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Mathematics The Empty Set, Finite and Infinite Sets, Equal Sets MCQ Questions for Class 11 General

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The Empty Set, Finite and Infinite Sets, Equal Sets Practice Questions

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Question 331/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(A_1={x:x\in \mathbb{N}, x\) (10) और (15) दोनों का गुणज है(}) है, तो \(A_1\) कैसा है?

If \(A_1={x:x\in \mathbb{N}, x\) is a multiple of both (10) and (15)(}), what type of set is \(A_1\)?

Explanation opens after your attempt

Correct Answer

C. अपरिमितInfinite

Step 1

Concept

(10) और (15) दोनों के गुणज (30) के गुणज होंगे। / Multiples common to (10) and (15) are multiples of (30).

Step 2

Why this answer is correct

\(30,60,90,\ldots\) की सूची अनंत है। / The list \(30,60,90,\ldots\) is endless.

Step 3

Exam Tip

बिना ऊपरी सीमा के संयुक्त गुणजों का समुच्चय भी अपरिमित होता है। / Without an upper bound, the set of common multiples is also infinite.

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Question 332/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(B_1={x:x\in \mathbb{N}, x\) (10) और (15) दोनों का गुणज है तथा (x<100}) है, तो \(B_1\) में कितने अवयव हैं?

If \(B_1={x:x\in \mathbb{N}, x\) is a multiple of both (10) and (15) and (x<100}), how many elements are in \(B_1\)?

Explanation opens after your attempt

Correct Answer

B. (3)

Step 1

Concept

(10) और (15) के संयुक्त गुणज (30) के गुणज हैं। / Common multiples of (10) and (15) are multiples of (30).

Step 2

Why this answer is correct

(100) से छोटे ऐसे गुणज (30,60,90) हैं। / Below (100), these are (30,60,90).

Step 3

Exam Tip

ऊपरी सीमा लगने पर संयुक्त गुणजों की गिनती समाप्त हो जाती है। / With an upper bound, the counting of common multiples ends.

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Question 333/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(C_1={x:x\in \mathbb{N}, x\) (12) और (18) दोनों का गुणनखंड है(}) है, तो \(C_1\) क्या है?

If \(C_1={x:x\in \mathbb{N}, x\) is a factor of both (12) and (18)(}), what is \(C_1\)?

Explanation opens after your attempt

Correct Answer

A. \(C_1={1,2,3,6}\)

Step 1

Concept

(12) के गुणनखंड (1,2,3,4,6,12) हैं। / Factors of (12) are (1,2,3,4,6,12).

Step 2

Why this answer is correct

(18) के गुणनखंड (1,2,3,6,9,18) हैं, इसलिए साझा गुणनखंड (1,2,3,6) हैं। / Factors of (18) are (1,2,3,6,9,18), so common factors are (1,2,3,6).

Step 3

Exam Tip

संयुक्त गुणनखंड हमेशा सीमित संख्या में होते हैं। / Common factors are always finite in number.

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Question 334/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(D_1={x:x\in \mathbb{N}, x\) (12) और (18) दोनों का गुणनखंड है तथा (x>6}) है, तो \(D_1\) क्या है?

If \(D_1={x:x\in \mathbb{N}, x\) is a factor of both (12) and (18) and (x>6}), what is \(D_1\)?

Explanation opens after your attempt

Correct Answer

C. \(D_1=\varnothing\)

Step 1

Concept

(12) और (18) के साझा गुणनखंड (1,2,3,6) हैं। / The common factors of (12) and (18) are (1,2,3,6).

Step 2

Why this answer is correct

इनमें कोई भी (6) से बड़ा नहीं है। / None of these is greater than (6).

Step 3

Exam Tip

साझा गुणनखंड मिलने के बाद अतिरिक्त असमानता जरूर लगाएँ। / After finding common factors, apply the extra inequality condition.

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Question 335/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(E_1={x:x\in \mathbb{Z}, x^2+4=4}\) है, तो \(E_1\) कैसा है?

If \(E_1={x:x\in \mathbb{Z}, x^2+4=4}\), what type of set is \(E_1\)?

Explanation opens after your attempt

Correct Answer

B. एकलSingleton

Step 1

Concept

\(x^2+4=4\) से \(x^2=0\) मिलता है। / \(x^2+4=4\) gives \(x^2=0\).

Step 2

Why this answer is correct

इसका एकमात्र पूर्णांक हल (x=0) है। / Its only integer solution is (x=0).

Step 3

Exam Tip

\(x^2=0\) में धनात्मक और ऋणात्मक दो अलग हल नहीं होते। / For \(x^2=0\), there are not two different positive and negative roots.

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Question 336/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(F_1={x:x\in \mathbb{R}, x^2+4=0}\) है, तो \(F_1\) क्या है?

If \(F_1={x:x\in \mathbb{R}, x^2+4=0}\), what is \(F_1\)?

Explanation opens after your attempt

Correct Answer

A. रिक्त समुच्चयEmpty set

Step 1

Concept

वास्तविक संख्या के लिए \(x^2\geq 0\) होता है। / For a real number, \(x^2\geq 0\).

Step 2

Why this answer is correct

इसलिए \(x^2+4=0\) किसी वास्तविक (x) के लिए संभव नहीं है। / Therefore, \(x^2+4=0\) is impossible for real (x).

Step 3

Exam Tip

वास्तविक वर्ग कभी ऋणात्मक नहीं होता, यह रिक्त समुच्चय पहचानने में मदद करता है। / A real square is never negative, which helps identify an empty set.

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Question 337/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(G_1={x:x\in \mathbb{Q}, x^2=2}\) है, तो \(G_1\) किस प्रकार का समुच्चय है?

