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Question #41528

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x)$ be a $7 \times 7$ matrix whose elements are given by $m_{i,j} = (-1)^k x^{6-k}$ , where $k \in \{0, 1, \dots, 6\}$ such that $k \equiv (j-i) \pmod 7$ for $1 \le i, j \le 7$ . If $f(x) = \det(M(x))$ , then the value of $\int_0^1 (f(x) + x f'(x)) \, dx$ is

Options

A.
0
B.
64
C.
128
D.
256

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Integrating f(x) + xf'(x) forms|Roots of Unity Properties|Determinant Product Rule

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of integration by parts (reverse product rule) and the spectral properties of circulant matrices.

A common trap is attempting to expand the $7 \times 7$ determinant algebraically or numerically, which is extremely time-consuming and prone to error.

Another trap lies in evaluating the product of the roots of unity $\prod (1+\omega^r)$ ; students often incorrectly assume it equals 1 or 0 due to cyclic symmetry, leading to answers like 128 or 0.

Recognizing the circulant eigenvalue formula $\lambda_r = \sum c_k \omega^{rk}$ and leveraging the polynomial roots of $z^n-1$ is the intended high-level heuristic.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free