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Question #40492

Difficulty: 8/10 | 🟢 Free

The Question

For each $n \in \mathbb{N}$ , let $\alpha_n$ and $\beta_n$ be the roots of the polynomial equation: $z^2 - \left( 2 + \frac{i}{n} + \frac{\sin(n^2)}{n^2} \right) z + \left( 1 + \frac{i}{n} + \frac{\sin(n^2)}{n^2} \right) = 0$ Let $a_n = \alpha_n^n + \beta_n^n$ . Find $\lim_{n \to \infty} a_n$ .

Options

A.
$1 + e^i$
B.
$2\cos 1$
C.
$\text{Does not exist}$
D.
$e^i$

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

Conjugate Root Theorem|Quadratic Roots in Sequences|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests two major JEE Advanced traps.

First, the Conjugate Root Theorem trap: students often blindly assume complex roots occur in conjugate pairs, which is only true for polynomials with real coefficients.

Doing so here yields the incorrect limit $2\cos 1$ .

Second, L'Hopital's Pitfall: evaluating the indeterminate limit $(1+u_n)^n$ by converting to a continuous variable $x$ and applying L'Hopital's rule fails because the derivative of the oscillating term $\sin(x^2)/x^2$ diverges.

Students must use Taylor expansion or the squeeze theorem to find the true limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Conjugate Root Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free