Class 10 Maths MCQ Test – Real Numbers (Chapter 1) | Full Chapter Test

Class 10 Mathematics – Chapter 1: Real Numbers (Detailed Notes)

These notes are designed for CBSE & GSEB students. Covers complete theory, formulas, and exam-oriented concepts.

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1. Introduction to Real Numbers

Real numbers include all numbers that can be represented on the number line.

• Natural Numbers (N): 1, 2, 3, ...
• Whole Numbers (W): 0, 1, 2, 3, ...
• Integers (Z): ..., -2, -1, 0, 1, 2, ...
• Rational Numbers (Q): Numbers of form \( \frac{p}{q} \), where \( q \neq 0 \)
• Irrational Numbers: Numbers which cannot be expressed as \( \frac{p}{q} \)

Examples of Irrational Numbers: \( \sqrt{2}, \sqrt{3}, \pi \)

Real Numbers = Rational + Irrational Numbers

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2. Euclid’s Division Lemma

Euclid’s Division Lemma states that for any two positive integers \( a \) and \( b \), there exist unique integers \( q \) and \( r \) such that:

\( a = bq + r \), where \( 0 \le r < b \)

It is the basis of finding HCF using Euclid’s Algorithm.

Euclid’s Division Algorithm (Steps)

• Step 1: Apply lemma \( a = bq + r \)
• Step 2: Replace \( a \) by \( b \) and \( b \) by \( r \)
• Step 3: Repeat until remainder becomes 0
• Step 4: Last non-zero remainder is HCF

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3. Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime numbers.

Prime Factorisation is Unique (except order)

Example:

\( 60 = 2^2 \times 3 \times 5 \)

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4. HCF and LCM

Using Prime Factorisation

• HCF: Product of smallest powers of common primes
• LCM: Product of highest powers of all primes

Important Formula

\( \text{HCF} \times \text{LCM} = \text{Product of two numbers} \)

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5. Revisiting Irrational Numbers

An irrational number cannot be written in the form \( \frac{p}{q} \).

Important Results:

• \( \sqrt{2} \), \( \sqrt{3} \), \( \sqrt{5} \) are irrational
• Sum of rational and irrational is irrational
• Product of non-zero rational and irrational is irrational

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6. Decimal Expansion of Rational Numbers

A rational number \( \frac{p}{q} \) has:

• Terminating Decimal: If \( q = 2^n \times 5^m \)
• Non-Terminating Recurring: If denominator has primes other than 2 or 5

Examples:

• \( \frac{1}{8} = 0.125 \) (Terminating)
• \( \frac{1}{3} = 0.333... \) (Recurring)

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7. Important Exam Points

• Always check remainder condition \( 0 \le r < b \)
• Use Euclid’s Algorithm for large numbers
• Prime factorisation must be correct
• Check denominator for decimal nature

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⬇️ Attempt the test given below after revising these notes

Class 10 Maths

Chapter 1: Real Numbers | CBSE Board MCQ Test

Time Left: 05:00

Student Name:

1. According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as:

product of primes in a unique way (ignoring order) product of any integers sum of primes difference of primes

2. The HCF of two numbers is 12 and their product is 1800. If their LCM is L, then L equals:

150 120 180 240

3. If the prime factorisation of a number is 2³ × 3² × 5, the total number of factors is:

24 12 18 16

4. Which of the following numbers has a terminating decimal expansion?

13/15 7/40 17/18 11/21

5. If √5 were rational, then it could be written as:

p/q where p and q are integers with no common factor p + q p² + q² pq

6. The decimal expansion of 29/125 is:

terminating non-terminating repeating non-terminating non-repeating irrational

7. In the proof that √2 is irrational, the contradiction arises because:

both p and q become odd both p and q become even p becomes odd q becomes prime

8. If the denominator of a rational number in lowest form has prime factors only 2 and 5, then its decimal expansion is:

terminating non-terminating repeating irrational undefined

9. Which of the following is irrational?

√16 √49 √7 √25

10. If HCF(a,b)=6 and LCM(a,b)=180, then a×b equals:

1080 900 540 360

Vedant Classes | CBSE Board Practice Test

Class 10 Maths Chapter 1: Real Numbers

This chapter is one of the most important topics in Class 10 Mathematics. It forms the foundation for number systems and helps students understand properties of integers, rational numbers, and irrational numbers. Concepts from this chapter are frequently used in higher mathematics and competitive exams.

