NDA Mathematics Question (2025) — Solution

Step-by-step solution

Let = vmatrix p+c & a & b \\ c & p+a & b \\ c & a & p+b vmatrix Applying the column operation C_1 C_1 + C_2 + C_3: = vmatrix p+a+b+c & a & b \\ p+a+b+c & p+a & b \\ p+a+b+c & a & p+b vmatrix Since p = a+b+c is the perimeter of the triangle, we have p+a+b+c = 2p. Substituting this into the determinant: = vmatrix 2p & a & b \\ 2p & p+a & b \\ 2p & a & p+b vmatrix Taking 2p common from the first column C_1: = 2p vmatrix 1 & a & b \\ 1 & p+a & b \\ 1 & a & p+b vmatrix Applying the row operations R_2 R_2 - R_1 and R_3 R_3 - R_1: = 2p vmatrix 1 & a & b \\ 0 & p & 0 \\ 0 & 0 & p vmatrix Expanding the determinant along the first column C_1: = 2p [1(p p - 0)] = 2p(p^2) = 2p^3 Answer: 2p^3