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Question #41727

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a random variable representing the number of successes in $n = 280$ independent Bernoulli trials. The probability of success in a single trial is given by $p$ , where $p = \int_0^{\pi/2} \frac{2\sin^{280} x + \sin(2x)}{2(\sin^{280} x + \cos^{280} x + \sin(2x))} \, dx$ Find the value of $\frac{16}{\pi^2} E[X^2 - X]$ .

Options

A.
78120
B.
78400
C.
39060
D.
19530

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Random Variable Expectation|Bernoulli Trials Logic|Definite Integral Properties (King's Property)

Why This Question Is Tricky

Common Mistakes

This question tests the application of King's Property in definite integrals to find a transcendental probability parameter $p$ , followed by the computation of the second factorial moment of a Binomial distribution.

The primary trap lies in the algebraic fatigue and potential for error if students attempt to calculate $E[X^2]$ and $E[X]$ separately, as the transcendental nature of $p = \pi/4$ creates messy intermediate terms (like $70\pi$ ) that must perfectly cancel.

Furthermore, the distracting constant '280' in the integral's exponents is designed to break perceived symmetry and intimidate, while option B ( $78400$ ) traps students who incorrectly approximate $E[X^2 - X]$ as $n^2p^2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Random Variable Expectation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free