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Mathematics Irrational numbers MCQ Questions for Class 10

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Question 451/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(x=\frac{\sqrt{10}+\sqrt{6}}{\sqrt{10}-\sqrt{6}}\), तो (x) का सरल रूप क्या है?

If \(x=\frac{\sqrt{10}+\sqrt{6}}{\sqrt{10}-\sqrt{6}}\), what is the simplified form of (x)?

Explanation opens after your attempt

Correct Answer

A. \(4+\sqrt{15}\)

Step 1

Concept

हर को परिमेय बनाने के लिए \(\sqrt{10}+\sqrt{6}\) से गुणा करें। / Multiply by \(\sqrt{10}+\sqrt{6}\) to rationalize the denominator.

Step 2

Why this answer is correct

ऊपर (\(\sqrt{10}+\sqrt{6}\)²=16+2\sqrt{60}) और नीचे (10-6=4) मिलता है, इसलिए मान \(4+\sqrt{15}\) है। / The numerator becomes (\(\sqrt{10}+\sqrt{6}\)²=16+2\sqrt{60}) and the denominator is (10-6=4), so the value is \(4+\sqrt{15}\).

Step 3

Exam Tip

संयुग्मी वाले भिन्नों में हर को पहले साफ करें। / In conjugate fractions, clear the denominator first.

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Question 452/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(x=\sqrt{5}+\sqrt{2}\) और \(y=\sqrt{5}-\sqrt{2}\), तो \(x^2+y^2\) का मान क्या है?

If \(x=\sqrt{5}+\sqrt{2}\) and \(y=\sqrt{5}-\sqrt{2}\), what is the value of \(x^2+y^2\)?

Explanation opens after your attempt

Correct Answer

A. (14)

Step 1

Concept

(x) और (y) संयुग्मी रूप में हैं। / (x) and (y) are conjugates.

Step 2

Why this answer is correct

(\(\sqrt{5}+\sqrt{2}\)²+\(\sqrt{5}-\sqrt{2}\)²) में बीच के अपरिमेय पद कट जाते हैं और मान (2(5+2)=14) मिलता है। / In (\(\sqrt{5}+\sqrt{2}\)²+\(\sqrt{5}-\sqrt{2}\)²), the middle irrational terms cancel and the value is (2(5+2)=14).

Step 3

Exam Tip

दो संयुग्मी वर्गों का योग लेते समय बीच वाले पद नहीं बचते। / When adding squares of conjugates, the middle terms vanish.

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Question 453/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

कौन-सा विकल्प \(\frac{\sqrt{12}+\sqrt{27}}{\sqrt{3}}\) का सही मान देता है?

Which option gives the correct value of \(\frac{\sqrt{12}+\sqrt{27}}{\sqrt{3}}\)?

Explanation opens after your attempt

Correct Answer

A. (5)

Step 1

Concept

\(\sqrt{12}=2\sqrt{3}\) और \(\sqrt{27}=3\sqrt{3}\) लिखें। / Write \(\sqrt{12}=2\sqrt{3}\) and \(\sqrt{27}=3\sqrt{3}\).

Step 2

Why this answer is correct

ऊपर का योग \(5\sqrt{3}\) है, इसलिए \(\frac{5\sqrt{3}}{\sqrt{3}}=5\)। / The numerator is \(5\sqrt{3}\), so \(\frac{5\sqrt{3}}{\sqrt{3}}=5\).

Step 3

Exam Tip

भाग से पहले समान मूल वाले पदों को जोड़ें। / Combine like surds before division.

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Question 454/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(\sqrt{m}+\sqrt{n}=7\) और (m,n) धनात्मक पूर्णांक हैं, तो कौन-सा युग्म निश्चित रूप से संभव है?

If \(\sqrt{m}+\sqrt{n}=7\) and (m,n) are positive integers, which pair is definitely possible?

Explanation opens after your attempt

Correct Answer

A. (m=9,n=16)

Step 1

Concept

\(\sqrt{9}=3\) और \(\sqrt{16}=4\) हैं। / \(\sqrt{9}=3\) and \(\sqrt{16}=4\).

Step 2

Why this answer is correct

इनका योग (3+4=7) है, इसलिए यह युग्म शर्त पूरी करता है। / Their sum is (3+4=7), so this pair satisfies the condition.

