Class 9 Mathematics Chapter 6: Cubes and Cube Roots — Previous Year Questions (2020-2025)

Reading your NCERT textbook once teaches *what* a perfect cube is. Solving 10 past-paper questions teaches *how examiners test* that concept. Over the last 5 years, CBSE has repeatedly asked: (1) Definition-based 1-mark questions disguised as true/false, (2) Cube root calculations where you must choose between prime factorisation and digit patterns, (3) Multi-step problems combining cubes, cube roots, and algebraic expressions. Students who skip past papers often waste time perfecting methods that aren't tested, or freeze when they see an unfamiliar number arrangement. Practicing actual exam questions removes that surprise factor. You learn which perfect cubes (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, and their multiples) appear most often. You discover that examiners love testing ∛(8 × 125) = ∛8 × ∛125 = 2 × 5 = 10 — not just ∛1000. This page gives you 13 real-style questions. Attempt each one before checking the solution; time yourself at 3 minutes per 1-mark, 5 minutes per 3-mark, and 8 minutes per 5-mark.

**Question 1:** Is 216 a perfect cube? If yes, find its cube root. **Answer:** Yes. 216 = 6³, so ∛216 = 6. **Explanation:** Prime factorisation: 216 = 2³ × 3³ = (2 × 3)³ = 6³. Examiners often ask this to check if you can group prime factors into triplets. **Question 2:** Which of the following is NOT a perfect cube? (a) 343 (b) 512 (c) 125 (d) 200 **Answer:** (d) 200. **Explanation:** 343 = 7³, 512 = 8³, 125 = 5³. For 200: 200 = 2³ × 5² — the exponent of 5 is 2, not a multiple of 3, so 200 is not a perfect cube. A perfect cube requires all prime factors to have exponents divisible by 3. **Question 3:** Define a perfect cube in one sentence. **Answer:** A perfect cube is a number that can be expressed as n³ for some integer n. **Question 4:** Is the cube root of a negative number defined? Explain briefly. **Answer:** Yes. Unlike square roots, cube roots of negative numbers are defined. For example, ∛(−8) = −2, because (−2)³ = −8. This is tested to distinguish cubes from squares. **Question 5:** True or False: ∛(x³ + y³) = ∛x³ + ∛y³ for all real x and y. **Answer:** False. **Explanation:** ∛(x³ + y³) ≠ ∛x³ + ∛y³ in general. Example: x = 1, y = 1. LHS = ∛2 ≈ 1.26. RHS = 1 + 1 = 2. Cube root is not distributive over addition.

**Question 1:** Find the cube root of 13,824 by prime factorisation. **Solution:** Step 1: Prime factorise 13,824. 13,824 ÷ 2 = 6,912 6,912 ÷ 2 = 3,456 3,456 ÷ 2 = 1,728 1,728 ÷ 2 = 864 864 ÷ 2 = 432 432 ÷ 2 = 216 216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 13,824 = 2⁹ × 3³ = (2³)³ × 3³ = (8 × 3)³ = 24³ Step 2: ∛13,824 = 24. **Question 2:** Simplify ∛(8 × 125 × 343). **Solution:** ∛(8 × 125 × 343) = ∛8 × ∛125 × ∛343 = 2 × 5 × 7 = 70. **Note:** Using the property ∛(a × b) = ∛a × ∛b saves calculation time when each factor is a perfect cube. **Question 3:** Find the smallest number by which 1,296 must be multiplied to make it a perfect cube. **Solution:** Prime factorisation: 1,296 = 2⁴ × 3⁴. For a perfect cube, all exponents must be divisible by 3. Current: 2⁴ (exponent 4), 3⁴ (exponent 4). Required: 2⁶ (exponent 6), 3⁶ (exponent 6). Multiply by: 2² × 3² = 4 × 9 = 36. Verify: 1,296 × 36 = 46,656 = 2⁶ × 3⁶ = (2² × 3²)³ = 36³. ✓ **Question 4:** If ∛x = 5, find the value of x. Also verify your answer. **Solution:** Cube both sides: x = 5³ = 125. Verify: ∛125 = 5. ✓ **Question 5:** Between which two integers does ∛250 lie? **Solution:** 6³ = 216, 7³ = 343. Since 216 < 250 < 343, we have 6 < ∛250 < 7. More precisely, ∛250 ≈ 6.3.

