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Question #59680

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ real matrix with distinct eigenvalues $\lambda_1 < \lambda_2 < \lambda_3$ forming an arithmetic progression. A dataset $S$ is constructed by evaluating the function $f(x) = \text{tr}(\text{adj}(xI - A))$ at $x \in \{\lambda_1, \lambda_2, \lambda_3\}$ . Let $\mu$ and $M_e$ be the mean and median of $S$ , respectively. If $MD_{\mu}$ and $MD_{M_e}$ denote the mean deviation of $S$ about its mean and median, which of the following is INCORRECT?

Options

A.
$\mu = \frac{1}{4}(\lambda_3 - \lambda_1)^2$
B.
$M_e = \frac{1}{2}(\lambda_3 - \lambda_1)^2$
C.
$MD_{\mu} = \frac{1}{3}(\lambda_3 - \lambda_1)^2$
D.
$MD_{M_e} = \frac{1}{2}(\lambda_3 - \lambda_1)^2$

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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

Mean Deviation about Mean vs. Median|Differentiation of Determinants|Characteristic Equation

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Boundary Condition

Tests behavior at limits like 0, 1, infinity, or empty sets.

Difficulty Boost : +4

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Recursive Logic

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

Difficulty Boost : +5

Common Mistakes

This problem constructs a lethal trap by disguising the derivative of the characteristic polynomial as the trace of the adjugate matrix.

Students unaware of Jacobi's formula, d/dx[det(xI - A)] = tr(adj(xI - A)), will attempt an O(n!) determinant expansion, exhausting their time.

For those who bypass the trap, the problem recursively demands the synthesis of spectral theory (roots of characteristic equations) with statistical measures (Mean Deviation about Mean vs.

Median) on a symbolically generated dataset, testing pure algebraic endurance and conceptual clarity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free