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Question #41081

Difficulty: 8/10 | 🟢 Free

The Question

An urn contains 5 Red and 3 Blue balls. Three balls are drawn successively without replacement. Let $p$ be the probability that the balls are drawn in the exact order Red-Blue-Red. Let $k = 28p$ .

A function $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = |x - k| + |x - 2k|$ .

Let $g(x) = f(f(x))$ . If $S$ is the product of all points $x \in \mathbb{R}$ where $g(x)$ is non-differentiable, find the value of $4S$ .

Options

A.
125
B.
200
C.
100
D.
150

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

Multiplication Theorem|Random Variable Mapping|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question fuses the Multiplication Theorem of probability with the calculus of piecewise functions, specifically targeting the 'Differentiability vs Continuity' trap in composite functions.

Students typically assume that if $f(x)$ is non-differentiable at $x=c$ , then $f(f(x))$ is also non-differentiable at $x=c$ .

However, the outer function can 'smooth out' the non-differentiability if it maps the neighborhood to a constant value.

Here, $g(x)$ is perfectly differentiable at the critical points of $f(x)$ ( $k$ and $2k$ ), and the true points of non-differentiability emerge only where $f(x) = 2k$ .

Failing to perform a rigorous LHD/RHD smoothness check leads to the incorrect product of 200 instead of 125.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Multiplication Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free