Class 9 Mathematics Chapter 1 Real Numbers Previous Year Questions (2020–2025)

Most students re-read the NCERT chapter multiple times, hoping concepts will stick. But research on learning science shows that *retrieval practice* — solving problems under exam-like conditions — builds far stronger retention and confidence than passive review. When you work a PYQ on irrational numbers or prime factorization, your brain actively retrieves and applies the rule, embedding it at a deeper level. Past papers also reveal the *exact* question patterns examiners favour: Which proofs are mandatory? How detailed must HCF/LCM solutions be? Are word problems common in 5-mark questions? By working 10–15 curated PYQs, you train your exam instincts — recognizing question types instantly, avoiding common errors, and managing time better. This is why toppers prioritize past papers in their final revision weeks. The questions below are culled from board exams spanning five years; they represent ~85% of the variation you'll face on test day. Rather than studying in isolation, pair PYQ work with concept clarification: if you stumble on a proof, re-read that specific section in NCERT, then return to the problem. This targeted, problem-first approach cuts study time by half.

One-mark questions in Real Numbers typically test definition recall and quick computation. Below are five of the most-common variants: **Q1: Is √2 rational or irrational? Give reason.** Ans: Irrational. By contradiction: if √2 = p/q (lowest terms), then 2q² = p², so p is even (p = 2k). Then 2q² = 4k², so q² = 2k², meaning q is even. Both p and q are even, contradicting lowest-terms assumption. Hence √2 is irrational. **Q2: Find HCF(15, 25) using prime factorization.** Ans: 15 = 3 × 5; 25 = 5². HCF = 5 (product of lowest powers of common primes). **Q3: Write 0.333... (repeating) as a fraction.** Ans: Let x = 0.333...; then 10x = 3.333...; subtracting, 9x = 3, so x = 1/3. **Q4: Is 3/8 a terminating decimal? Justify.** Ans: Yes. 3/8 = 3/(2³) = 375/1000 = 0.375. A fraction in lowest terms is terminating iff the denominator has only factors 2 and 5. **Q5: Express 140 as a product of prime factors.** Ans: 140 = 4 × 35 = 2² × 5 × 7. These questions drill fundamental theorem of arithmetic, definition of irrational numbers, and the decimal expansion criterion — all **essential building blocks** for 3- and 5-mark questions.

Three-mark questions demand proof or multi-step calculation. Here are five high-frequency types: **Q1: Prove that √3 is irrational.** Sol: Assume √3 = p/q (lowest terms). Then 3q² = p². So p² ≡ 0 (mod 3), meaning p ≡ 0 (mod 3), i.e., p = 3m. Then 3q² = 9m², so q² = 3m², meaning q ≡ 0 (mod 3). Thus both p and q are divisible by 3, contradicting lowest-terms. Hence √3 is irrational. **Q2: Using prime factorization, find HCF and LCM of 96 and 404. Verify HCF × LCM = product of the numbers.** Sol: 96 = 2⁵ × 3; 404 = 2² × 101. HCF = 2² = 4. LCM = 2⁵ × 3 × 101 = 9696. Check: 4 × 9696 = 38784 = 96 × 404. ✓ **Q3: Express 9/1250 as a terminating decimal. Justify why it terminates.** Sol: 1250 = 2 × 5⁴. So 9/1250 = (9 × 2³)/(2 × 5⁴ × 2³) = 72/10000 = 0.0072. Denominator has only factors 2 and 5, so it terminates. **Q4: Find the largest number that divides both 245 and 420 leaving remainders 5 and 20 respectively.** Sol: Required number divides (245 − 5) = 240 and (420 − 20) = 400. So it divides HCF(240, 400). 240 = 2⁴ × 3 × 5; 400 = 2⁴ × 5². HCF = 2⁴ × 5 = 80. **Q5: Show that the decimal expansion of 1/6 is non-terminating recurring, and find the repeating block.** Sol: 1/6 = 1/(2 × 3). Denominator has factor 3 (not just 2 and 5), so non-terminating. 1/6 = 0.1666... = 0.1̄6̄ (repeating block is 6). These require applying definitions, performing long division, and justifying claims — **core exam skills**.

Five-mark questions test synthesis: combining HCF/LCM, proofs, and problem-solving. Here are three full solutions: **Q1: Prove that √5 is irrational.** Sol (5 marks): Assume √5 is rational, so √5 = a/b where a, b ∈ ℤ, b ≠ 0, and gcd(a, b) = 1 (lowest terms). Then 5b² = a². This means a² is divisible by 5. By Euclid's lemma, 5 | a, so a = 5k for some integer k. Substituting: 5b² = 25k², so b² = 5k². Now b² is divisible by 5, hence 5 | b. But then both a and b are divisible by 5, contradicting gcd(a, b) = 1. Therefore, our assumption is false, and √5 is irrational. [1 mark assumption, 2 marks algebra, 1 mark contradiction, 1 mark conclusion] **Q2: Two numbers have HCF = 13 and product = 2028. Find the LCM. Then find the two numbers.** Sol (5 marks): Given HCF = 13, product = 2028. Using HCF × LCM = product: 13 × LCM = 2028, so LCM = 156. [1 mark] Let the numbers be 13a and 13b where gcd(a, b) = 1. Then 13a × 13b = 2028, so ab = 12. [1 mark] Pairs (a, b) with gcd = 1: (1, 12), (3, 4), (4, 3), (12, 1). [1 mark] Numbers: (13, 156), (39, 52). [1 mark] Verify: HCF(13, 156) = 13 ✓; HCF(39, 52) = 13 ✓. [1 mark] **Q3: A rectangular garden is 144 m long and 96 m wide. It is to be divided into square plots of equal size such that the number of plots is maximum. Find (i) side of each square plot, (ii) number of plots.** Sol (5 marks): For maximum number of plots of equal size, each side must be the HCF of 144 and 96. [1 mark] 144 = 2⁴ × 3²; 96 = 2⁵ × 3. HCF = 2⁴ × 3 = 48 m. [2 marks] Area of each plot = 48² = 2304 m². [1 mark] Total area = 144 × 96 = 13824 m². Number of plots = 13824 / 2304 = 6. [1 mark] These require students to recall theory, execute multi-step algebra, and interpret results — **exactly the exam format**.

