NDA Mathematics Question (2025) — Solution

Step-by-step solution

The given equation is abx^2 + bcx + ca = cax^2 + abx + bc. Rearranging the terms, we get: (ab - ca)x^2 + (bc - ab)x + (ca - bc) = 0 a(b - c)x^2 + b(c - a)x + c(a - b) = 0 The sum of the coefficients is: a(b - c) + b(c - a) + c(a - b) = ab - ac + bc - ab + ac - bc = 0 Since the sum of the coefficients is 0, one root of the equation is x = 1. It is given that the roots of the equation are equal, so the other root must also be x = 1. The product of the roots is given by c(a - b) a(b - c) . Equating the product of the roots to 1 1 = 1: c(a - b) a(b - c) = 1 ca - cb = ab - ac 2ca = ab + bc Dividing both sides by abc, we obtain: 2 b = 1 c + 1 a This implies that 1 a , 1 b , 1 c are in AP. Therefore, a, b, c are in HP.