UPSC NDA I 2023 Mathematics Solved Paper

Concept:The equation is a determinant that simplifies to a polynomial equation. The sum of its roots is found by expanding the determinant and factoring.Explanation:We are given the determinant equation:$\begin{vmatrix} 0 & x-a & x-b \\ 0 & 0 & x-c \\ x+b & x+c & 1 \end{vmatrix} = 0$Expand along the first column (column 1).The first element is $0$, and the second element is $0$.The third element is $(x+b)$, with its minor determinant $\begin{vmatrix} x-a & x-b \\ 0 & x-c \end{vmatrix}$.The expansion gives $0 + 0 + (x+b) \cdot [(x-a)(x-c) - 0 \cdot (x-b)] = 0$.Simplify: $(x+b)(x-a)(x-c) = 0$.So the roots are the values of $x$ that make each factor zero: $x+b=0$, $x-a=0$, $x-c=0$.Thus the roots are $x = -b$, $x = a$, $x = c$.The sum of the roots is $(-b) + a + c = a - b + c$.Answer:$a - b + c$ (Option B).