Class 10 Maths Chapter 1 Important Questions (Real Numbers) – Most Repeated CBSE Questions with Solutions & PDF 2026

Hello students hope you are doing well and i am also good.Students preparing for the CBSE Board Exam 2026 often look for Class 10 Maths Chapter 1 Important Questions to revise the chapter quickly. The chapter Real Numbers is one of the most important topics in the Class 10 Mathematics syllabus now i have started to provide chapter wise important questions ans their solutions.

In this Artical we have provided the most repeated Class 10 Maths Chapter 1 Important Questions (Real Numbers) with step-by-step solutions for CBSE Board Exam 2026.

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Real Numbers – Chapter Overview

• Euclid Division Lemma
• HCF and LCM
• Fundamental Theorem of Arithmetic
• Irrational Numbers

Check some other questions in hindi content you should check Class 10 Maths Chapter 1 Important Questions with Solutions PDF

Most Repeated Important Questions

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Class 10 Maths Chapter 1 Important Questions with Solutions

Question 1

Use Euclid’s Division Algorithm to find the HCF of 135 and 225.

Solution:

Step 1: Write the numbers

we have to find HCF of 135 and 225 by using Euclid’s Division Algorithm.

Formula:

$$Dividend = Divisor times Quotient + Remainder$$

Larger number = 225 And smaller number = 135.

Step 2: First Division:

Now Divide 225 by 135 we find

225=135×1+90

• Quotient = 1
• Remainder = 90

So the New divisor is 90.

similary

Step 3: Second Division

135=90×1+45

• Quotient = 1
• Remainder = 45

now the Divisor is 45

Step 4: Third Division

90=45×2+0

• Quotient = 2
• Remainder = 0

Jab remainder 0 ho jata hai, tab last non-zero divisor HCF hota hai.

Step 5: Final Answer

Since last Non Zero Divisor is 45 that is HCF(135, 225) = 45

Question 2

Prove that √5 is irrational.

Solution:

We will prove this by the method of contradiction.

Step 1: Assume the opposite

Assume that √5 is rational.

This means it can be written in the form:$$sqrt{5} = frac{p}{q}$$5​=qp​

where p and q are integers and q ≠ 0, and p and q have no common factor (coprime).

Step 2: Square both sides

5​=qp​

Squaring both sides,$$5 = frac{p^2}{q^2}$$5=q2p2​

Multiply both sides by $q^2$q2:$$5q^2 = p^2$$5q2=p2

Step 3: Analyze the equation

From$$p^2 = 5q^2$$p2=5q2

This means p² is divisible by 5, so p must also be divisible by 5.

Let$$p = 5k$$p=5k

for some integer k.

Step 4: Substitute the value of p

p2=(5k)2=25k2

Put this in the equation:$$5q^2 = 25k^2$$5q2=25k2

Divide both sides by 5:$$q^2 = 5k^2$$q2=5k2

This means q² is divisible by 5, so q is also divisible by 5.

Step 5: Contradiction

We found that both p and q are divisible by 5.

But we assumed that p and q have no common factor.

This is a contradiction.

Step 6: Conclusion

Therefore our assumption is wrong.

Hence,$$sqrt{5} text{ is irrational.}$$Final Statement: √5 is an irrational number.

Question 3

Find the HCF and LCM of 306 and 657 using the Euclid’s Division Algorithm. Verify that

HCF×LCM=Product of the two numbers

Solution:

Step 1: Apply Euclid’s Division Algorithm

Take the larger number 657 and divide it by 306.$$657 = 306 times 2 + 45$$

Step 2: Next Division

Now divide 306 by 45.$$306 = 45 times 6 + 36$$

Solve similar to question 1

Step 3: next Divisions

• $36=9×4+0$

When remainder becomes 0, the last non-zero divisor is the HCF.

$$text{HCF} = 9$$

Step 4: Find LCM

Formula:$$text{LCM} = frac{text{Product of the numbers}}{text{HCF}}$$$$text{LCM} = frac{306 times 657}{9}$$$$text{LCM} = frac{201042}{9}$$$$text{LCM} = 22338$$

Step 5: Verification

HCF×LCM = Product of numbers=306×657=201042

Final Answer:

• HCF=9
• LCM=22338

Releted:

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Exam Pattern Based Questions

CBSE Board Exam Pattern Questions

Question 4

Prove that 3 + 2√5 is irrational

Solution:

This is also an important question for board exam and this repeated question ,we will solve this by the method of contradiction.

