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Question #40979

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & -1 & -1 & -1 \\ -1 & 2 & -1 & -1 \\ -1 & -1 & 2 & -1 \\ -1 & -1 & -1 & 2 \end{pmatrix}$ be a $4 \times 4$ matrix. Define a function $f: \mathbb{R} \to \mathbb{R}$ as $f(x) = \text{tr}(\text{adj}(xI - A))$ , where $I$ is the $4 \times 4$ identity matrix. A statistics dataset $S$ is formed by evaluating $f(x)$ at the first five natural numbers, i.e., $S = \{f(1), f(2), f(3), f(4), f(5)\}$ . If $M_1$ and $M_2$ denote the mean deviation about the mean and the mean deviation about the median of the dataset $S$ respectively, then the value of $5(M_1 - M_2)$ is:

Options

A.
12
B.
24
C.
36
D.
48

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

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

Why This Question Is Tricky

Common Mistakes

The problem tests the student's ability to connect matrix calculus with spectral theory and statistics.

The primary trap is attempting to compute the adjoint of a $4 \times 4$ matrix with variables or expanding its determinant directly, which is computationally lethal.

By recognizing Jacobi's formula or the eigenvalue properties of the adjoint, $f(x)$ simplifies to the derivative of the characteristic polynomial.

Furthermore, identifying $A$ as a shifted all-ones matrix allows instant computation of its eigenvalues, bypassing the determinant entirely.

Finally, it tests the precise definitions of mean deviation about the mean versus the median, penalizing any confusion between the two statistical measures.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free