UPSC NDA 2 2016 Mathematics Solved Paper

Concept:For a system of linear equations to have a unique solution, the coefficient matrix must be non-singular, i.e., its determinant is not zero.Explanation:The coefficient matrix is:$A = \begin{bmatrix} 2 & 3 & 5 \\ 7 & 3 & -2 \\ 2 & 3 & \lambda \end{bmatrix}$For a unique solution, $\det(A) \neq 0$.Compute the determinant:$\det(A) = 2(3\lambda + 6) - 3(7\lambda + 4) + 5(21 - 6) \neq 0$Simplify: $6\lambda + 12 - 21\lambda - 12 + 75 \neq 0$This gives $-15\lambda + 75 \neq 0$, so $\lambda \neq 5$.The condition for $\mu$ does not appear in the determinant; hence $\mu$ can be any real number.Answer:$\lambda \neq 5$ and $\mu$ has any real value.