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Question #41213

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ z \in \mathbb{C} : |z| \le 3 \}$ . For any $z \in S$ , a matrix $A$ is defined as: $A = \begin{pmatrix} [|z|] & -1 & 1 \\ 1 & [|z-3|] & -1 \\ -1 & 1 & [|z-4i|] \end{pmatrix}$ where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $V$ be the volume of the parallelepiped whose coterminous edges are the column vectors of the matrix $\text{Adj}(A)$ . Find the maximum possible value of $V$ .

Options

A.
10816
B.
15129
C.
9604
D.
104

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Adjoint-Determinant Relation|Scalar Triple Product Volume Logic|Geometrical Interpretation of Modulus|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of matrix adjoint properties ( $|\text{Adj}(A)| = |A|^{n-1}$ ) with complex number geometry and the Greatest Integer Function (GIF).

The primary trap lies in the 'independent maximization fallacy': students often find the maximum possible values of $[|z-3|]$ and $[|z-4i|]$ independently ( $6$ and $7$ ) and incorrectly assume they can occur simultaneously, leading to a determinant of $123$ and a trap volume of $15129$ .

Rigorous parameterization of $z$ on the circle $|z|=3$ is required to find the true coupled maximum of the piecewise constant determinant function.

A secondary trap is forgetting the square in the volume formula, leading to option D.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Adjoint-Determinant Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free