Let R be the relation over the set A of all straight lines in a plane such that \(l_1\,R\,l_2 \Leftrightarrow l_1\) is parallel to \(l_2\). Then R is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → An Equivalence relation
Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (1), Option (2), Option (3) and Option (4) are all correct, as explained below.
As usual for this relation, every line is taken to be parallel to itself.
Reflexive: \(l \parallel l\), so \((l, l) \in R\) for every line l
Symmetric: \(l_1 \parallel l_2 \Rightarrow l_2 \parallel l_1\)
Transitive: \(l_1 \parallel l_2\) and \(l_2 \parallel l_3 \Rightarrow l_1 \parallel l_3\)
A relation that is reflexive, symmetric and transitive is an equivalence relation.
Option (1) Symmetric — Should also be correct, though not the answer given by NTA. If \(l_1\) is parallel to \(l_2\), then \(l_2\) is parallel to \(l_1\).
Option (2) An Equivalence relation — Correct. R has all three properties: reflexive, symmetric and transitive.
Option (3) Transitive — Should also be correct, though not the answer given by NTA. Two lines parallel to the same line are parallel to each other.
Option (4) Reflexive — Should also be correct, though not the answer given by NTA. Every line is parallel to itself.
Advice for the exam: When more than one option is correct, mark the one that answers the question most completely, here Option (2), An Equivalence relation, which already includes the other three properties. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.