Class 9 Mathematics Chapter 5: Squares and Square Roots Previous Year Questions (2020–2025)

Many Class 9 students make the mistake of reading the textbook repeatedly, hoping concepts will 'stick.' In reality, Squares and Square Roots rewards active problem-solving. Past papers reveal which concepts the examiner prioritizes: long division method for √2, √3, √5; Pythagorean triplet identification (3–4–5, 5–12–13, 8–15–17); and square number properties (perfect squares end in 0, 1, 4, 5, 6, or 9). By solving actual exam questions, you train your brain to: • Recognize 'Pythagorean triplet' as a signal to check if a² + b² = c² • Know when to use prime factorisation vs. long division for square roots • Spot trap answers designed to catch careless errors • Build speed: 1-mark questions should take ≤1 minute, 3-mark ≤3 minutes Each question you solve from a past paper is a 'real' signal about what the examiner values. You're not guessing what might appear—you're seeing what *has* appeared. This targeted practice compresses months of vague study into weeks of sharp, focused revision.

1-mark questions test recall and quick calculation. These five patterns recur almost every year: **Q1: Is 2352 a perfect square?** Answer: No. Working: Find prime factorisation: 2352 = 2⁴ × 3 × 7². For a perfect square, all prime factors must have even exponents. Here, 3 appears once (odd exponent), so 2352 is not a perfect square. **Q2: How many perfect squares lie between 40 and 120?** Answer: 4 (they are 49, 64, 81, 100). Working: √40 ≈ 6.3, √120 ≈ 11.0. Integers from 7 to 11 are 7, 8, 9, 10, 11. Their squares: 49, 64, 81, 100, 121. Only 49, 64, 81, 100 lie between 40 and 120. **Q3: Write a Pythagorean triplet where one number is 6.** Answer: (6, 8, 10) or (6, 10, 8) depending on position. [Note: Use the formula 2m, m²−1, m²+1 with m=3: 6, 8, 10.] Working: Check: 6² + 8² = 36 + 64 = 100 = 10². ✓ **Q4: √196 = ?** Answer: 14. Working: 14² = 196, so √196 = 14. Or: 196 = 4 × 49 = 2² × 7², so √196 = 2 × 7 = 14. **Q5: Is 87 a perfect square? Justify.** Answer: No. Working: √87 ≈ 9.3, so 9² = 81 and 10² = 100. Since 87 lies between 81 and 100, it is not a perfect square.

3-mark questions combine two or three concepts or require a structured method. Here are the patterns that dominate: **Q1: Find √1521 by prime factorisation.** Solution: 1521 = 3 × 507 = 3 × 3 × 169 = 9 × 169 = 3² × 13² √1521 = 3 × 13 = 39. Verification: 39² = 1521. ✓ **Q2: Find √2 correct to 3 decimal places using long division method.** Solution: √2 = 1.414... By long division: - 2.000000 ÷ 1 gives quotient 1, remainder 1 - Bring down 00: 100. Next divisor: 24. 24 × 4 = 96, remainder 4 - Bring down 00: 400. Next divisor: 281. 281 × 1 = 281, remainder 119 - Continue to 3 decimal places: √2 ≈ 1.414 **Q3: Verify that (20, 21, 29) is a Pythagorean triplet.** Solution: 20² = 400 21² = 441 29² = 841 Check: 400 + 441 = 841? Yes, 841 = 841. ✓ Therefore, (20, 21, 29) is a Pythagorean triplet. **Q4: A number when multiplied by 8 gives 2048. Find the original number. Is the result a perfect square?** Solution: Let the number be x. Then 8x = 2048, so x = 2048 ÷ 8 = 256. Is 256 a perfect square? 256 = 2⁸ = (2⁴)² = 16². Yes, 256 is a perfect square. **Q5: Find the smallest number by which 252 must be multiplied to make it a perfect square.** Solution: 252 = 4 × 63 = 4 × 9 × 7 = 2² × 3² × 7¹ For a perfect square, all exponents must be even. The exponent of 7 is 1 (odd), so multiply by 7. 252 × 7 = 1764 = 2² × 3² × 7² = 42². Answer: Multiply by 7.

5-mark questions test deep understanding, multi-step reasoning, and often combine 2–3 concepts. Expect these types: **Q1: A school has 2116 students. They want to arrange them in rows and columns such that the number of rows equals the number of columns. How many students should be in each row? Verify your answer.** Solution: For a square arrangement: number of rows = number of columns = √2116. Prime factorisation: 2116 = 4 × 529 = 4 × 23² = 2² × 23² √2116 = 2 × 23 = 46 Each row (and each column) has 46 students. Verification: 46 × 46 = 2116. ✓ **Q2: Find √5929 using long division method. Then express 5929 as a sum of two perfect squares.** Solution: √5929 = 77 (by long division, grouping digits from right: 59|29) Verification: 77² = 5929. ✓ Express as sum of two perfect squares: 5929 = 77² = (70 + 7)² — but use Pythagorean triplet approach. Note: If (a, b, c) is a Pythagorean triplet with c² = 5929, find a and b. Example: Use (a, b, 77) where a² + b² = 77² = 5929. Trying: 36² + 65² = 1296 + 4225 = 5521 (no). Use systematic search or factorisation. 5929 = 77² = (7 × 11)² — no simpler pair found in standard triplets. (Answer depends on specific triplet data provided.) **Q3: Prove that (m²−1)/2, m, (m²+1)/2 form a Pythagorean triplet for odd m > 1. Verify for m = 5.** Solution: Let a = (m²−1)/2, b = m, c = (m²+1)/2. Prove a² + b² = c²: a² + b² = [(m²−1)/2]² + m² = (m⁴−2m²+1)/4 + m² = (m⁴−2m²+1+4m²)/4 = (m⁴+2m²+1)/4 = [(m²+1)/2]² = c². ✓ For m = 5: a = (25−1)/2 = 12, b = 5, c = (25+1)/2 = 13 Check: 12² + 5² = 144 + 25 = 169 = 13². ✓ (5, 12, 13) is a Pythagorean triplet. To deepen your command of such problems, start a 3-day free trial at cbsetutor.ai, where live tutors guide you through past papers step-by-step.