Jaise Question 1 kiya gya tha aap ise contradiction method se bhi kar sakte hai iska dusara method bhi bta rhe hai

Step 1 (Easy Steps)

Assume that$$3 + 2sqrt{5}$$

is a rational number.

let 3 + 2√5 = r

Step 2: Subtract 3 from both sides

$$3 + 2sqrt{5} – 3$$$$2sqrt{5}$$

2√5 = r-3

If $3 + 2sqrt{5}$ is rational, then $2sqrt{5}$ will also be rational.

Step 3: Divide by 2

So √5 will also be rational.

Step 4: Contradiction

But we already know that √5 is irrational. This is a contradiction.

Final Answer:
$3 + 2sqrt{5}$​ is an irrational number.

Note- you can solve all other questions like root 3, root 5 are irrational.

Key Concepts

Important Concepts from Real Numbers

1.Euclid Division Algorithm :

When we divide a larger number by a smaller number, we get a quotient and a remainder.

It can be written as:$$text{Dividend} = text{Divisor} times text{Quotient} + text{Remainder}$$

Where:

1. Dividend → The number that is divided
2. Divisor → The number we divide by
3. Quotient → The result of division
4. Remainder → The number left after division

2.Fundamental Theorem of Arithmetic:

Every composite number can be written as a product of prime numbers, and this factorization is unique (except for the order of the factors).

In simple words: It means that any number greater than 1 can be broken into prime numbers, and there is only one correct way to do it.

3.Irrational Numbers

Irrational numbers are numbers that cannot be written in the form of a fraction $frac{p}{q}$​, where $p$pand $q$are integers and $q neq 0$

The decimal form of irrational numbers never ends and never repeats.

Examples of Irrational Numbers

• $sqrt{2}$
• $sqrt{3}$
• $sqrt{5}$
• $pi$

Example:$$sqrt{2} = 1.4142135ldots$$

The decimal continues forever and does not repeat, so it is an irrational number.

Key Points:

1. Cannot be written as a simple fraction.
2. Decimal expansion is non-terminating and non-repeating.
3. They are part of the real number system.

Practice Questions

1. Use Euclid’s Division Algorithm to find the HCF of 135 and 225.
2. Find the HCF and LCM of 306 and 657 and verify that HCF×LCM=Product of the two numbers
3. Prove that √5 is an irrational number.
4. Prove that 3 + 2√5 is an irrational number.
5. Find the LCM and HCF of 8, 12 and 15 using prime factorisation method.

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FAQ

Q1. Is Real Numbers chapter important for CBSE board exam?

Ans: Yes, Real Numbers is a very important chapter for the CBSE Class 10 Maths board exam. Questions from this chapter appear almost every year. Students are usually asked problems based on Euclid’s Division Algorithm, HCF & LCM, Fundamental Theorem of Arithmetic, and Irrational Numbers.

Q2. How many questions come from Chapter 1 Real Numbers?

Ans: Generally, 1–2 questions come from the Real Numbers chapter in the CBSE Class 10 board exam. These questions may carry 2 to 4 marks and are often based on HCF using Euclid’s Division Algorithm or proofs of irrational numbers.

Q3. Where can I download Class 10 Maths Chapter 1 Important Questions PDF?

Ans: You can download the Class 10 Maths Chapter 1 Important Questions PDF from this page by clicking the “Download PDF” button provided above. The PDF includes most repeated board questions with step-by-step solutions for quick revision.

Conclusion:

In this article we shared Class 10 Maths Chapter 1 Important Questions with Solutions for board exam preparation. Students should practice these questions regularly to score better marks in exams.

All important questions will be updated here time to time kindly visit for updates and keep join our telegram channel for quick and important updates.

Last update: 07 march 2026

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