Class 11 Mathematics - Relations And Functions - Real valued functions, domain and range of these functions Expert Quiz

Mathematics Relations And Functions Real valued functions, domain and range of these functions Expert Class 11 Level 27

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Question 1/3 Expert Mathematics Relations and Functions Relations Class 11 Level 27

वास्तविक संख्याओं के समुच्चय \(\mathbb{R}\) पर (aRb) तभी जब \(a-b\in\mathbb{Z}\)। (R) की प्रकृति क्या है?

On the set of real numbers \(\mathbb{R}\), (aRb) if and only if \(a-b\in\mathbb{Z}\). What is the nature of (R)?

Explanation opens after your attempt
Correct Answer

A. तुल्यता संबंधEquivalence relation

Step 1

Concept

Since \(a-a=0\in\mathbb{Z}\), \(a-b\in\mathbb{Z}\) implies \(b-a\in\mathbb{Z}\), and the sum of integers is an integer. Hence it is an equivalence relation.

Step 2

Why this answer is correct

The correct answer is A. तुल्यता संबंध / Equivalence relation. Since \(a-a=0\in\mathbb{Z}\), \(a-b\in\mathbb{Z}\) implies \(b-a\in\mathbb{Z}\), and the sum of integers is an integer. Hence it is an equivalence relation.

Step 3

Exam Tip

\(a-a=0\in\mathbb{Z}\), \(a-b\in\mathbb{Z}\) से \(b-a\in\mathbb{Z}\), और पूर्णांकों का योग पूर्णांक है। इसलिए यह तुल्यता संबंध है।

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Question 2/3 Expert Mathematics Relations and Functions Relations Class 11 Level 27

समुच्चय \(\mathbb{R}\) पर (aRb) तभी जब \(a^2=b^2\)। (R) के बारे में सही कथन कौन सा है?

On \(\mathbb{R}\), (aRb) if and only if \(a^2=b^2\). Which statement about (R) is correct?

Explanation opens after your attempt
Correct Answer

A. यह तुल्यता संबंध हैIt is an equivalence relation

Step 1

Concept

Since \(a^2=a^2\), equality is symmetric, and \(a^2=b^2\), \(b^2=c^2\) imply \(a^2=c^2\). Hence (R) is an equivalence relation.

Step 2

Why this answer is correct

The correct answer is A. यह तुल्यता संबंध है / It is an equivalence relation. Since \(a^2=a^2\), equality is symmetric, and \(a^2=b^2\), \(b^2=c^2\) imply \(a^2=c^2\). Hence (R) is an equivalence relation.

Step 3

Exam Tip

\(a^2=a^2\), equality symmetric होती है, और \(a^2=b^2\), \(b^2=c^2\) से \(a^2=c^2\)। इसलिए (R) तुल्यता संबंध है।

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Question 3/3 Expert Mathematics Relations and Functions Relations Class 11 Level 27

\(\mathbb{R}\) पर (aRb) तभी जब \(a^3\leq b^3\)। यह relation किस relation के समान है?

On \(\mathbb{R}\), (aRb) if and only if \(a^3\leq b^3\). This relation is equivalent to which relation?

Explanation opens after your attempt
Correct Answer

A. \(a\leq b\)

Step 1

Concept

The function \(x^3\) is strictly increasing, so \(a^3\leq b^3\) is exactly equivalent to \(a\leq b\). Use monotonicity to identify relation properties quickly.

Step 2

Why this answer is correct

The correct answer is A. \(a\leq b\). The function \(x^3\) is strictly increasing, so \(a^3\leq b^3\) is exactly equivalent to \(a\leq b\). Use monotonicity to identify relation properties quickly.

Step 3

Exam Tip

Function \(x^3\) strictly increasing है, इसलिए \(a^3\leq b^3\) exactly \(a\leq b\) के बराबर है। Monotonicity से relation की property जल्दी पहचानें।

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