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Question #66807

Difficulty: 8/10 | 🟢 Free

The Question

The mean and variance of $10$ observations are calculated as $7$ and $15$ respectively. Later, it is discovered that one observation was incorrectly recorded as $12$ instead of the correct value $2$ . Let the correct mean and correct variance of these observations be $\mu$ and $\sigma^2$ respectively. Consider a $3 \times 3$ matrix $M$ with real entries such that $\text{Tr}(M) = \mu$ , $\text{Tr}(M^2) = \sigma^2$ , and $\text{Tr}(M^3) = \mu^2$ . Find the value of $\det(M) + \text{Tr}(\text{adj}(M))$ .

Options

A.
15
B.
17
C.
19
D.
21

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

Inverse Existence Criterion|Matrix Trace Properties|Correction of Incorrect Mean/Variance

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 recursive propagation of mathematical logic across two distinct domains: Statistics and Linear Algebra.

The primary trap lies in the variance correction formula; students frequently forget to subtract the square of the new mean when calculating the corrected variance, leading to an incorrect trace matrix input.

Furthermore, in the linear algebra phase, the problem punishes students who attempt to construct a specific matrix $M$ .

The optimal path requires recognizing that the trace of powers of a matrix corresponds to the sum of powers of its eigenvalues, thereby transforming the matrix problem into an application of Newton's sums and symmetric polynomials to find the determinant and the trace of the adjugate.

Quick Facts About This Question

Question Information

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