NDA Mathematics Question (2025) — Solution

Step-by-step solution

Applying the column operation C_1 C_1 + C_2 + C_3, the determinant becomes: vmatrix x+1+ + ^2 & & ^2 \\ x+1+ + ^2 & x+ ^2 & 1 \\ x+1+ + ^2 & 1 & x+ vmatrix = 0 Since 1 + + ^2 = 0 for a non-real cube root of unity, the equation simplifies to: vmatrix x & & ^2 \\ x & x+ ^2 & 1 \\ x & 1 & x+ vmatrix = 0 Taking x common from the first column: x vmatrix 1 & & ^2 \\ 1 & x+ ^2 & 1 \\ 1 & 1 & x+ vmatrix = 0 This implies x = 0 is a root of the equation. Answer: x = 0