Real Numbers : Assessment for classes IX and X

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25 Question MCQ Chapter on Real Numbers for CBSE, ICSE, Madhyamik and other State Boards Class IX & X

👨‍🏫 Author: Singh

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Real Numbers Quiz

Test your understanding of rational, irrational numbers and their properties. The study of Real Numbers in Classes IX and X establishes the foundation for all higher-level mathematics.Real Numbe: The set including all Rational and Irrational numbers. Rational Numbers: Numbers expressible as p/q, q is non-zero . Their decimal expansions are either terminating or non-terminating recurring.Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating non-recurring.Number Line: Every point on a number line represents a unique real number.

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Question 1

Which of the following is an irrational number?

A\(\sqrt{25}\) B\(\sqrt{2} + \sqrt{3}\) C\(\frac{3}{7}\) D\(0.\overline{3}\)

Question 2

The decimal expansion of \(\frac{13}{625}\) will terminate after how many decimal places?

A2 B3 C4 D5

Question 3

Which of the following has a terminating decimal expansion?

A\(\frac{77}{210}\) B\(\frac{23}{30}\) C\(\frac{125}{441}\) D\(\frac{129}{2^2 \times 5^7}\)

Question 4

The simplified value of \(\sqrt{12} + \sqrt{27} - \sqrt{75}\) is:

A\(\sqrt{3}\) B\(2\sqrt{3}\) C\(3\sqrt{3}\) D\(0\)

Question 5

If \(p\) and \(q\) are co-prime numbers, then \(p^2\) and \(q^2\) are:

ACo-prime BNot co-prime CEven numbers DOdd numbers

Question 6

A rational number between \(\sqrt{2}\) and \(\sqrt{3}\) is:

A\(\frac{\sqrt{2} + \sqrt{3}}{2}\) B\(\frac{\sqrt{2} \times \sqrt{3}}{2}\) C\(1.5\) D\(1.8\)

Question 7

The value of \(1.\overline{36} + 0.\overline{9}\) is:

A\(2.\overline{27}\) B\(2.\overline{26}\) C\(2.\overline{3}\) D\(2.3\)

Question 8

Which of the following is not an irrational number?

A\(\sqrt{8} \times \sqrt{2}\) B\(\sqrt{18} \div \sqrt{2}\) C\(\sqrt{5} + \sqrt{10}\) D\(\sqrt{7} \times \sqrt{28}\)

Question 9

If \(n\) is a natural number, then \(\sqrt{n}\) is:

AAlways rational BAlways irrational CRational if \(n\) is a perfect square DIrrational if \(n\) is a perfect square

Question 10

The product of a non-zero rational and an irrational number is:

AAlways rational BAlways irrational CCan be rational or irrational DAlways an integer

Question 11

Which of the following rational numbers have terminating decimal representation?

A\(\frac{16}{125}\) B\(\frac{27}{250}\) C\(\frac{343}{2^3 \times 5^3}\) DAll of these

Question 12

The ascending order of \(\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5}\) is:

A\(\sqrt[3]{2} < \sqrt[6]{5} < \sqrt{3}\) B\(\sqrt[6]{5} < \sqrt[3]{2} < \sqrt{3}\) C\(\sqrt[6]{5} < \sqrt{3} < \sqrt[3]{2}\) D\(\sqrt{3} < \sqrt[3]{2} < \sqrt[6]{5}\)

Question 13

The decimal expansion of the rational number \(\frac{14587}{1250}\) will terminate after:

AOne decimal place BTwo decimal places CThree decimal places DFour decimal places

Question 14

After rationalizing the denominator of \(\frac{7}{3\sqrt{3} - 2\sqrt{2}}\), we get:

A\(3\sqrt{3} + 2\sqrt{2}\) B\(3\sqrt{3} - 2\sqrt{2}\) C\(7(3\sqrt{3} + 2\sqrt{2})\) D\(7(3\sqrt{3} - 2\sqrt{2})\)

Question 15

An irrational number between \(\frac{1}{7}\) and \(\frac{2}{7}\) is:

A\(\frac{1}{2}\left(\frac{1}{7} + \frac{2}{7}\right)\) B\(\sqrt{\frac{1}{7} \times \frac{2}{7}}\) C\(\frac{1}{\sqrt{14}}\) D\(\sqrt{0.14}\)

Question 16

If \(x = 2 + \sqrt{3}\), then \(x + \frac{1}{x}\) equals:

A\(4\) B\(2\sqrt{3}\) C\(-4\) D\(-2\sqrt{3}\)

Question 17

Which of the following is a rational number?

