Sets, Relations & Functions
Sets and Relations
nta_pyq_2025_jan
Grade 11
Question:
Define a relation R on the interval [0, \pi2 ) by xRy if and only if sec 2 x - tan 2 y = 1. Then R is :
both reflexive and transitive but not symmetric
an equivalence relation
reflexive but neither symmetric not transitive
both reflexive and symmetric but not transitive
Step-by-Step Solution
Key Concept: Apply the core result for equivalence relations and set operations and simplify using the given constraints.
2 2 sec x - tan x = 1 ( on replacing y with x) (2) \Rightarrow Reflexive 2 2 sec x - tan y = 1 2 2 \Rightarrow 1 + tan x + 1 - sec y = 1 2 2 \Rightarrow sec y - tan x = 1 \Rightarrow symmetric 2 2 sec x - tan y = 1 2 2 sec y - tan z = 1 Adding both 2 2 2 2 \Rightarrow sec x - tan y + sec y - tan z = 1 + 1 2 2 sec x + 1 - tan z = 2 2 2 sec x - tan z = 1 \Rightarrow Transitive hence equivalence releation
Correct Answer: 2