Step 3

Exam Tip

पूर्णांक योग पाने के लिए पहले पूर्ण वर्ग वाले विकल्प जाँचें। / To get an integer sum, first check the perfect-square options.

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Question 455/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(x=2+\sqrt{3}\), तो \(x^2+\frac{1}{x^2}\) का मान क्या है?

If \(x=2+\sqrt{3}\), what is the value of \(x^2+\frac{1}{x^2}\)?

Explanation opens after your attempt

Correct Answer

A. (14)

Step 1

Concept

\(\frac{1}{2+\sqrt{3}}=2-\sqrt{3}\) होता है। / \(\frac{1}{2+\sqrt{3}}=2-\sqrt{3}\).

Step 2

Why this answer is correct

इसलिए \(x+\frac{1}{x}=4\), अतः (x^-2+\frac{1}{x^-2}=(4)²-2=14)। / Hence \(x+\frac{1}{x}=4\), so \(x^2+\frac{1}{x^2}=4^2-2=14\).

Step 3

Exam Tip

पहले \(x+\frac{1}{x}\) निकालना लंबी गणना बचाता है। / Finding \(x+\frac{1}{x}\) first saves long calculation.

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Question 456/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

कौन-सा विकल्प \(0.303003000300003\ldots\) की प्रकृति सही बताता है?

Which option correctly describes the nature of \(0.303003000300003\ldots\)?

Explanation opens after your attempt

Correct Answer

C. असांत अनावर्ती अपरिमेयNon-terminating non-recurring irrational

Step 1

Concept

यह दशमलव समाप्त नहीं होता। / This decimal does not terminate.

Step 2

Why this answer is correct

इसमें शून्यों की संख्या बदलती रहती है, इसलिए कोई निश्चित आवर्ती समूह नहीं बनता। / The number of zeros keeps changing, so no fixed repeating block is formed.

Step 3

Exam Tip

असांत और अनावर्ती दशमलव अपरिमेय होता है। / A non-terminating non-recurring decimal is irrational.

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Question 457/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(a=\sqrt{6}+\sqrt{2}\) और \(b=\sqrt{6}-\sqrt{2}\), तो \(a^2-b^2\) का मान क्या है?

If \(a=\sqrt{6}+\sqrt{2}\) and \(b=\sqrt{6}-\sqrt{2}\), what is the value of \(a^2-b^2\)?

Explanation opens after your attempt

Correct Answer

A. \(8\sqrt{3}\)

Step 1

Concept

(a^-2-b^-2=(a-b)(a+b)) लगाएँ। / Use (a^-2-b^-2=(a-b)(a+b)).

Step 2

Why this answer is correct

\(a-b=2\sqrt{2}\) और \(a+b=2\sqrt{6}\), इसलिए गुणन \(4\sqrt{12}=8\sqrt{3}\) है। / \(a-b=2\sqrt{2}\) and \(a+b=2\sqrt{6}\), so the product is \(4\sqrt{12}=8\sqrt{3}\).

Step 3

Exam Tip

पहचान सूत्र से हल तेज और साफ होता है। / Identities make the solution quicker and cleaner.

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Question 458/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

किस विकल्प में (x) अपरिमेय है, पर \(x+\frac{1}{x}\) परिमेय है?

In which option is (x) irrational but \(x+\frac{1}{x}\) rational?

Explanation opens after your attempt

Correct Answer

A. \(x=3+\sqrt{8}\)

Step 1

Concept

\(3+\sqrt{8}=3+2\sqrt{2}\) अपरिमेय है। / \(3+\sqrt{8}=3+2\sqrt{2}\) is irrational.

Step 2

Why this answer is correct

इसका व्युत्क्रम \(3-\sqrt{8}\) है, क्योंकि (\(3+\sqrt{8}\)\(3-\sqrt{8}\)=1)। इसलिए योग (6) परिमेय है। / Its reciprocal is \(3-\sqrt{8}\), because (\(3+\sqrt{8}\)\(3-\sqrt{8}\)=1). Hence the sum is (6), which is rational.

