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Question #41850

Difficulty: 8/10 | 🟢 Free

The Question

A point $(x,y)$ is chosen at random from the sample space $S$ consisting of all points with integer coordinates that satisfy the constraints $x \ge 0$ , $y \ge 0$ , and $2x + y \le 10$ . Let $E$ be the event that the chosen point lies in the region defined by $y \ge x^2 - 3x + 4 \text{sgn}(x)$ , where $\text{sgn}(\cdot)$ denotes the signum function. Find the probability of $E$ .

Options

A.
$\frac{2}{3}$
B.
$\frac{5}{9}$
C.
$\frac{8}{9}$
D.
$\frac{13}{25}$

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

Multiplication Principle Scope|Combinatorial Probability|Signum Function Discontinuity|Unbounded Region Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously evaluate piecewise functions at boundary conditions within a combinatorially defined sample space.

The specific trap lies in the evaluation of the signum function, $\text{sgn}(x)$ , at the origin.

Because the sample space is restricted by the Linear Programming constraint $x \ge 0$ , students commonly and incorrectly assume that $\text{sgn}(x) = 1$ across the entire domain, failing to recognize the jump discontinuity at $x = 0$ where $\text{sgn}(0) = 0$ .

This erroneous omission of the boundary case leads to an undercounting of valid integer coordinates on the y-axis, trapping them into selecting the incorrect probability of 5/9.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Multiplication Principle Scope
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free