A\(\pi\) B\(\sqrt{2}\) C\(0.\overline{101}\) D\(\sqrt{5} + 1\)

Question 18

The simplest rationalizing factor of \(\sqrt[3]{54}\) is:

A\(\sqrt[3]{2}\) B\(\sqrt[3]{3}\) C\(\sqrt[3]{4}\) D\(\sqrt[3]{9}\)

Question 19

If \(\frac{p}{q}\) is a rational number with terminating decimal expansion, where \(p\) and \(q\) are co-prime, then \(q\) must be of the form:

A\(2^m \times 3^n\) B\(2^m \times 5^n\) C\(3^m \times 5^n\) D\(2^m \times 3^n \times 5^k\)

Question 20

The sum of two irrational numbers is:

AAlways irrational BAlways rational CCan be rational or irrational DAlways an integer

Question 21

The value of \(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}}\) is:

A\(1\) B\(9\) C\(10\) D\(99\)

Question 22

If \(a\) and \(b\) are rational numbers and \(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{3}} = a + b\sqrt{3}\), then \(a + b =\)

A\(11\) B\(-11\) C\(13\) D\(-13\)

Question 23

Which of the following is not true?

AEvery integer is a rational number BEvery whole number is a natural number CEvery irrational number is a real number DEvery rational number is a real number

Question 24

The product \(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32}\) equals:

A\(\sqrt{2}\) B\(2\) C\(\sqrt[12]{2}\) D\(\sqrt[12]{32}\)

Question 25

The value of \(0.\overline{6} + 0.\overline{7} + 0.\overline{4}\) is:

A\(2\) B\(\frac{17}{9}\) C\(\frac{53}{22}\) D\(\frac{59}{22}\)

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh

Problems on Algebra : ICSE and CBSE

Algebraic Identities Problem Set

Master Algebra through Practice Problems. Prime Maths

About This Problem Set

This collection of problems is designed for Class IX students following CBSE, ICSE, and State Board curricula. The problems cover fundamental algebraic identities, inequalities, factorization, and polynomial expansions. Work through these problems systematically to strengthen your understanding of algebraic concepts and their applications.

Problem 1

Problem 1 of 22

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Practice these problems to master algebraic identities. Remember to attempt each problem before checking the hint!

Designed for Class IX Mathematics

Designed by Vinod Singh, 9038126497

Indices Interactive Chapter Test: ICSE and CBSE

Prime Maths

25 Question MCQ Chapter on Indices & Exponents for ICSE, CBSE, WBBSE and other State Boards

👨‍🏫 Author: Vinod Singh

📞 WA: +91-9038126497

Prime Maths

Indices & Exponents Quiz

Test your understanding of mathematical indices

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Question 1

\( \big( 125^{-4}\times 256^{-3/2}\big)^{-1/6} \times \big( (x^{2}y^{-4})^{1/3} \times \sqrt{y^{3}x^{-5}}\big)^{12}\times (x^{2})^{11}\)

A\( 10y^{2}\) B\( 100y^{2}\) C\( 10x^{2}\) D\( 100x^{2}\)

Question 2

The value of \( (5^{0} + 3^{0} + 2^{0}) ÷\bigg( \frac{2^{n}(2^{n-1})^n}{2^{n+1}.2^{n-1}} \times \bigg( \frac{\sqrt [3] {8^{n}}}{4}\bigg)^{-n} \bigg) \) is:

A0 B1 C2 D3

Question 3

Simplify: \( \big(\frac{1728}{729}\big)^{\frac{2}{3}}\times \frac{3^{2}}{\sqrt{144}}\times \sqrt{5} \)

A\( \frac{4\sqrt{5}}{3} \) B\( \frac{4\sqrt{5}}{7} \) C\( \frac{\sqrt{5}}{3} \) D\( \frac{7\sqrt{5}}{3} \)

Question 4

If \(5^{10x}=4900 \) and \( 2^{ \sqrt{y}}=25 \), the value of \(4^ { \sqrt{y}}\times 5^{5x-5} \) is:

A12 B16 C14 D13

Question 5

If \(\big( \frac{2a}{b}\big)^{2x-4} \) = \(\big( \frac{b}{2a}\big)^{2x-4} \), then the value of \( x \) is:

A2 B4 C3 D-2

Question 6

\( \big[1-\big(1-\big(1-n\big)^{-1}\big)^{-1} \big]^{-1} = \) where \(n \neq 0,1 \)

A\( n \) B\( 2n \) C\( -n \) D\( \frac{1}{n} \)

