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Question #41473

Difficulty: 8/10 | 🟢 Free

The Question

Let $w \in \mathbb{C}$ satisfy $|w + 2| \le 1$ . If the equations $z^2 + wz + 1 + i = 0$ and $z^2 + iz + w + 1 = 0$ share at least one common root, let $f(x) = x^2 + \text{Re}(w)x + \text{Im}(w)$ for $x \in \mathbb{R}$ . Find the area of the region bounded by the curves $y = |f(x)|$ and $y = 2$ .

Options

A.
$\frac{32 - 16\sqrt{2}}{3}$
B.
$\frac{32}{3}$
C.
$\frac{16\sqrt{2}}{3}$
D.
$\frac{16 - 8\sqrt{2}}{3}$

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

The Sign-Consistency Principle|Condition for Common Roots|Geometry of Complex Inequalities

Why This Question Is Tricky

Common Mistakes

This problem tests the 'Sign-Consistency Principle' in integration alongside the 'Geometry of Complex Inequalities'.

The trap lies in evaluating the area integral: students often ignore the modulus sign of the quadratic $f(x)$ and integrate directly over $[-1, 3]$ , yielding the distractor $32/3$ .

The complex parameter $w$ acts as a rigorous filter; equating the polynomials gives two cases, but the geometric constraint $|w + 2| \le 1$ uniquely isolates $w = -2 - i$ , forcing the specific quadratic that requires splitting the integral at its roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free