Real Numbers — Class 10 MCQs with Answers
Class 10 CBSE Mathematics · Chapter 1
108 practice questions · 31 Easy · 39 Medium · 38 Hard · Updated
Jump to the 6 solved questions
Practise the most important Class 10 CBSE Mathematics questions from Chapter 1, "Real Numbers". You get 11 timed quizzes made from 108 NCERT-based MCQs, with answers and explanations. The questions are split into 31 Easy, 39 Medium and 38 Hard. Warm up on the basics, then move on to the exam-level questions that top scorers in CBSE Board exams and the JEE & NEET foundation years get right.
To score well in "Real Numbers", focus on fast problem-solving, formula recall and step-by-step working. Each MCQ here is timed and uses exam-style marking (+4 correct, −1 wrong, 0 skipped). This helps you stay accurate when time is short, just as real papers need. Every question has a short explanation, so a wrong answer becomes a quick lesson. It is the fastest way to fix gaps before a test.
Use this chapter for focused revision. Start with the Easy set to check your basics on Real Numbers, then move to Medium and Hard to practise applying them. Your accuracy, streaks and XP save automatically. This chapter also adds to your overall Class 10 Mathematics mastery score. Try the 6 sample questions below: tap an option to see the answer and a worked explanation. Sign in free to start practising.
Real Numbers — important questions & MCQs with answers (Class 10 Mathematics)
6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.
0 of 6 answered
- Q1 of 6Easy
The HCF of 12 and 18 is:
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Answer: 6
Explanation: Prime factorise both: 12 = 2²×3 and 18 = 2×3². HCF uses the lowest power of each common prime, 2¹×3¹ = 6 — not the highest power, which would instead give the LCM (36).
- Q2 of 6Easy
Which of the following is irrational?
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Answer: √2
Explanation: √4 = 2, 22/7, and 0.333... = 1/3 are all ratios of integers, so they're rational. √2 cannot be written as p/q since 2 is not a perfect square, making it irrational.
- Q3 of 6Medium
Which of these has a terminating decimal expansion?
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Answer: 13/3125
Explanation: A fraction in lowest terms terminates only if its denominator has no prime factors besides 2 and 5. 30, 21 and 455 all carry a 3, 7 or 13, but 3125 = 5⁵ is pure powers of 5, so 13/3125 terminates.
- Q4 of 6Medium
Assertion (A): If HCF(120, 144) = 24, then LCM(120, 144) = 720. Reason (R): For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
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Answer: Both A and R are true and R is the correct explanation of A
Explanation: A is true: LCM = (120 × 144) ÷ 24 = 17280 ÷ 24 = 720. R is true, and A is found directly from R.
- Q5 of 6Hard
Which method is used to prove that √2 is irrational?
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Answer: Contradiction
Explanation: The standard proof assumes √2 = p/q in lowest terms and derives that both p and q must be even, contradicting the 'lowest terms' assumption — this is proof by contradiction (assume the opposite, derive a contradiction), not induction or a direct construction.
- Q6 of 6Hard
Assertion (A): For the numbers 4, 6 and 10, HCF × LCM is not equal to 4 × 6 × 10. Reason (R): For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
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Answer: Both A and R are true but R is not the correct explanation of A
Explanation: A is true: HCF = 2 and LCM = 60, so HCF × LCM = 120, while 4 × 6 × 10 = 240. R is true but holds only for two numbers, so it does not explain the result for three numbers.
Real Numbers notes: key concepts (Class 10 Mathematics)
Every composite number breaks into primes in exactly one way, and that single fact settles HCF, LCM and the irrationality of numbers such as √2. This chapter also explains when a fraction's decimal expansion terminates and when it repeats forever.
- Fundamental Theorem of Arithmetic
- Every composite number can be written as a product of primes, and that factorisation is unique apart from the order of the factors.
- Prime factorisation
- Expressing a number as a product of primes, such as 140 = 2² × 5 × 7. It is the starting point for both HCF and LCM.
