Before You Begin

Question #67312

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in the $xy$ -plane defined by the inequalities: $x - 3y \le 3$ $x - y \ge 5$ Let $S$ be the set of all real values of $\alpha$ such that the matrix $A = \begin{pmatrix} x & y \\ 1 & \alpha \end{pmatrix}$ is singular for at least one point $(x,y) \in R$ . If $a$ and $b$ are the infimum and supremum of the set $S$ respectively, let $M = \begin{pmatrix} 6a & b \\ -b & 6a \end{pmatrix}$ Find the value of $\det(M^4) - \text{Tr}(M^4)$ .

Options

A.
24
B.
639
C.
16
D.
8

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Matrix Equality|Singular vs. Non-Singular Threshold|Unbounded Region Trap|Graphical Feasible Region

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of Linear Programming and Matrix Singularity, specifically targeting the 'Unbounded Region Trap'.

Students must recognize that the singularity condition translates to finding the range of the slope y/x over the feasible region.

The trap lies in the region being unbounded; students who rely solely on algorithmic vertex checking will only find the infimum at (6,1) and fail to find the supremum.

Furthermore, those who check the asymptotic slopes of the bounding rays might incorrectly conclude the infimum is 1/3 (from x-3y=3) instead of the true global minimum 1/6 at the vertex.

This effectively filters out candidates who do not perform a rigorous domain analysis of the objective function as variables approach infinity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Matrix Equality
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free