Step 3

Exam Tip

जिन संयुग्मियों का गुणन (1) हो, वहाँ व्युत्क्रम तुरंत मिल सकता है। / When conjugates multiply to (1), the reciprocal is easy to identify.

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Question 459/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

कौन-सा विकल्प \(\sqrt{18}+\sqrt{50}-\sqrt{8}\) का सही सरल रूप है?

Which option is the correct simplified form of \(\sqrt{18}+\sqrt{50}-\sqrt{8}\)?

Explanation opens after your attempt

Correct Answer

A. \(6\sqrt{2}\)

Step 1

Concept

\(\sqrt{18}=3\sqrt{2}\), \(\sqrt{50}=5\sqrt{2}\), और \(\sqrt{8}=2\sqrt{2}\)। / \(\sqrt{18}=3\sqrt{2}\), \(\sqrt{50}=5\sqrt{2}\), and \(\sqrt{8}=2\sqrt{2}\).

Step 2

Why this answer is correct

\(3\sqrt{2}+5\sqrt{2}-2\sqrt{2}=6\sqrt{2}\)। / \(3\sqrt{2}+5\sqrt{2}-2\sqrt{2}=6\sqrt{2}\).

Step 3

Exam Tip

चिह्नों को ध्यान से रखकर गुणांक जोड़ें या घटाएँ। / Keep the signs carefully while adding or subtracting coefficients.

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Question 460/550 Expert Mathematics Real Numbers Irrational numbers Class 10 Level 14

यदि \(x=\sqrt{7}-\sqrt{3}\), तो \(x^2\) किसके बराबर है?

If \(x=\sqrt{7}-\sqrt{3}\), what is \(x^2\) equal to?

Explanation opens after your attempt

Correct Answer

A. \(10-2\sqrt{21}\)

Step 1

Concept

((a-b)²=a^-2-2ab+b^-2) का प्रयोग करें। / Use ((a-b)²=a^-2-2ab+b^-2).

Step 2

Why this answer is correct

\(x^2=7-2\sqrt{21}+3=10-2\sqrt{21}\)। / \(x^2=7-2\sqrt{21}+3=10-2\sqrt{21}\).

Step 3

Exam Tip

अंतर के वर्ग में बीच वाले पद का ऋण चिह्न न भूलें। / Do not forget the negative sign of the middle term in the square of a difference.

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Mathematics Irrational numbers MCQ Questions for Class 10

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यदि \(x=\frac{\sqrt{10}+\sqrt{6}}{\sqrt{10}-\sqrt{6}}\), तो (x) का सरल रूप क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(x=\sqrt{5}+\sqrt{2}\) और \(y=\sqrt{5}-\sqrt{2}\), तो \(x^2+y^2\) का मान क्या है?

Concept

Why this answer is correct

Exam Tip

कौन-सा विकल्प \(\frac{\sqrt{12}+\sqrt{27}}{\sqrt{3}}\) का सही मान देता है?

Concept

Why this answer is correct

Exam Tip

यदि \(\sqrt{m}+\sqrt{n}=7\) और (m,n) धनात्मक पूर्णांक हैं, तो कौन-सा युग्म निश्चित रूप से संभव है?

Concept

Why this answer is correct

Exam Tip

यदि \(x=2+\sqrt{3}\), तो \(x^2+\frac{1}{x^2}\) का मान क्या है?

Concept

Why this answer is correct

Exam Tip

कौन-सा विकल्प \(0.303003000300003\ldots\) की प्रकृति सही बताता है?

Concept

Why this answer is correct

Exam Tip

यदि \(a=\sqrt{6}+\sqrt{2}\) और \(b=\sqrt{6}-\sqrt{2}\), तो \(a^2-b^2\) का मान क्या है?

Concept

Why this answer is correct

Exam Tip

किस विकल्प में (x) अपरिमेय है, पर \(x+\frac{1}{x}\) परिमेय है?

Concept

Why this answer is correct

Exam Tip

कौन-सा विकल्प \(\sqrt{18}+\sqrt{50}-\sqrt{8}\) का सही सरल रूप है?

Concept

Why this answer is correct

Exam Tip

यदि \(x=\sqrt{7}-\sqrt{3}\), तो \(x^2\) किसके बराबर है?

Concept

Why this answer is correct

Exam Tip

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