- HCF (Highest Common Factor)
- The largest number dividing two numbers exactly; found by taking the SMALLEST power of each common prime.
- LCM (Lowest Common Multiple)
- The smallest number both numbers divide into; found by taking the HIGHEST power of every prime appearing in either.
- Relation between HCF and LCM
- For any two numbers, HCF × LCM equals the product of the numbers. It lets you find one when the other three are known.
- Irrational number
- A number that cannot be written as p/q with integers p and q. Its decimal expansion neither terminates nor repeats.
- Proving irrationality
- √2 is shown irrational by contradiction: assume √2 = p/q in lowest terms, and derive that p and q share a factor 2.
- Terminating decimal condition
- A fraction in lowest terms terminates exactly when its denominator has the form 2ⁿ5ᵐ — no other prime may appear.
- Non-terminating recurring decimal
- If the denominator in lowest terms contains any prime other than 2 or 5, the expansion repeats forever.
- Rational vs irrational operations
- The sum or product of a rational and an irrational is always irrational, which is how expressions like 3 + √5 are classified.
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Real Numbers — FAQs
What are the key concepts in Class 10 Mathematics Real Numbers?+
Every composite number breaks into primes in exactly one way, and that single fact settles HCF, LCM and the irrationality of numbers such as √2. This chapter also explains when a fraction's decimal expansion terminates and when it repeats forever. Key ideas include Fundamental Theorem of Arithmetic, Prime factorisation, HCF (Highest Common Factor), LCM (Lowest Common Multiple), Relation between HCF and LCM, Irrational number.
What does Class 10 Mathematics Chapter 1 (Real Numbers) cover on XamBaaz?+
It has 108 NCERT-based MCQs on "Real Numbers": 31 Easy, 39 Medium and 38 Hard. Together they make 11 timed quizzes, and you never get the same set twice. Every question has an instant explanation. They help you prepare 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. One year from the day you pay. You stay in Class 10 till 31 March; on 1 April your account moves up to Class 11 and the rest of your year carries over. No chapter is charged separately.
How should I revise "Real Numbers" for the exam?+
Start with the Easy quiz to check your basics, then try Medium and Hard to practise applying them. There are 11 timed quizzes on this chapter, so you can come back for a fresh set instead of one you have seen. Read each explanation, retry the questions you miss, and track your accuracy until it stays high.
Are these "Real Numbers" MCQs available with answers?+
Yes. 6 sample questions are on this page with answers: tap an option, or "Show answer", to see the correct option and a step-by-step explanation. Sign in free with Google to start practising, with instant scoring.
Is there negative marking in the "Real Numbers" quizzes?+
Yes. The timed quizzes use exam-style marking: +4 for a right answer, −1 for a wrong one and 0 for a skip. MHT-CET and CBSE board papers have no negative marking. Our mocks for those are scored their way.
What are the important questions from Real Numbers (Class 10 Mathematics)?+
The questions that matter most test Fundamental Theorem of Arithmetic, Prime factorisation, HCF (Highest Common Factor), LCM (Lowest Common Multiple), Relation between HCF and LCM, Irrational number. This page shows 6 solved important MCQs with answers and explanations. Sign in to practise all 108 questions on the chapter as timed quizzes.
Is there an online quiz for Real Numbers?+
Yes — Class 10 Mathematics Real Numbers has timed online quizzes at Easy, Medium and Hard levels, with instant scoring and a worked explanation on every question. The first quiz on the chapter is free.
Are there question answers and extra questions for Real Numbers (Class 10 Mathematics)?+
Yes. This page has 6 solved questions on Real Numbers, each with the correct answer and a short explanation. Sign in to practise all 108 questions on the chapter as timed quizzes.
Are there assertion–reason and case study questions for Real Numbers (Class 10 Mathematics)?+
Yes. The Real Numbers question bank has 6 assertion–reason questions and 12 case-based (case study) questions in the CBSE board format, alongside the regular MCQs. Each comes with the answer and a worked explanation. Sign in to practise them in timed quizzes.