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.

---

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

---

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

---

3. Fundamental Theorem of Arithmetic

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

Prime Factorisation is Unique (except order)

Example:

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

---

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

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

---

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

---

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)

---

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

---

⬇️ 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