NDA 1 2025 Maths, question 9

Topic: Matrices and determinants

Consider the following statements in respect of the determinant $\Delta = \left| \begin{matrix} k(k + 2) & 2k + 1 & 1 \\ 2k + 1 & k + 2 & 1 \\ 3 & 3 & 1 \end{matrix} \right|$ .

I. $\Delta$ is positive if $k > 0$ .

II. $\Delta$ is negative if $k < 0$ .

III. $\Delta$ is zero if $k = 0$ .

How many of the statements given above are correct?

(a) None
(b) OneCorrect answer
(c) Two
(d) All three

Solution.

Answer: (b) One

Apply $R_{1} \rightarrow R_{1} - R_{3}$ and $R_{2} \rightarrow R_{2} - R_{3}$ , then expand along the last column.
This gives $\Delta = \left( k^{2} + 2k - 3 \right)(k - 1) - (2k - 2)^{2} = (k - 1)^{3}$ .
I is false: for example, $k = 1$ gives zero, not a positive determinant. II is true because $k < 0$ implies $(k - 1)^{3} < 0$ .
III is false since $k = 0$ gives $- 1$ .
Exactly one statement is correct.