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Question #40623

Difficulty: 8/10 | 🟢 Free

The Question

Let $i = \sqrt{-1}$ . Define the complex numbers $A$ and $B$ as: $A = \frac{1}{16} \sum_{k=0}^{10} \binom{10}{k} i^k, \quad B = -\frac{5}{16} \sum_{k=0}^{8} \binom{8}{k} i^k$ Let $z_1$ and $z_2$ be the roots of the quadratic equation $z^2 - Az + B = 0$ . Find the value of $|z_1 - \bar{z_2}|^2 + |z_2 - \bar{z_1}|^2$ .

Options

A.
0
B.
16
C.
32
D.
40

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

Binomial Coefficient Summation Properties|Quadratic Equations with Complex Coefficients|Conjugate Root Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of binomial sums with complex terms and the properties of quadratic equations with complex coefficients.

The primary trap lies in the interpretation of the discriminant and root symmetries.

Students often memorize that a strictly positive real discriminant ( $\Delta = 16 > 0$ ) implies two distinct real roots, leading them to incorrectly calculate the symmetric sum as if $z_1, z_2 \in \mathbb{R}$ (yielding 32).

Alternatively, seeing a complex coefficient, they might falsely assume the Conjugate Root Theorem applies, setting $z_2 = \bar{z_1}$ (yielding 0).

Both heuristics fail because the coefficients are non-real, invalidating the standard real-coefficient discriminant conditions and conjugate root symmetries.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Coefficient Summation Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free