If \(G_1={x:x\in \mathbb{Q}, x^2=2}\), what type of set is \(G_1\)?

Explanation opens after your attempt

Correct Answer

C. रिक्तEmpty

Step 1

Concept

\(x^2=2\) के वास्तविक हल \(\sqrt{2}\) और \(-\sqrt{2}\) हैं। / The real solutions of \(x^2=2\) are \(\sqrt{2}\) and \(-\sqrt{2}\).

Step 2

Why this answer is correct

ये परिमेय संख्याएँ नहीं हैं, इसलिए \(G_1\) में कोई अवयव नहीं होगा। / These are not rational numbers, so \(G_1\) has no element.

Step 3

Exam Tip

\(\mathbb{Q}\) की शर्त हो तो अपरिमेय हल स्वीकार नहीं करें। / When the domain is \(\mathbb{Q}\), do not accept irrational solutions.

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Question 338/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

यदि \(H_1={x:x\in \mathbb{R}, x^2=2}\) और \(I_1={-\sqrt{2},\sqrt{2}}\) हैं, तो सही कथन कौन-सा है?

If \(H_1={x:x\in \mathbb{R}, x^2=2}\) and \(I_1={-\sqrt{2},\sqrt{2}}\), which statement is correct?

Explanation opens after your attempt

Correct Answer

A. \(H_1=I_1\)

Step 1

Concept

वास्तविक संख्याओं में \(x^2=2\) के दो हल \(-\sqrt{2}\) और \(\sqrt{2}\) हैं। / In real numbers, \(x^2=2\) has two solutions, \(-\sqrt{2}\) and \(\sqrt{2}\).

Step 2

Why this answer is correct

यही दोनों अवयव \(I_1\) में हैं। / These are exactly the elements of \(I_1\).

Step 3

Exam Tip

मूल चिह्न वाले अवयव भी समुच्चय के वैध अवयव हो सकते हैं। / Elements involving radicals can also be valid set elements.

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Question 339/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

कौन-सा समुच्चय परिमित है लेकिन रिक्त नहीं है?

Which set is finite but not empty?

Explanation opens after your attempt

Correct Answer

C. \( {x:x\in \mathbb{N}, x\mid 24} \)

Step 1

Concept

\(x\mid 24\) का अर्थ है (x), (24) का गुणनखंड है। / \(x\mid 24\) means (x) is a factor of (24).

Step 2

Why this answer is correct

(24) के प्राकृतिक गुणनखंड सीमित और उपस्थित हैं, इसलिए समुच्चय परिमित और रिक्त नहीं है। / Natural factors of (24) exist and are limited, so the set is finite and non-empty.

Step 3

Exam Tip

निश्चित संख्या के गुणनखंडों का समुच्चय हमेशा सीमित होता है। / The set of factors of a fixed number is always finite.

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Question 340/603 Hard Mathematics Sets The Empty Set, Finite and Infinite Sets, Equal Sets Class 11 Level 4

कौन-सा समुच्चय अपरिमित है लेकिन सभी पूर्णांक नहीं है?

Which set is infinite but not the set of all integers?

Explanation opens after your attempt

Correct Answer

A. \( {x:x\in \mathbb{Z}, x>0} \)

Step 1

Concept

(x>0) वाले पूर्णांक \(1,2,3,\ldots\) हैं। / Integers with (x>0) are \(1,2,3,\ldots\).

Step 2

Why this answer is correct

यह सूची अपरिमित है, पर इसमें ऋणात्मक पूर्णांक नहीं हैं। / This list is infinite, but it does not include negative integers.

Step 3

Exam Tip

अपरिमित होना और सभी पूर्णांक होना एक ही बात नहीं है। / Being infinite is not the same as being the set of all integers.

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Mathematics The Empty Set, Finite and Infinite Sets, Equal Sets MCQ Questions for Class 11 General

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The Empty Set, Finite and Infinite Sets, Equal Sets Practice Questions

यदि \(A_1={x:x\in \mathbb{N}, x\) (10) और (15) दोनों का गुणज है(}) है, तो \(A_1\) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि \(B_1={x:x\in \mathbb{N}, x\) (10) और (15) दोनों का गुणज है तथा (x<100}) है, तो \(B_1\) में कितने अवयव हैं?

Concept

Why this answer is correct

Exam Tip

यदि \(C_1={x:x\in \mathbb{N}, x\) (12) और (18) दोनों का गुणनखंड है(}) है, तो \(C_1\) क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(D_1={x:x\in \mathbb{N}, x\) (12) और (18) दोनों का गुणनखंड है तथा (x>6}) है, तो \(D_1\) क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(E_1={x:x\in \mathbb{Z}, x^2+4=4}\) है, तो \(E_1\) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि \(F_1={x:x\in \mathbb{R}, x^2+4=0}\) है, तो \(F_1\) क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(G_1={x:x\in \mathbb{Q}, x^2=2}\) है, तो \(G_1\) किस प्रकार का समुच्चय है?

Concept

Why this answer is correct

Exam Tip

यदि \(H_1={x:x\in \mathbb{R}, x^2=2}\) और \(I_1={-\sqrt{2},\sqrt{2}}\) हैं, तो सही कथन कौन-सा है?

Concept

Why this answer is correct

Exam Tip

कौन-सा समुच्चय परिमित है लेकिन रिक्त नहीं है?

Concept

Why this answer is correct

Exam Tip

कौन-सा समुच्चय अपरिमित है लेकिन सभी पूर्णांक नहीं है?

Concept

Why this answer is correct

Exam Tip

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