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Question #41014

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \\begin{cases} ax + b & \\text{if } x \\ge 0 \\\\ cx^2 + d & \\text{if } x < 0 \\end{cases}$ and $g(x) = x^2 - 1$ . A 4-tuple $(a, b, c, d)$ is chosen uniformly at random from the set $S = \\{-2, -1, 0, 1, 2\\}^4$ . Let $A$ be the event that the composite function $h(x) = f(g(x))$ is continuous at $x = 1$ , and $B$ be the event that $h(x)$ is differentiable at $x = 1$ . Which of the following statements is correct?

Options

A.
$P(A \\cap B^c) = \\frac{4}{25}$
B.
$P(B|A) = \\frac{1}{25}$
C.
$P(A|B) = \\frac{1}{5}$
D.
Events $A$ and $B$ are independent

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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

Mutually Exclusive vs Independent|Differentiability vs Continuity|Composition Continuity

Why This Question Is Tricky

Common Mistakes

This question bridges probability and calculus by defining events based on the continuity and differentiability of a piecewise composite function.

The primary trap tests the logical implication that differentiability implies continuity ( $B \\subset A$ ), meaning the events cannot be probabilistically independent unless trivial.

Students often mistakenly calculate $P(A \\cap B) = P(A)P(B)$ , failing to recognize the set-theoretic subset relationship.

Additionally, evaluating the composite function $f(g(x))$ at the critical point $x=1$ requires careful tracking of the inner function's sign to correctly apply the piecewise definitions, penalizing students who blindly apply the chain rule without checking one-sided limits.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Mutually Exclusive vs Independent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free