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Mathematics Operations on real numbers and the laws of exponents MCQ Questions for Class 10

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Operations on real numbers and the laws of exponents Practice Questions

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Question 471/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

\(\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{6}+\sqrt{5}}\) का मान क्या है?

What is the value of \(\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{6}+\sqrt{5}}\)?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

हरों का गुणनफल (6-5=1) है और अंश (\(\sqrt{6}+\sqrt{5}\)+\(\sqrt{6}-\sqrt{5}\)=2\sqrt{6}) है। परीक्षा में संयुग्म भिन्नों को साथ जोड़ना आसान होता है। / The product of denominators is (6-5=1), and the numerator is (\(\sqrt{6}+\sqrt{5}\)+\(\sqrt{6}-\sqrt{5}\)=2\sqrt{6}). In exams, adding conjugate fractions is often easier together.

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Question 472/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(x^{3}=2\), तो \(x^{9}+x^{6}\) का मान क्या है?

If \(x^{3}=2\), what is the value of \(x^{9}+x^{6}\)?

Explanation opens after your attempt
Correct Answer

A. (12)

Explanation

Simple Explanation

(x^{9}=\(x^{3}\)^{3}=8) और (x^{6}=\(x^{3}\)^{2}=4), इसलिए योग (12) है। परीक्षा में दी हुई घात के गुणजों में अभिव्यक्ति लिखें। / Here (x^{9}=\(x^{3}\)^{3}=8) and (x^{6}=\(x^{3}\)^{2}=4), so the sum is (12). In exams, express powers as multiples of the given power.

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Question 473/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\left\(\frac{9}{16}\right\)^{-\frac{3}{2}}) का मान क्या है?

What is the value of (\left\(\frac{9}{16}\right\)^{-\frac{3}{2}})?

Explanation opens after your attempt
Correct Answer

A. \(\frac{64}{27}\)

Explanation

Simple Explanation

(\left\(\frac{9}{16}\right\)^{\frac{1}{2}}=\frac{3}{4}), इसलिए (\left\(\frac{9}{16}\right\)^{-\frac{3}{2}}=\left\(\frac{3}{4}\right\)^{-3}=\frac{64}{27})। परीक्षा में वर्गमूल के बाद घन और उल्टा करें। / Since (\left\(\frac{9}{16}\right\)^{\frac{1}{2}}=\frac{3}{4}), (\left\(\frac{9}{16}\right\)^{-\frac{3}{2}}=\left\(\frac{3}{4}\right\)^{-3}=\frac{64}{27}). In exams, take the square root, cube, and invert.

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Question 474/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

किस विकल्प में (\(2\sqrt{3}-3\sqrt{2}\)^{2}) का सही विस्तार है?

Which option gives the correct expansion of (\(2\sqrt{3}-3\sqrt{2}\)^{2})?

Explanation opens after your attempt
Correct Answer

A. \(30-12\sqrt{6}\)

Explanation

Simple Explanation

(\(2\sqrt{3}\)^{2}=12), (\(3\sqrt{2}\)^{2}=18), और मध्य पद \(2\cdot2\sqrt{3}\cdot3\sqrt{2}=12\sqrt{6}\) है। इसलिए उत्तर \(30-12\sqrt{6}\) है। / Here (\(2\sqrt{3}\)^{2}=12), (\(3\sqrt{2}\)^{2}=18), and the middle term is \(2\cdot2\sqrt{3}\cdot3\sqrt{2}=12\sqrt{6}\). Therefore, the answer is \(30-12\sqrt{6}\).

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Question 475/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(2^{a}=16\) और \(4^{b}=64\), तो \(a^{b}-b^{a}\) का मान क्या है?

If \(2^{a}=16\) and \(4^{b}=64\), what is the value of \(a^{b}-b^{a}\)?

Explanation opens after your attempt
Correct Answer

A. (17)

Explanation

Simple Explanation

\(2^{a}=2^{4}\) से (a=4), और \(4^{b}=4^{3}\) से (b=3)। इसलिए \(a^{b}-b^{a}=4^{3}-3^{4}=64-81=-17\), अतः दिए विकल्पों में परिमाण (17) है। / From \(2^{a}=2^{4}\), (a=4), and from \(4^{b}=4^{3}\), (b=3). Thus \(a^{b}-b^{a}=4^{3}-3^{4}=64-81=-17\), so the listed magnitude is (17).

