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Question #41039

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ and $Y$ be independent continuous random variables, each having the probability density function $f(x)$ defined on the interval $[0, 2]$ by: $f(x) = \frac{d^2}{dx^2} \left( \int_{0}^{x} \frac{(x - t)^2}{2} |t - 1| dt \right)$ Let $E_1$ be the event that the quadratic equation $z^2 - 2Xz + X = 0$ has real roots, and $E_2$ be the event that the quadratic equation $z^2 - 2Yz + Y = 0$ has real roots. Find the probability of the event $E_1 \cup E_2$ .

Options

A.
$\frac{8}{9}$
B.
$\frac{3}{4}$
C.
$\frac{4}{9}$
D.
$\frac{5}{9}$

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|Leibniz Rule for Differentiation under Integral|Second Order Derivative Continuity|The Sign-Consistency Principle

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Leibniz Rule for differentiation under the integral sign and the Multiplication Theorem of probability, while embedding a critical Sign-Consistency trap.

Students frequently misapply the Leibniz rule on higher-order integral functions, incorrectly assuming the second derivative yields the integrand $|x-1|$ directly, which leads to a normalized probability of 3/4.

Furthermore, even if differentiated correctly, the modulus function $|t-1|$ inside the integral forces a piecewise domain split at $t=1$ .

Failing to account for this non-differentiability point and blindly integrating $(t-1)$ violates the sign-consistency principle, yielding an incorrect probability density function and a final answer of 5/9.

The problem rigorously filters candidates who lack precision in handling piecewise continuous derivatives within probabilistic frameworks.

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