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Question #40993

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} 2x+1 & x+2 & x \\ x-1 & 2x-1 & 2x \\ 5x+1 & 4x+3 & 4x+1 \end{vmatrix}$ . A data set $S$ consists of the values $y_n = f'(n)$ for $n = 1, 2, \dots, 25$ . If the mean and variance of the data set $S$ are $\mu$ and $\sigma^2$ respectively, find the value of $\mu + \sigma^2$ .

Options

A.
1949
B.
1872
C.
1950
D.
1849

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

Standard Deviation and Coefficient of Variation|Sigma Notation Properties|Sum of Powers of Natural Numbers|Differentiation of Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the strategic use of determinant properties before applying calculus operations, combined with statistical properties of linear sequences.

The primary trap lies in differentiating the determinant directly using the product rule (sum of three determinants).

This leads to severe algebraic bloat and a high probability of calculation errors.

By observing the linear relationship between the rows ( $R_3 = 2R_1 + R_2 + [0,0,1]$ ), students can reduce the determinant to a simple quadratic polynomial.

Furthermore, it tests the property that the variance of a linearly transformed dataset $y_n = an+b$ is $a^2 \text{Var}(n)$ , bypassing the need to manually compute large sums of squares.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free