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Question #41110

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the family of equations $E_a : a x^2 + (a^2 + 4) x + 4a = 0$ for $a \in \mathbb{R}$ . A relation $R$ is defined on $S$ such that $E_a R E_b$ if and only if $E_a$ and $E_b$ have the exact same set of real roots. Let $[E_a]$ denote the equivalence class of $E_a$ under $R$ . Find the number of values of $a \in \mathbb{R}$ for which $[E_a]$ contains exactly one element.

Options

A.
1
B.
2
C.
3
D.
4

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

Transitivity of Identity|Condition for Common Roots|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation' trap within the context of equivalence relations and root sets.

Students typically factor the quadratic assuming a ≠ 0, finding the root set {-a, -4/a}.

Equating sets gives b = 4/a, leading them to find only the repeated root cases a = ±2 (where the class size is 1).

By implicitly assuming the leading coefficient is non-zero, they fail to evaluate the singular case a = 0.

At a = 0, the equation collapses to a linear space with root set {0}, forming an isolated equivalence class [E_0].

Missing this transition from quadratic to linear space causes students to incorrectly answer 2 instead of 3.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Transitivity of Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free