Question 7

If \( 2016 = 2^{a}\times 3^{b} \times 5^{c}, \) then the value of \(3^{a}\times 2^{-b} \times 5^{-c} \) is

A\( \frac{243}{4} \) B\( \frac{247}{4} \) C\( \frac{241}{4} \) D\( \frac{234}{4} \)

Question 8

If \( 2^{x}=3^{y}=6^{-z} \) then the value of \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z} =\)

A0 B1 C-1 D\( \frac{1}{2} \)

Question 9

If \( a^{x}=b^{y}=c^{z} \) and \(b^{2} = ac \) then the value of \( \frac{2xz}{z+x}\) is

Ay B2x Cx D-x

Question 10

The simplest value of \( \frac{(x^{a+b})^{2} . (x^{b+c})^{2} . (x^{c+a})^{2}}{(x^{a}.x^{b}.x^{c})^{4}} \) is

A0 B1 C2 D-1

Question 11

Simplest value of \( \big( \frac{2^{-1}\times 3^{2}}{2^{2}\times 3^{-4}}\big)^{7/2} \times \big( \frac{2^{-2}\times 3^{3}}{2^{3}\times 3^{-5}}\big)^{-5/2} \) is

A12 B22 C2 D11

Question 12

The value of \(x\) satisfying \( 9^{x}-9^{x-1}=648 \) is

A3 B\( \frac{1}{3} \) C2 D-3

Question 13

The simplest value of \( \big[ 5\big(8^{1/3}+27^{1/3}\big)^{3}\big]^{\frac{1}{4}} \) is

A5 B6 C8 D12

Question 14

If \( y= x^{1/3}-x^{-1/3}, \) then \( y^{3}+3y = \)

A\( x-x^{-1} \) B\( x^{2} - x^{-2} \) C\( x^{2}+x^{-2} \) D\( x+x^{-1} \)

Question 15

If \( (10^{11}+25)^{2}-(10^{11}-25)^{2} = 10^{n}\) then the value of \( n \) is

A11 B13 C17 D7

Question 16

The least value of the expression \( 4^{x}+4^{1-x} \) is

A2 B4 C8 D16

Question 17

Value of \(x\) and \(y\) satisfying the equations \( 5^{x}-3^{y}=16; \quad 5^{x-1}+3^{y+1}=32\) are

A\( x=2 \quad y = 2 \) B\( x=2 \quad y = -2 \) C\( x=-2 \quad y = 2 \) D\( x=-2 \quad y = -2 \)

Question 18

If \(\big( x^{n^{2}}\big)^{n}= \big( x^{2^{n}}\big)^{2}\), then the value of \( \sqrt[n+1]{n^{3}} \) is

A2 B4 C1 D\( 2^{4} \)

Question 19

If \( x = 3+3^{2/3}+3^{1/3}\) then the value of \( x^3-9x^2+18x-12 \) is

A0 B1 C3 D-3

Question 20

If \( a^{x}=bc, \quad b^{y}= ca, \quad c^{z}=ab\), then the value of \(\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1} \quad \) is

A1 B2 C0 D-1

Question 21

The value of \(x\) satisfying \( 4^{x}-3^{x-1/2}=3^{x+1/2}-2^{2x-1}\) is

A\( \frac{3}{2} \) B2 C\( \frac{1}{2} \) D\( \frac{2}{3} \)

Question 22

The value of \( \frac{\big(2^{2n}-3.2^{2n-2}\big)\big(3^{n}-2.3^{n-2}\big)}{3^{n-4}\big( 4^{n+3}-2^{2n}\big)} \quad \) is

A\( \frac{1}{4} \) B\( \frac{3}{4} \) C\( \frac{-3}{4}\) D\( \frac{1}{3} \)

Question 23

If \( a^{x}=b^{y}\) and \( b^{x}=a^{y} \quad ( ab \neq 1)\) then which of the following relation is true

A\( a=b^{2} \) B\( a=b \) C\( b=a^{2} \) D\( ab=2 \)

Question 24

Solve for \(x \) and \( y: \quad 2^{x}+2^{y}=12; \quad x+y= 5\)

A\( x=3, \quad y = 2 \) B\( x=2, \quad y = 3 \) C\( x=4, \quad y = 1\) D\( x=1, \quad y = 4 \)

Question 25

The value of \( \bigg( \frac{9^{n+1/4}.\sqrt{3.3^{n}}}{3.\sqrt{3^{-n}}}\bigg)^{1/n}\) is

A3 B9 C27 D81

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh