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Question #41703

Difficulty: 8/10 | 🟢 Free

The Question

A manufacturing plant produces three types of components, with daily production quantities denoted by the column vector $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ . The production constraints are modeled by the matrix equation $(A + I)X = B$ , where $I$ is the $3 \times 3$ identity matrix, $B = \begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix}$ , and $A$ is a $3 \times 3$ matrix such that $A^2 = I$ . If it is given that $A B = \begin{pmatrix} -2 \\ 21 \\ 19 \end{pmatrix}$ , and the profit function is $P = 10x + 15y + 20z$ , what is the maximum possible daily profit?

Options

A.
120
B.
85
C.
0
D.
\text{No feasible solution exists}

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

Word Problem Modeling|System of Linear Equations (Cramer/Inverse)|Idempotent and Involutory Properties

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the consistency of a linear system using matrix polynomial properties, specifically involutory matrices ( $A^2=I$ ).

The trap lies in the word problem format, which strongly suggests a standard Linear Programming Problem (LPP) with a feasible region and an optimal profit.

Students who blindly attempt to isolate $X$ by assuming $(A+I)$ is invertible will fail to realize that $\det(A+I)$ can be zero.

By pre-multiplying by the annihilating factor $(A-I)$ , the system reduces to $0 = AB - B$ .

Since the given vectors yield a non-zero result, the system is inconsistent, meaning the feasible region is empty.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Word Problem Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free