Class 9 Mathematics Chapter 12 Factorisation: Complete Previous Year Questions (2020–2025)

Reading Chapter 12 once is passive. Solving a 2022 board paper question on factorisation 6ab + 9b² – 3b involves *decision-making*: Do I extract the GCD first? Should I recognise a perfect square after factoring? These micro-decisions train your brain in a way textbook examples do not. Past papers reveal the CBSE's testing priorities: • 1-mark questions test *speed*—can you factor 2x + 4 in 30 seconds? • 3-mark questions test *method clarity*—show your regrouping steps or lose marks. • 5-mark questions test *proof and application*—you must divide polynomials correctly and justify each step. Students who solve even 5 previous year questions score 8–10 marks higher than those who only revise theory. The reason: repetition builds automaticity. Once you've factorised 4a² – 9 using (a² – b²) = (a + b)(a – b) five times across different papers, you no longer *think* about it—you see it instantly. This speed is the difference between finishing all questions and running out of time. Your exam is marked by a human examiner following a fixed rubric—not an algorithm. Seeing *how* examiners expect you to lay out factorisation steps (especially for regrouping and algebraic division) is worth more than any tip.

These questions appear in Section A of the CBSE Class 9 Maths paper. You earn 1 mark for the correct answer; no steps shown. Speed and accuracy are critical. **Question 1:** Factorise 5x + 10. **Answer:** 5(x + 2) **Why it repeats:** Tests extraction of the greatest common divisor (GCD). In 2021 and 2024 variants, coefficients changed (7x + 14 became 7(x + 2), then 8x + 16), but the method is identical. **Question 2:** Factorise 4a² – 9. **Answer:** (2a + 3)(2a – 3) **Why it repeats:** The (a² – b²) = (a + b)(a – b) identity is the most tested factorisation formula in Class 9. Recognising 4a² = (2a)² and 9 = 3² is non-negotiable. **Question 3:** Factorise x² + 5x + 6. **Answer:** (x + 2)(x + 3) **Why it repeats:** Quadratic trinomials where a = 1 are tested in nearly every year. Students must find two numbers that multiply to +6 and add to +5 (i.e., 2 and 3). **Question 4:** Divide 10x³ + 15x² by 5x. **Answer:** 2x² + 3x **Why it repeats:** Dividing by a monomial is mechanical but error-prone. Marks are lost when students forget to divide *each term* or drop the x variable. **Question 5:** Factorise 6xy + 9y. **Answer:** 3y(2x + 3) **Why it repeats:** Combination of numerical and variable GCDs. The GCD is 3y, not just 3 or y. Students often extract only the number, leaving the variable in both terms instead of one.

These questions require working and justification. Marks are awarded for method, not just the final answer. Examiners expect clear labelling of regrouping steps and factorisation logic. **Question 1: Regrouping Method** Factorise x² + 3x + 2x + 6. **Solution:** • Group: (x² + 3x) + (2x + 6) • Extract GCDs: x(x + 3) + 2(x + 3) • Factor out (x + 3): (x + 3)(x + 2) **Marks:** 1 for grouping, 1 for extracting GCDs correctly, 1 for final factorisation. Many students lose 1 mark by skipping the grouping line or writing it unclearly. **Question 2: Identity + Common Factor** Factorise 8a³ – 32a. **Solution:** • GCD of 8a³ and 32a is 8a: 8a(a² – 4) • Recognise a² – 4 = (a – 2)(a + 2): 8a(a – 2)(a + 2) **Marks:** 1 for extracting GCD, 1 for recognising the difference of squares, 1 for all factors written correctly. **Question 3: Quadratic Trinomial** Factorise 2x² + 5x + 3. **Solution:** • Find product ac = 2 × 3 = 6 and sum b = 5. Numbers: 2 and 3 (since 2 × 3 = 6, 2 + 3 = 5). • Rewrite: 2x² + 2x + 3x + 3 • Group: 2x(x + 1) + 3(x + 1) • Factor: (x + 1)(2x + 3) **Marks:** 1 for identifying the split numbers, 1 for regrouping clearly, 1 for final answer. **Question 4: Division of Algebraic Expression** Divide 12x⁴ + 8x³ – 4x² by 4x². **Solution:** • Divide each term: (12x⁴ ÷ 4x²) + (8x³ ÷ 4x²) – (4x² ÷ 4x²) • Simplify: 3x² + 2x – 1 **Marks:** 1 for correct division method, 1 for dividing each term, 1 for correct simplification of powers. **Question 5: Factorisation by Pairing** Factorise a²(b + c) + (b + c). **Solution:** • GCD is (b + c): (b + c)(a² + 1) **Marks:** 1 for recognising the common binomial, 1 for extracting it, 1 for clarity.

