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Question #40591

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a matrix of order $3 \times 3$ such that $|A| = 2$ . Let $\lambda \in \mathbb{R}$ be the real root of the equation $|-\lambda A| + \lambda^2 |A| = 24$ Consider the following set $S$ of 5 linear equations in variables $x, y, z$ : \begin{align*} E_1&: x + 2y + 3z = 1 \ E_2&: 2x + 3y + 4z = 2 \ E_3&: 3x + 4y + 5z = 3 \ E_4&: 4x + 5y + 6z = \lambda^2 \ E_5&: 5x + 6y + 7z = \lambda + 7 \end{align*} Find the number of non-empty subsets of $S$ that form a consistent system of equations.

Options

A.
31
B.
20
C.
16
D.
15

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

Total Selection (at least one)|System of Linear Equations|Determinant Scaling Property|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the determinant scaling property $|kA| = k^n|A|$ and the consistency of 3D linear systems.

The primary trap is the 'Modulus Trap' where students incorrectly assume $|-\lambda A| = \lambda^3|A|$ by ignoring the negative sign for odd-order matrices.

This sign-error leads to the false root $\lambda = 2$ .

For $\lambda = 2$ , $E_5$ becomes $5x+6y+7z=9$ , which is parallel to the intersection line of the first four equations but does not contain it.

Consequently, any system containing $E_5$ and at least two other equations becomes inconsistent, leading students to incorrectly calculate 20 consistent subsets instead of the true 31.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Total Selection (at least one)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free