JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

Given f(x) = 2x^2 - 3x + 2 3x^2 + x + 3 To check for one-one, let f(x) = f(y) 2x^2 - 3x + 2 3x^2 + x + 3 = 2y^2 - 3y + 2 3y^2 + y + 3 Cross-multiplying and simplifying, we get: 11x^2y - 11xy^2 - 11x + 11y = 0 11xy(x - y) - 11(x - y) = 0 11(x - y)(xy - 1) = 0 This implies x = y or xy = 1. Since f(x) = f(1/x) for x 0 (for example, f(2) = f(1/2) = 4 17 ), the function is many-one (not one-one). To check for onto, let y = 2x^2 - 3x + 2 3x^2 + x + 3 (3y - 2)x^2 + (y + 3)x + (3y - 2) = 0 For x R , the discriminant D 0: (y + 3)^2 - 4(3y - 2)^2 0 (y + 3 - 6y + 4)(y + 3 + 6y - 4) 0 (7 - 5y)(7y - 1) 0 (5y - 7)(7y - 1) 0 y [ 1 7 , 7 5 ] Since the range is a proper subset of the codomain R , the function is not onto. Thus, f is neither one-one nor onto. Answer: neither one-one nor onto