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Question #41835

Difficulty: 8/10 | 🟢 Free

The Question

Three distinct numbers $a, b,$ and $c$ are chosen randomly from the set $S = \{0, 1, 2, 3, 4, 5\}$ . What is the probability that the equation $ax^2 + bx + c = 0$ has exactly two distinct real roots?

Options

A.
$\frac{17}{60}$
B.
$\frac{9}{20}$
C.
$\frac{7}{24}$
D.
$\frac{4}{15}$

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

Combinatorial Probability|Quadratic Roots in Sequences|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the 'Extraneous Solution Rejection' trap in the context of combinatorial probability.

Students commonly associate 'two distinct real roots' exclusively with the discriminant condition $\Delta > 0$ , forgetting the fundamental prerequisite that the leading coefficient must be non-zero ( $a \neq 0$ ) for the equation to be quadratic.

If $a=0$ , the equation collapses to a linear form $bx+c=0$ , which has only one root.

Blindly applying $b^2 - 4ac > 0$ includes 20 extraneous cases where $a=0$ (since $b^2 > 0$ is true), leading to the incorrect probability of $9/20$ .

Recognizing the degree collapse is a classic JEE Advanced discriminator.

Quick Facts About This Question

Question Information

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