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Question #41106

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all ellipses $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ) such that their eccentricity $e$ satisfies $\int_{0}^{e} |3t - 1| \, dt = \frac{5}{6}$ A relation $R$ is defined on $S$ by $E_1 R E_2$ if and only if the area of $E_1$ is strictly less than the area of $E_2$ . Which of the following statements is true about the relation $R$ ?

Options

A.
$R$ is an equivalence relation.
B.
$R$ is transitive, but neither reflexive nor symmetric.
C.
$R$ is symmetric and transitive, but not reflexive.
D.
$R$ is reflexive and transitive, but not symmetric.

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

Reflexivity in Empty Sets|Standard Ellipse/Circle Area|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' combined with 'Reflexivity in Empty Sets'.

Students must correctly evaluate the absolute value integral to find the eccentricity.

The trap lies in the fact that the calculated eccentricities ( $e=1, -1/3$ ) fall outside the valid geometric range for an ellipse ( $0 < e < 1$ ), making the set of such ellipses empty.

Students who fail to check the existence of the locus will blindly apply the properties of the strict inequality relation ( $<$ ) and incorrectly conclude it is transitive but not reflexive or symmetric.

However, on an empty set, the empty relation is vacuously reflexive, symmetric, and transitive, making it an equivalence relation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Reflexivity in Empty Sets
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free