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Learning Outcomes

After completing this chapter, students will be able to:

• Understand the concept of real numbers and their classification.
• Apply Euclid’s Division Lemma to find HCF of two numbers.
• Use the Euclidean Algorithm for efficient calculations.
• Find HCF and LCM of given numbers and understand their relationship.
• Understand the Fundamental Theorem of Arithmetic.
• Express numbers as product of prime factors.
• Determine whether a number has a terminating or non-terminating decimal expansion.

Exam Tip: Questions based on HCF, LCM, and Euclid’s Division Lemma are frequently asked in CBSE and GSEB exams.

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Real Life Applications

Concepts of real numbers are used in many practical situations:

• Distribution Problems: Dividing items into equal groups using HCF.
• Time Management: Finding common time intervals using LCM.
• Measurement: Simplifying ratios and measurements in daily life.
• Computer Science: Algorithms like Euclidean Algorithm are widely used.
• Engineering: Prime factorization is used in cryptography and coding systems.

Important Concept: The decimal expansion of rational numbers helps in identifying whether a number is terminating or recurring.

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Common Mistakes Students Make

• Confusing HCF with LCM.
• Not applying Euclid’s Division Lemma correctly.
• Skipping steps in prime factorization.
• Forgetting the condition for terminating decimals.
• Errors in division during Euclidean Algorithm.
• Not writing proper mathematical steps in exams.

Pro Tip: Always show complete steps while solving HCF and LCM problems to score full marks.

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Conclusion

The chapter “Real Numbers” is a high-scoring and concept-based chapter. Regular practice of problems and clear understanding of algorithms will help students perform well in exams. Focus on stepwise solutions and revise important theorems regularly.

Class 10 Maths Chapter 1: Real Numbers

10 Detailed Solved Examples

Example 1: Euclid’s Division Lemma

Find the HCF of 135 and 225 using Euclid’s Division Algorithm.

Solution:
225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0

HCF = 45

Example 2: HCF and LCM

Find HCF and LCM of 12 and 18.

Solution:
Prime factorization:
12 = 2² × 3
18 = 2 × 3²

HCF = 2 × 3 = 6
LCM = 2² × 3² = 36

Example 3: Verification

Verify that HCF × LCM = Product of numbers for 12 and 18.

Solution:
HCF = 6, LCM = 36
LHS = 6 × 36 = 216
RHS = 12 × 18 = 216

Verified ✔

Example 4: Fundamental Theorem of Arithmetic

Express 360 as product of prime factors.

Solution:
360 = 2³ × 3² × 5

Example 5: Terminating Decimal

Check whether 13/125 has terminating decimal expansion.

Solution:
125 = 5³
Denominator has only 2 or 5 → Terminating decimal ✔

Example 6: Non-Terminating Decimal

Check whether 7/15 is terminating.

Solution:
15 = 3 × 5
Contains factor other than 2 or 5 → Non-terminating recurring

Example 7: Largest Number

Find the greatest number that divides 306 and 657 leaving remainder 9.

Solution:
Numbers become: 297 and 648
HCF = 27

Answer = 27

Example 8: LCM Application

Find smallest number divisible by 6, 8 and 12.

Solution:
LCM of 6, 8, 12 = 24

Example 9: Euclid Application

Use Euclid’s lemma to find HCF of 867 and 255.

Solution:
867 = 255 × 3 + 102
255 = 102 × 2 + 51
102 = 51 × 2 + 0

HCF = 51

Example 10: Decimal Conversion

Convert 1/8 into decimal.

Solution:
1 ÷ 8 = 0.125 (Terminating decimal)

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10 Important FAQs

Q1: What are real numbers?
All rational and irrational numbers together form real numbers.

Q2: What is Euclid’s Division Lemma?
It states that a = bq + r, where 0 ≤ r < b.

Q3: Difference between HCF and LCM?
HCF is greatest common divisor, while LCM is smallest common multiple.

Q4: What is Fundamental Theorem of Arithmetic?
Every composite number can be expressed as product of primes.

Q5: When is a decimal terminating?
If denominator has only 2 and/or 5 as prime factors.

Q6: What is Euclidean Algorithm?
A method to find HCF using repeated division.

Q7: Why is prime factorization important?
It helps in finding HCF, LCM, and solving number problems.

Q8: What are irrational numbers?
Numbers which cannot be expressed as p/q.

Q9: Relationship between HCF and LCM?
Product of numbers = HCF × LCM.

Q10: Why is this chapter important?
It forms the base for algebra, number theory, and competitive exams.