can simplify the larger angles in the numerator: ( 15 22 ) = ( - 7 22 ) = - ( 7 22 ) ( 21 22 ) = ( - 22 ) = - ( 22 ) ( 27 22 ) = ( + 5 22 ) = - ( 5 22 ) Substituting these back into the numerator product: _ k=1 ^ 5 (3 _k/2) = ( 3 22 ) ( 9 22 ) (- ( 7 22 ) ) (- ( 22 ) ) (- ( 5 22 ) ) _ k=1 ^ 5 (3 _k/2) = - [ ( 22 ) ( 3 22 ) ( 5 22 ) ( 7 22 ) ( 9 22 ) ] Notice that the term in the brackets is exactly the denominator product. Therefore, the ratio of the numerator product to the denominator product is -1. = (-1)^5 (-1) = (-1) (-1) = 1 We need to find the value of 5 - ^2: 5 - (1)^2 = 5 - 1 = 4 Answer: 4