CBSE Class 10 Mathematics (Basic) Chapter 1 Real Numbers Important Questions 2027
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25 questions from Real Numbers appeared in 6 of the 7 CBSE papers we analysed (2022–2026) — here are the topics that matter, the questions that repeat, and every previous-year question from this chapter.
Key facts
- Number Systems unit — 6 marks in the official syllabus
- Asked in 6 of 7 analysed papers (2022–2026)
- About 3.6 questions per paper (OR alternatives counted separately)
- 8 question patterns repeated across papers
Important topics in Real Numbers
| # | Topic | Asked in | Importance |
|---|---|---|---|
| 1 | Proving irrationality (root 2, root 3, root 5 type) | Board 2023Board 2024Board 2025Board 2026Sample 2025Sample 2026 | ★★☆☆☆ |
| 2 | Fundamental Theorem of Arithmetic / prime factorisation | Board 2023Board 2024Board 2025Board 2026Sample 2026 | ★★☆☆☆ |
| 3 | HCF-LCM relationship and word problems | Board 2024Board 2025Board 2026Sample 2025 | ★☆☆☆☆ |
| 4 | HCF and LCM using prime factorisation | Board 2023Board 2024Sample 2025Sample 2026 | ★☆☆☆☆ |
Repeated questions from Real Numbers
1. Prove sqrt(n) irrational
3×Board 2025Board 2026Sample 20253 marks
Prove that √3 is an irrational number.
2. HCF/LCM of two numbers
3×Board 2023Board 2024Sample 20261/2 marks
Find the smallest number which is divisible by both 306 and 657.
3. Show expression is composite
2×Board 2024Sample 20262 marks
Show that the number 2 × 5 × 7 × 11 + 11 × 13 is a composite number.
4. LCM of coprime numbers
2×Board 2025Board 20261 marks
If the HCF of two positive integers a and b is 1, then their LCM is : (A) a + b (B) a (C) b (D) ab
5. Identify rational/irrational
2×Board 2025Board 20261 marks
The number 3 + √2 is : (A) a rational number (B) an irrational number (C) an integer (D) a natural number
6. AR: HCF divides LCM
2×Board 2025Board 20261 marks
Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b. Reason (R) : HCF of any two natural numbers divides both the numbers. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.
7. Complete factor tree
2×Board 2025Board 20263 marks
The factor tree of a number x is shown below : x branches into 2 and y; y branches into 2 and 210; 210 branches into a and 70; 70 branches into 2 and 35; 35 branches into 5 and b. Find the values of x, y, a and b. Hence, write the product of the prime factors of the number x so obtained.
8. Prove a+b*sqrt(n) irrational
2×Board 2023Board 20242/3 marks
Prove that 6 - 4√5 is an irrational number, given that √5 is an irrational number.
Real Numbers questions by paper
| Paper | Questions | Marks |
|---|---|---|
| Board 2022 | 0 | 0 |
| Board 2023 | 3 | 6 |
| Board 2024 | 4 | 8 |
| Board 2025 | 5 | 9 |
| Sample 2025 | 4 | 6 |
| Board 2026 | 5 | 9 |
| Sample 2026 | 4 | 8 |
Counts include both options of internal-choice (OR) questions.
Previous year questions: Real Numbers
Board 2026 5 questions
If the HCF of two positive integers a and b is 1, then their LCM is : (A) a + b (B) a (C) b (D) ab
The number 3 + √2 is : (A) a rational number (B) an irrational number (C) an integer (D) a natural number
Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b. Reason (R) : HCF of any two natural numbers divides both the numbers. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.
Prove that √3 is an irrational number.
The factor tree of a number x is shown below : x branches into 2 and y; y branches into 2 and 210; 210 branches into a and 70; 70 branches into 2 and 35; 35 branches into 5 and b. Find the values of x, y, a and b. Hence, write the product of the prime factors of the number x so obtained.
Sample 2026 4 questions
The exponent of 3 in the prime factorization of 2025 is
A) 1
B) 2
C) 3
D) 4
Show that the number 2 × 5 × 7 × 11 + 11 × 13 is a composite number.
Find the smallest number which is divisible by both 306 and 657.
Show that √2 − √5 is an irrational number.
Board 2025 5 questions
If the HCF of two positive integers a and b is 1, then their LCM is : (A) a + b (B) a (C) b (D) ab
The number 3 + √2 is : (A) a rational number (B) an irrational number (C) an integer (D) a natural number
Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b. Reason (R) : HCF of any two natural numbers divides both the numbers. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.
Prove that √3 is an irrational number.
The factor tree of a number x is shown (branches: x → 2 and y; y → 2 and 210; 210 → a and 70; 70 → 2 and 35; 35 → 5 and b). Find the values of x, y, a and b. Hence, write the product of the prime factors of the number x so obtained.
Sample 2025 4 questions
HCF of (33 × 52 × 2), (32 × 53 × 22) and (34 × 5 × 23) is (A) 450 (B) 90 (C) 180 (D) 630
If H.C.F(420, 189) = 21 then L.C.M(420, 189) is (A) 420 (B) 1890 (C) 3780 (D) 3680
Assertion (A): (2 + √3)√3 is an irrational number. Reason (R): Product of two irrational numbers is always irrational.
Prove that √5 is an irrational number.
Board 2024 4 questions
HCF (132, 77) is : (a) 11 (b) 77 (c) 22 (d) 44
Prove that 6 - 4√5 is an irrational number, given that √5 is an irrational number.
Show that 11 × 19 × 23 + 3 × 11 is not a prime number.
Two alarm clocks ring their alarms at regular intervals of 20 minutes and 25 minutes respectively. If they first beep together at 12 noon, at what time will they beep again together next time ?
Board 2023 3 questions
The prime factorisation of natural number 288 is : (a) 24 × 33 (b) 24 × 32 (c) 25 × 32 (d) 25 × 31
Find the LCM and HCF of 92 and 510, using prime factorisation.
Prove that 3 + 7√2 is an irrational number, given that √2 is an irrational number.
Frequently asked questions
What are the most important topics in CBSE Class 10 Mathematics (Basic) Chapter 1 Real Numbers?
By how often and how recently they were asked: Proving irrationality (root 2, root 3, root 5 type); Fundamental Theorem of Arithmetic / prime factorisation; HCF-LCM relationship and word problems.
How many marks does Real Numbers carry in the CBSE Class 10 Mathematics (Basic) exam?
Real Numbers is part of the Number Systems unit, which carries 6 marks in the official syllabus. In the analysed papers it had about 3.6 questions per paper.
Do questions from Real Numbers repeat in board exams?
Yes — 8 question patterns from this chapter appeared in two or more past board or official sample papers. They are listed on this page with the years they were asked.