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Question #41707

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, \omega, \omega^2\}$ , where $\omega = e^{i2\pi/3}$ . Let $\mathcal{F}$ be the set of all quadratic equations of the form $z^2 + az + b = 0$ , where $a, b \in S$ . If two distinct equations are chosen at random from $\mathcal{F}$ , the probability that they have at least one common root is $\frac{p}{q}$ , where $p$ and $q$ are coprime positive integers. The value of $p+q$ is:

Options

A.
13
B.
17
C.
19
D.
37

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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

Sample Space Partitioning|Condition for Common Roots|Quadratic Equations with Complex Coefficients

Why This Question Is Tricky

Common Mistakes

This question tests the condition for common roots in polynomials with complex coefficients, combined with sample space partitioning.

The primary trap is attempting to use the standard resultant condition $(b_1-b_2)^2 + (a_1-a_2)(a_1b_2-a_2b_1) = 0$ algebraically, which leads to a massive, error-prone casework of 36 pairs.

The elegant Advanced-level bypass is to analyze the geometric properties of the roots of unity, realizing that the common root $\alpha = -\Delta b / \Delta a$ must be a unit vector, and specifically proving that negative roots of unity require a distance of 1 between elements of $S$ , which is geometrically impossible.

This restricts the sample space of common roots to $S$ itself, collapsing the problem into a simple set-matching exercise.

Quick Facts About This Question

Question Information

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