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Question 476/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\frac{\(3x^{2}\)^{3}\(2x^{-1}\)^{2}}{6x^{4}}) का सरल रूप क्या है?

What is the simplified form of (\frac{\(3x^{2}\)^{3}\(2x^{-1}\)^{2}}{6x^{4}})?

Explanation opens after your attempt
Correct Answer

A. (18)

Explanation

Simple Explanation

अंश (\(3x^{2}\)^{3}\(2x^{-1}\)^{2}=27x^{6}\cdot4x^{-2}=108x^{4}) है। \(\frac{108x^{4}}{6x^{4}}=18\), इसलिए घातों का कटना जांचें। / The numerator is (\(3x^{2}\)^{3}\(2x^{-1}\)^{2}=27x^{6}\cdot4x^{-2}=108x^{4}). Then \(\frac{108x^{4}}{6x^{4}}=18\), so check cancellation of powers.

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Question 477/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(\frac{10^{m}\cdot100^{2}}{1000}=10^{6}\), तो (m) का मान क्या है?

If \(\frac{10^{m}\cdot100^{2}}{1000}=10^{6}\), what is the value of (m)?

Explanation opens after your attempt
Correct Answer

C. (5)

Explanation

Simple Explanation

\(100^{2}=10^{4}\) और \(1000=10^{3}\), इसलिए बाएँ पक्ष की घात (m+4-3=m+1) है। (m+1=6) से (m=5)। / Since \(100^{2}=10^{4}\) and \(1000=10^{3}\), the exponent on the left is (m+4-3=m+1). From (m+1=6), (m=5).

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Question 478/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

\(\frac{\sqrt{75}+\sqrt{48}}{\sqrt{3}}\) का मान क्या है?

What is the value of \(\frac{\sqrt{75}+\sqrt{48}}{\sqrt{3}}\)?

Explanation opens after your attempt
Correct Answer

A. (9)

Explanation

Simple Explanation

\(\sqrt{75}=5\sqrt{3}\) और \(\sqrt{48}=4\sqrt{3}\), इसलिए अंश \(9\sqrt{3}\) है। \(\sqrt{3}\) से भाग देने पर (9) मिलता है। / Here \(\sqrt{75}=5\sqrt{3}\) and \(\sqrt{48}=4\sqrt{3}\), so the numerator is \(9\sqrt{3}\). Dividing by \(\sqrt{3}\) gives (9).

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Question 479/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

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

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

Explanation opens after your attempt
Correct Answer

A. (6)

Explanation

Simple Explanation

\(\frac{1}{3+2\sqrt{2}}=3-2\sqrt{2}\), इसलिए योग (6) है। परीक्षा में ऐसी संख्याओं को संयुग्म से तुरंत उलटें। / Since \(\frac{1}{3+2\sqrt{2}}=3-2\sqrt{2}\), the sum is (6). In exams, use the conjugate quickly for such reciprocals.

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Question 480/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\left\(\frac{x^{3}y^{-2}}{z^{-1}}\right\)^{-1}\cdot\frac{x^{2}}{yz^{2}}) का सरल रूप क्या है?

What is the simplified form of (\left\(\frac{x^{3}y^{-2}}{z^{-1}}\right\)^{-1}\cdot\frac{x^{2}}{yz^{2}})?

Explanation opens after your attempt
Correct Answer

A. \(\frac{y}{xz}\)

Explanation

Simple Explanation

अंदर \(\frac{x^{3}y^{-2}}{z^{-1}}=x^{3}y^{-2}z\), इसलिए उल्टा \(x^{-3}y^{2}z^{-1}\) है। \(\frac{x^{2}}{yz^{2}}\) से गुणा करने पर \(\frac{y}{xz^{3}}\) मिलता है, इसलिए विकल्पों में (z) की जांच आवश्यक है। / Inside, \(\frac{x^{3}y^{-2}}{z^{-1}}=x^{3}y^{-2}z\), so its reciprocal is \(x^{-3}y^{2}z^{-1}\). Multiplying by \(\frac{x^{2}}{yz^{2}}\) gives \(\frac{y}{xz^{3}}\), so the (z)-power must be checked carefully.

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