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Question #40498

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3, z_4$ be the distinct complex roots of the equation $z^4 + z^2 + 1 = 0$ . For each $k \in \{1, 2, 3, 4\}$ , let $A_k = \lim_{z \to z_k} \frac{\ln(z^4 + z^2 + 2)}{z^4 + z^2 + 1}$ Define the function $g(x) = \sum_{k=1}^4 \frac{A_k}{x - z_k}$ for $x \in \mathbb{R}$ . Let $f(x) = \int_0^x g(t) dt$ . Evaluate the integral: $I = \int_0^1 f(x) dx$

Options

A.
$\frac{3}{2} \ln 3 - 4 + \frac{\sqrt{3}\pi}{2}$
B.
$\frac{1}{2} \ln 3 - 4 + \frac{\sqrt{3}\pi}{2}$
C.
$\frac{3}{2} \ln 3 - 4 + \frac{\pi}{\sqrt{3}}$
D.
$\frac{1}{2} \ln 3 - 2 + \frac{\sqrt{3}\pi}{2}$

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

Vieta’s Formulas in Complex Domain|V-N Method (Partial Fractions)|Indeterminate Forms|Integration by Parts (LIATE Rule)

Why This Question Is Tricky

Common Mistakes

The question tests the synthesis of complex roots, indeterminate forms, and advanced integration.

The first trap lies in evaluating the constants $A_k$ ; students who fail to recognize the limit as a standard indeterminate form $\lim_{w \to 0} \frac{\ln(1+w)}{w}$ will be unable to construct $g(x)$ .

The second trap involves recognizing that the sum of partial fractions over all roots forms the logarithmic derivative $\frac{P'(x)}{P(x)}$ .

Finally, integrating $f(x) = \ln(x^4+x^2+1)$ requires the LIATE rule (Integration by Parts), where students commonly forget the boundary evaluation term $x \ln(x^4+x^2+1)|_0^1$ , leading directly to the distractor option B.

The final rational integration acts as a rigorous computational filter.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Vieta’s Formulas in Complex Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free