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Question #41722

Difficulty: 8/10 | 🟢 Free

The Question

Let $p$ be a valid probability of success for a Bernoulli trial such that the function $f(x) = 16x^4 - 48x^3 + 46x^2 - 15x$ attains a local maximum at $x=p$ . An experiment consists of repeatedly performing independent Bernoulli trials with success probability $p$ until 2 consecutive successes occur. Let $E$ be the expected number of trials required. Find the value of $9E$ .

Options

A.
24
B.
28
C.
54
D.
180

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

Bernoulli Trials Logic|First Derivative Test Trap|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate calculus with probability while navigating two distinct traps.

First, the Extraneous Solution Rejection trap requires students to discard the critical point $x = 5/4$ because a valid probability must satisfy $p \in [0,1]$ .

Second, the First Derivative Test Trap forces students to classify the remaining critical points ( $1/4$ and $3/4$ ) using the second derivative, ensuring they select the local maximum ( $3/4$ ) rather than the local minimum ( $1/4$ ).

Finally, it requires the application of the total expectation theorem to find the expected number of trials for consecutive successes.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bernoulli Trials Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free