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Question #41361

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha, \beta \in \mathbb{C}$ be the roots of the quadratic equation $z^2 - (4-2i)z + (4-3i) = 0$ . Consider the complex-valued function $f(t) = t^3 - (\alpha+\beta)t^2 + \alpha\beta t$ defined for a real parameter $t \in [0, 2]$ . The function $f(t)$ traces a curve in the complex plane. If the tangent vector to this curve at $t = c$ (where $0 < c < 2$ ) is parallel to the chord joining the endpoints $f(0)$ and $f(2)$ , find the value of $15c$ .

Options

A.
8
B.
10
C.
12
D.
15

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

LMVT Tangent Parallelism|Quadratic Equations with Complex Coefficients|Vieta’s Formulas in Complex Domain

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Vieta's formulas in the complex domain and the geometric interpretation of the Mean Value Theorem (MVT).

The primary trap lies in blindly applying the algebraic LMVT formula $f'(c) = \frac{f(b)-f(a)}{b-a}$ to a complex-valued function.

If a student does this, they obtain $(3c^2 - 8c + 4) + i(4c - 3) = i$ , which yields a contradictory system ( $c=2/3$ or $2$ for the real part, but $c=1$ for the imaginary part), leading to confusion.

The theorem fails in the complex plane.

To resolve this, the student must rigorously decompose the derivative into real and imaginary components and apply the geometric condition of parallelism (cross product of vectors is zero, or in this case, the real part must vanish), successfully bypassing the trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	LMVT Tangent Parallelism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free