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Question #40631

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n$ be the set of all polynomials $Q(x)$ of degree $n$ with real coefficients such that the coefficient of $x^n$ is $1$ . Let $m_n = \min_{Q \in S_n} \int_{0}^{1} (Q(x))^2 dx$ . Find the value of $\lim_{n \to \infty} n \cdot m_n \cdot \binom{2n}{n}^2$ .

Options

A.
$\frac{1}{4}$
B.
$\frac{1}{2}$
C.
1
D.
2

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

Coefficient of x^r in Expansion|Symmetry Utilization|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of orthogonal polynomials, integral inequalities, and exact coefficient extraction.

The trap lies in attempting to bound the integral using standard inequalities (like Cauchy-Schwarz on the polynomial coefficients), which yields a loose lower bound that fails to evaluate the limit.

The 'Lethal Collision' occurs because students must realize that the minimum norm is achieved by the orthogonal projection (shifted Legendre polynomial), requiring them to extract its exact leading coefficient via Rodrigues' formula and evaluate the integral using repeated integration by parts to find the exact limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Coefficient of x^r in Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free