The 2024–25 CBSE rationalized syllabus has streamlined Class 9 Mathematics, and examiners are shifting focus. Key changes to expect: **Increased emphasis on application and reasoning:** Questions no longer ask just 'Find √529.' Instead: 'A rectangular plot has area 529 m². If one side is a whole number, find both dimensions. Is the plot a square?' This integrates perfect squares with geometry and real-world context. **Greater focus on Pythagorean triplets in geometry:** Expect more link-ups with right triangles in coordinate geometry and trigonometry (Class 10 preview). Questions like: 'Plot the points (0,0), (3,4), (3,0) on a grid. Is the triangle right-angled? Verify using the Pythagorean theorem.' **Long division method moving to 'understanding' rather than 'rote calculation':** Instead of 'Find √7 to 2 decimal places,' expect: 'Explain why √7 lies between 2 and 3, then narrow it down.' This tests conceptual depth. **Harder, multi-step Pythagorean questions:** Look for: 'Generate three different Pythagorean triplets using the formula 2m, m²−1, m²+1. Are all of them primitive triplets? Justify.' This combines recall, generation, and proof. **Mixed number types:** Questions now blend perfect squares, primes, and composites: 'Among 100, 101, 102, ..., 200, identify all perfect squares and verify each is not a Pythagorean triple hypotenuse.' Stay ahead by solving papers from 2020 onwards, which already reflect this shift toward conceptual and applied thinking.

Time management and accuracy are critical in math exams. Here's a battle-tested strategy for Chapter 5: **Before the exam (15 minutes review):** • Memorize perfect squares from 1² to 20² (1, 4, 9, 16, ..., 400). • Memorize the five most common Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29). • Recall the prime factorisation rule: exponents must all be even for perfect squares. • Know the long division method by heart: pair digits from right, find quotient, bring down next pair. **During the exam (question order):** 1. **Scan and mark:** Read all questions. Lightly mark 1-mark questions (≤1 min each), 3-mark (≤3 min), 5-mark (≤5 min). 2. **Start with 1-mark questions:** Quick wins build confidence. If 'Is 1936 a perfect square?' factorize in seconds: 1936 = 16 × 121 = 4² × 11² → Yes. 3. **Move to 3-mark questions:** Use prime factorisation or long division, write all steps. Partial credit rewards working. 4. **Tackle 5-mark questions last:** These need careful reading. Underline 'Verify,' 'Prove,' 'Express.' Plan before writing. 5. **Verification habit:** After every square root or Pythagorean check, square your answer. (39² = 1521? Yes → confident.) **Common pitfalls to avoid:** • Forgetting that √4 = 2, not ±2 (in this context, principal square root only). • Writing (3, 4, 5) as (5, 3, 4)—order matters for clarity. • Miscounting digit pairs in long division (group right-to-left, not left-to-right). • Assuming 'multiply to make perfect square' means multiply each factor—it means multiply the whole number by the missing factor(s). **Last 5 minutes:** Don't start new questions. Review your working. Correct arithmetic errors. Ensure all 5-mark solutions have proper justification, not just an answer.

Carry these five facts into your exam as 'anchors': **1. Perfect Square Properties:** A number is a perfect square if and only if all exponents in its prime factorisation are even. Example: 72 = 2³ × 3². Exponent of 2 is 3 (odd) → not a perfect square. Multiply by 2 to get 144 = 2⁴ × 3² → perfect square. **2. Pythagorean Triplet Formula:** For any integer m > 1, the triplet (2m, m²−1, m²+1) is always Pythagorean. Quick check: (2×3, 9−1, 9+1) = (6, 8, 10). Verify: 36 + 64 = 100. ✓ **3. Long Division Pattern for Square Roots:** Pair digits from right to left. At each step, find the largest digit d such that (20×quotient + d) × d ≤ current remainder + brought-down pair. Write d as next quotient digit. Repeat. **4. Interval Narrowing:** If n² < N < (n+1)², then n < √N < n+1. Use this to estimate square roots of non-perfect squares. Example: 9 < 10 < 16, so 3 < √10 < 4. **5. Count of Perfect Squares Between a and b:** Find the smallest integer k such that k² ≥ a, and largest m such that m² ≤ b. Count = m − k + 1. Example: Between 10 and 100: smallest k is 4 (since 4² = 16), largest m is 10 (since 10² = 100). Count = 10 − 4 + 1 = 7. (16, 25, 36, 49, 64, 81, 100.)