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Question #40484

Difficulty: 8/10 | 🟢 Free

The Question

Let $\omega$ be a non-real cube root of unity. Consider the polynomial expansion $P(x) = (1 + x\omega)^{20} + (1 + x\omega^2)^{20} = \sum_{r=0}^{20} A_r x^r$ If $x = \frac{2}{3}$ , find the value of $r$ for which the modulus of the $r$ -th degree term, $|A_r x^r|$ , is maximum.

Options

A.
6
B.
8
C.
9
D.
10

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

Roots of Unity Properties|Numerically Greatest Term|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Sign Dependency' trap in Numerically Greatest Term (NGT) problems.

Students typically apply the standard ratio test ( $T_{r+1}/T_r \ge 1$ ) to the binomial coefficients, which incorrectly yields $r=8$ .

However, the complex roots of unity introduce a fluctuating modulus factor that equals 2 when $r$ is a multiple of 3, and 1 otherwise.

This breaks the monotonicity of the sequence, forcing the student to manually check the amplified critical points ( $r=6, 9$ ) against the standard peak.

The true maximum shifts to $r=9$ because the $2\times$ amplification overcomes the natural decay of the binomial sequence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Roots of Unity Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free