**Question 1:** A cubical box has a side length of ∛64 cm. Find: (a) the exact side length, (b) the volume in cm³, (c) the surface area in cm². **Solution:** (a) Side length = ∛64 = 4 cm. [Since 4³ = 64.] (b) Volume = (side)³ = 4³ = 64 cm³. [Or directly, volume of a cube with side ∛64 is (∛64)³ = 64 cm³.] (c) Surface area of cube = 6 × (side)² = 6 × 4² = 6 × 16 = 96 cm². **Question 2:** Two numbers are in the ratio 2 : 3. If their cubes are in the ratio 8 : 27, verify this relationship and find the numbers if the first is 10. **Solution:** Let numbers be 2k and 3k. Cubes: (2k)³ : (3k)³ = 8k³ : 27k³ = 8 : 27. ✓ If first number = 10, then 2k = 10, so k = 5. Second number = 3k = 3(5) = 15. Verify cubes: 10³ = 1,000 and 15³ = 3,375. Ratio = 1,000 : 3,375 = 8 : 27. ✓ **Question 3:** Find the cube root of 17,576 using prime factorisation, then use it to solve: If a cube has volume 17,576 cm³, what is its edge length? Also calculate the total surface area of this cube. **Solution:** Step 1: Prime factorisation of 17,576. 17,576 = 2³ × 13³ = (2 × 13)³ = 26³. ∛17,576 = 26. Step 2: Edge length of cube = 26 cm. Step 3: Surface area = 6 × 26² = 6 × 676 = 4,056 cm². --- **General 5-Mark Strategy:** Examiners combine three mini-tasks: (1) Factorisation or digit recognition to find ∛n, (2) Application to geometry (volume/surface area) or ratios, (3) Verification or a related calculation. Time your work: 2 minutes for factorisation, 3 minutes for application and verification.

**Old Pattern (2020-2025):** Section A: Five 1-mark questions (definition, true/false, identification of perfect cubes). Section B: Six 3-mark questions (cube roots via prime factorisation, simplification of cube root expressions). Section C: Two 5-mark questions (word problems, multi-step calculations). Total: 40 marks over 2 hours. Weightage to Chapter 6 ≈ 8–10 marks per paper. **New 2026-27 CBSE Pattern:** Increased emphasis on case study and application-based questions. Expect fewer pure definition questions; more questions like: "A factory produces cubical containers. The volume of one container is 729 cm³. A new design increases volume to 4,913 cm³. Find the ratio of edge lengths and the percentage increase in surface area." This requires cube roots, ratios, percentages, and geometry—all in one question. **Key Shift:** Problem-solving will be more integrated. Isolated "Find ∛x" questions will decrease. Instead, you'll see cube roots embedded in real scenarios (water tanks, shipping boxes, cost calculations). Practising only textbook problems won't be enough; you must solve past papers *and* think about how this chapter connects to Numbers, Algebra, and Geometry. **What This Means for You:** Don't just memorize ∛1728 = 12. Understand *why* it matters: if a cubic tank holds 1,728 litres, and you need to double the volume to 13,824 litres, the new edge length is 24 cm (not 24, but exactly 24 because 13,824 = 24³). That's the mindset examiners now test.

**For 1-Mark Questions (1.5 min per question):** 1. If asked "Is n a perfect cube?" — mentally check if all prime factors have exponents divisible by 3. If yes, cube root is the product of primes with exponents ÷ 3. 2. If asked to compare or identify — recall the first 10 perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Spot patterns: cubes of multiples of 10 end in 000 (e.g., 1000, 8000); cubes of odd numbers are odd. **For 3-Mark Questions (5 min per question):** 1. Always prime-factorise first, even if the number looks large. Write each prime factor clearly. 2. Group factors into triplets (exponent 3). If exponents are already multiples of 3, cube root is immediate. 3. Use the property ∛(a × b × c) = ∛a × ∛b × ∛c only if *each* is a perfect cube. Don't split arbitrarily. 4. Double-check: cube your answer. If ∛n = m, verify m³ = n. **For 5-Mark Questions (8 min per question):** 1. Read the problem twice. Underline all numerical data and what's asked. 2. Separate into sub-parts: (a) Find cube root, (b) Apply (volume, ratio, etc.), (c) Verify or extend. 3. Show all working. Marks are awarded for method, not just the final answer. 4. If the number is ugly (e.g., 19,683), don't panic—systematic factorisation always works. 19,683 = 3⁹ = (3³)³ = 27³, so ∛19,683 = 27. **Common Mistakes to Avoid:** - Forgetting that ∛(−8) = −2 (cube root preserves sign). - Assuming ∛(a + b) = ∛a + ∛b (it doesn't). - Stopping at ∛1728 = 12 without simplifying further if asked for a simplified form. - Not showing prime factorisation steps—examiners need to see your method.

After working through these 13 questions, do the following: 1. **Retake one 5-mark question without notes.** Time yourself. Aim for 8 minutes or less and full marks. 2. **Attempt your school's pre-board or mock paper** (Chapters 1–6 combined). This tests whether you can spot cube-root problems mixed with other topics. 3. **Create a small flashcard set** of the first 10 perfect cubes and their cube roots. Spend 2 minutes daily for a week. Instant recall saves exam time. 4. **Watch for CBSE official sample papers** released mid-year. Solve them under timed conditions. They often hint at the year's emphasis (e.g., more geometry-combined questions). 5. **Join cbsetutor.ai's live doubt sessions** (available for premium members) where instructors solve real past-paper questions and explain common errors in real-time. Consistency beats intensity. Solve 2–3 new past-paper questions daily for 10 days rather than 13 in one sitting. This spacing helps memory retention and reduces exam-day anxiety.