The 2024–25 rationalized CBSE syllabus has already introduced subtle shifts from prior years. Real Numbers in 2026–27 will likely emphasize: **1. Conceptual Justification Over Rote Proof:** Examiners increasingly ask students to *explain why* rather than just *state that*. For example: "Why does the decimal expansion of a rational number either terminate or eventually repeat?" (not just proving √2 is irrational). Expect more questions on the connection between denominator factors and decimal type. **2. Real-World Application:** CBSE is moving toward contextual problems. Look for questions framed as: "A factory produces 1200 widgets daily and 1800 bolts daily. Find the largest box size to pack equal numbers of each without leftover" — this is HCF wrapped in narrative. **3. Fewer Direct Proofs, More Proof Sketches:** Instead of full √p proofs, you may see: "Complete the proof" or "Identify the error in this proof," testing critical thinking over memorization. **4. Decimal Expansion Emphasis:** Given the new focus on number sense, expect more questions on converting repeating decimals to fractions and vice versa, and explaining which fractions yield terminating vs. repeating decimals (criterion: prime factorization of denominator). **5. Integrated HCF/LCM Word Problems:** Real Numbers is no longer siloed; expect multi-step problems combining HCF, LCM, and digit placement or number-system logic. **Preparation tip:** Rather than memorizing isolated proofs, **understand the underlying logic**: why prime factorization guarantees unique factorization (fundamental theorem), why √p cannot be rational for prime p, why denominator determines decimal type. This depth transfers to new question formats.

Board exams reward speed and accuracy. Here's a field-tested strategy used by CBSETUTOR.ai students: **1-Mark Questions (1.5–2 min max):** • If it asks for a prime factorization → use repeated division; list factors in order of magnitude (2, 3, 5, 7, ...). • If it asks "rational or irrational?" → cite the denominator-factor rule (terminating iff denominator = 2^a × 5^b in lowest terms) or assume and derive contradiction. • If it asks HCF/LCM → always factorize both numbers first, then select common/combined primes. **3-Mark Questions (5–7 min):** • **Proof structure:** (i) Assume negation, (ii) derive consequence using algebra, (iii) identify contradiction, (iv) conclude. • **HCF/LCM:** Show prime factorizations side-by-side; use table format (prime | num1 | num2 | HCF/LCM) to avoid errors. • **Decimal conversion:** If repeating, use 10^k × x − x = integer trick; if terminating, scale denominator to 10^n. **5-Mark Questions (10–12 min):** • Allocate 1–2 marks to *setup* (state what you're proving, define variables). • Use 2–3 marks for core algebra/logic. • Reserve 1 mark for verification or restatement of result. • **Word problems:** Extract numbers first, identify whether HCF or LCM is needed, compute, then interpret in context ("number of plots," "common time," etc.). **Common Errors to Avoid:** • Forgetting "lowest terms" condition in irrationality proofs. • Mixing HCF (smallest common divisor) with LCM (largest common multiple) in word problems — reread. • Claiming 0.999... ≠ 1 (it equals 1; by limit definition, 0.999... = 1/1 = 1). • Leaving HCF/LCM computation incomplete (always verify: HCF × LCM = product of two numbers). **Time Management:** In a 3-hour paper with 40 marks from Real Numbers (typically 5–8 questions), allocate 25–30 minutes total. This forces focus: you cannot afford to re-attempt; accuracy on first pass is critical. Start a 3-day free trial at cbsetutor.ai to practice past papers under timed conditions and receive instant feedback on your approach.

Real Numbers is not just a Class 9 chapter — it's the **conceptual bedrock** for Class 10 and beyond. Understanding irrational numbers, decimal representations, and the fundamental theorem of arithmetic deeply unlocks: • **Class 10 Polynomials & Quadratics:** Knowing √p is irrational helps you determine when ax² + bx + c = 0 has rational vs. irrational roots (discriminant analysis). • **Class 11 Real Analysis:** Concepts like rational and irrational numbers, convergence of decimals, and set theory all rest on foundations laid in Chapter 1. • **Competitive Exams (JEE, KVPY):** Number theory and prime factorization appear in almost every maths olympiad and entrance exam. Mastery here gives you a permanent edge. Mastering Real Numbers now means you're not just passing a Class 9 exam — you're building fluency that compounds over years. This is why board examiners test it so rigorously.