Class 10 Mathematics — Chapter 1: Real Numbers
69 practice questions · 23 Easy · 23 Medium · 23 Hard
Practise Class 10 Mathematics Chapter 1, "Real Numbers", with 69 NCERT-aligned multiple-choice questions. The set is split into 23 Easy, 23 Medium and 23 Hard questions, so you can warm up on the fundamentals and then push into the exam-level problems that separate top scorers in CBSE Board exams and the JEE & NEET foundation years.
"Real Numbers" is one of the chapters where problem-solving speed, formula recall and step-by-step reasoning really pays off. Each MCQ on this chapter is timed and uses exam-grade marking (+2 correct, −1 wrong, 0 skipped), training the same accuracy-under-pressure that real papers demand. Every question carries a short explanation, so a wrong answer becomes a quick lesson rather than a dead end — the fastest way to close gaps before a test.
Use this chapter as targeted revision: attempt the Easy set first to confirm your basics on Real Numbers, then move to Medium and Hard to test application and problem-solving. Your accuracy, streaks and XP save automatically, and the chapter feeds into your overall Class 10 Mathematics mastery score. A few sample questions are shown below; sign in free to practise all 69.
Key concepts: Real Numbers (Class 10 Mathematics)
Real numbers are built from primes. This chapter uses prime factorisation to find HCF and LCM, and proves that numbers like √2 are irrational.
- Fundamental Theorem of Arithmetic
- Every composite number can be written as a product of primes in exactly one way, apart from the order of the factors.
- HCF & LCM by prime factorisation
- HCF = product of the smallest power of each common prime; LCM = product of the greatest power of each prime appearing in the numbers.
- HCF–LCM relationship
- For any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b. This holds for two numbers only.
- Irrational numbers
- A number that cannot be written as p/q (q ≠ 0). Examples: √2, √3, √5. Their decimal form is non-terminating and non-repeating.
- Proving irrationality
- Use proof by contradiction: assume the number is rational (p/q in lowest terms), derive that p and q share a common factor, contradicting 'lowest terms'.
Key formulas — Real Numbers
💡 Exam tips for Real Numbers
- The HCF × LCM = product rule works for two numbers only — never for three.
- To prove √p is irrational (p prime), assume √p = a/b in lowest terms and reach a contradiction.
Sample questions
The prime factorisation of 156 is:
156 = 2 × 78 = 2 × 2 × 39 = 2² × 3 × 13.
Which of these has a terminating decimal expansion?
p/q (lowest terms) terminates iff q has only 2 and/or 5 as prime factors. 3125 = 5⁵.
Which method is used to prove that √2 is irrational?
Assume √2 = p/q in lowest terms; show p and q must both be even — contradicting 'lowest terms'.
Real Numbers — FAQs
What are the key concepts in Class 10 Mathematics Real Numbers?+
Real numbers are built from primes. This chapter uses prime factorisation to find HCF and LCM, and proves that numbers like √2 are irrational. Key ideas include Fundamental Theorem of Arithmetic, HCF & LCM by prime factorisation, HCF–LCM relationship, Irrational numbers, Proving irrationality.
What does Class 10 Mathematics Chapter 1 (Real Numbers) cover on XamBaaz?+
It covers 69 NCERT-aligned MCQs on "Real Numbers" — 23 Easy, 23 Medium and 23 Hard — each with a timed quiz and an instant explanation, suitable for CBSE Board exams and the JEE & NEET foundation years.
Are these "Real Numbers" questions free to practise?+
Yes — sign in with Google to practise "Real Numbers" free. Full unlimited access is ₹999/year (limited-time launch price), with no per-chapter charges.
How should I revise "Real Numbers" for the exam?+
Start with the Easy quiz to confirm your fundamentals, then attempt Medium and Hard for application-level practice. Review each explanation, retry the questions you miss, and track your accuracy on this chapter until it is consistently high.
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