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Question

For any two real numbers θ and ϕ, we define θRϕ, if and only if sec2θtan2ϕ=1.

The relation R is


A

reflexive but not transitive

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B

symmetric but not reflexive

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C

both reflexive and symmetric but not transitive

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D

an equivalence relation

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Solution

The correct option is D

an equivalence relation


Explanation for the correct option:

Step 1. For Reflexive:

Given relation is defined as

θRϕ such that sec2θtan2ϕ=1

When θRθ

sec2θtan2θ=1

1=1, which is true.

Thus, it is reflexive.

Step 2. For Symmetric:

When θRϕ

sec2θtan2ϕ=1

(1+tan2θ)(sec2ϕ1)=1

2+tan2θsec2ϕ=1

sec2ϕtan2θ=1

ϕRθ

Thus, it is symmetric.

Step 3. For Transitive:

When θRϕ and ϕRψ, then

sec2θtan2ϕ=1

and sec2ϕtan2ψ=1

Now, θRψ

Then, sec2θtan2ψ=1

sec2θtan2ψ+1=1+1

sec2θtan2ψ+sec2ϕtan2ϕ=1+1

θRϕandϕRψ

Thus, it is transitive.

It is an equivalence relation

Hence, Option ‘D’ is Correct.


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