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Question #41059

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be two independent events associated with a random experiment such that $P(A) = P(B) = p$ , where $p \in (0, 1)$ . A curve $y = f(x)$ in the first quadrant passes through the point $(1, 1)$ and satisfies the differential equation $(P(A \cup B) x^2 + P(A \cap B) y^2) dx - P(A) xy dy = 0$ If the curve passes through the point $(2, 2\sqrt{2})$ , then the value of $p$ is

Options

A.
$\frac{1}{4}$
B.
$\frac{1}{3}$
C.
$\frac{1}{2}$
D.
$\frac{2}{3}$

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

Mutually Exclusive vs Independent|Homogeneous Function Homogeneity|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of probability axioms and homogeneous differential equations.

The core trap lies in the implicit assumption of mutual exclusivity versus independence.

If a student incorrectly assumes $A$ and $B$ are mutually exclusive (a common error when rushing), $P(A \cap B) = 0$ , causing the $y^2$ term to vanish.

The differential equation's structure collapses into $2x dx - y dy = 0$ , eliminating the parameter $p$ entirely and leading to a contradictory curve $2x^2 - y^2 = 1$ that cannot pass through $(2, 2\sqrt{2})$ .

Correctly applying independence preserves the homogeneity and the parameter $p$ , requiring careful integration and solving the resulting transcendental equation $p = 2^{2p-2}$ .

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