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Question #40478

Difficulty: 8/10 | 🟢 Free

The Question

A point $P$ in the complex plane traces a locus given by $z(t) = x(t) + i y(t)$ , where $x(t) = (t^2 - 2t)\cos t - (t^2 - 4)\sin t$ $y(t) = (t^2 - 2t)\sin t + (t^2 - 4)\cos t$ for $t \in [-2, 3]$ . Let $M$ and $m$ be the absolute maximum and absolute minimum values of $|z(t)|$ , respectively. Find the value of $M^2 + m^2$ .

Options

A.
64
B.
16
C.
98
D.
34

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Euler's Form and Rotations|Locus of a Moving Point|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

This question tests the ability to recognize Euler's form hidden in trigonometric expansions and optimize the modulus of a complex locus.

The core trap lies in the distinction between a stationary point and an absolute extremum on a closed interval.

Students often find the derivative of the distance function, identify the local maximum at $t=0$ ( $|z|^2=16$ ), and incorrectly assume it is the absolute maximum, leading to the trap answer 16.

The true absolute maximum occurs at the boundary $t=-2$ ( $|z|^2=64$ ), forcing a rigorous check of boundary conditions alongside critical points.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Euler's Form and Rotations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free