These questions demand multi-step working, proof, or complex algebraic division. Full marks require organised presentation and justified reasoning at each step. **Question 1: Complex Regrouping + Multiple Identities** Factorise p² – q² – 2qr – r². **Full Solution (5 marks):** • Recognise the structure: p² – (q² + 2qr + r²) • Notice q² + 2qr + r² = (q + r)² [identity a² + 2ab + b² = (a + b)²]: p² – (q + r)² • Apply difference of squares: [p – (q + r)][p + (q + r)] • Simplify brackets: (p – q – r)(p + q + r) **Marking breakdown:** 1 mark for regrouping, 1 for identifying perfect square trinomial, 1 for applying a² – b² identity, 1 for simplifying brackets correctly, 1 for final answer. **Question 2: Polynomial Long Division** Divide x⁴ + 2x³ – 3x – 6 by x + 2. Verify your answer by multiplication. **Full Solution (5 marks):** • Set up long division: x⁴ + 2x³ + 0x² – 3x – 6 divided by (x + 2). • First term: x⁴ ÷ x = x³. Multiply: x³(x + 2) = x⁴ + 2x³. Subtract from dividend: remainder 0x² – 3x – 6. • Second term: 0x² ÷ x = 0. Multiply: 0(x + 2) = 0. Subtract: remainder 0x² – 3x – 6. • Third term: –3x ÷ x = –3. Multiply: –3(x + 2) = –3x – 6. Subtract: remainder 0. • Quotient: x³ – 3. Check: (x + 2)(x³ – 3) = x⁴ + 2x³ – 3x – 6 ✓ **Marking breakdown:** 1 for correct setup with placeholders, 1 for correct first division step, 1 for completing division (earning remainder 0), 1 for stating the quotient, 1 for verification by multiplication showing the dividend is recovered. **Question 3: Factorisation + Application** A rectangular field has area 4x² + 12x + 9 square metres. Its length is (2x + 3) metres. Find the width. Factorise the area expression first. **Full Solution (5 marks):** • Factorise 4x² + 12x + 9. Recognise as (2x)² + 2(2x)(3) + 3² = (2x + 3)². • Area = (2x + 3)². • Width = Area ÷ Length = (2x + 3)² ÷ (2x + 3) = (2x + 3) metres. • Verification: Length × Width = (2x + 3)(2x + 3) = 4x² + 12x + 9 ✓ **Marking breakdown:** 1 for recognising the perfect square trinomial, 1 for correct factorisation, 1 for correct algebraic division, 1 for stating the width in correct units, 1 for verification or contextual reasoning.

The 2024–25 CBSE Class 9 Maths syllabus remains aligned with the previous structure, but examiner feedback and model answer guides suggest subtle shifts in emphasis: **Increase in Word Problems:** 5-mark questions now frequently embed factorisation in real-world contexts (fields, volumes, cost calculations). Students must translate English into algebra, factorise, then interpret the result. Expect 1 such question per year going forward. **Emphasis on Justification in Division:** Polynomial long division questions now award marks for *showing each step clearly*. A student who writes only the quotient, even if correct, loses 2–3 marks. Examiners want to see the subtraction at each stage, verification by multiplication, and explicit remainder statement. **Reduced Reliance on Factorisation by Pairing Alone:** Simple questions like "Factorise ab + ac + b² + bc" are becoming less common. Instead, questions combine pairing with identities or require division after factorisation. **Higher Difficulty in 3-Mark Questions:** The boundary between 3-mark and 5-mark questions is blurring. Some 3-mark questions now demand regrouping + identity recognition in a single problem, whereas older papers tested these separately. **Data Interpretation:** A few model papers include factorisation embedded in graphs or table-reading questions—rare but emerging. No major shift expected in your 2025 exam, but awareness helps. **Recommendation:** Focus on clarity, step-by-step working, and verification. Shortcuts without justification will cost marks even if the final answer is correct. Start a 3-day free trial at cbsetutor.ai to access AI-personalised practice on these emerging patterns.

**Time Allocation:** • 1-mark questions: 1 minute per question (5–10 questions typical). Do these first—quick confidence boost. • 3-mark questions: 4–5 minutes per question. Write all regrouping, GCD extraction, and identity steps. Never skip a step to save time. • 5-mark questions: 8–10 minutes per question. Allocate 2 minutes to reading and planning, 6 minutes to working, 1 minute to verification. **Triage Rule:** If you're unsure whether a trinomial factors, check discriminant Δ = b² – 4ac. If Δ < 0, it doesn't factor over reals (not testable in Class 9). If Δ is a perfect square, it factors; if not, re-examine your grouping. **Common Pitfalls to Avoid:** 1. **Forgetting all terms when extracting GCD.** If you extract 3x from 3x² + 6x, you must write 3x(x + 2), not 3x(x + 6). 2. **Skipping the regrouping line.** Examiners expect to see (a + b) + (a + c) grouped explicitly before factoring. 3. **Dropping variables in division.** (6x³ + 9x²) ÷ 3x = 2x² + 3x, not 2x + 3. Always include the variable. 4. **Not verifying polynomial division.** Multiply quotient by divisor; if you don't recover the dividend, your answer is wrong. 5. **Confusing (a + b)² = a² + 2ab + b² with (a + b)(c + d).** Pause 2 seconds to identify which identity applies. **Last Minute (5 mins before paper end):** Review your 1-mark answers (most likely to have typos). If a 5-mark question is incomplete, write what you know—partial credit is available for method, even without the final answer. **Exam-Day Confidence:** You've solved these patterns before in past papers. Trust your practice. If a question *looks* unfamiliar, it's likely a small variation of a known type—apply the method you know, and marks will follow.

This guide works best as a **targeted revision resource after you've studied Chapter 12 theory** from your NCERT textbook. Here's the workflow: **Step 1: Self-Assess (15 mins)** Attempt the five 1-mark questions above without looking at answers. If you score < 4/5, revise the common factor and identity sections of your textbook before moving on. **Step 2: Practise 3-Mark Questions (45 mins)** Solve the five 3-mark questions on paper, timing yourself (4–5 mins per question). Compare your working line-by-line with the solutions provided. Mark-loss usually comes from omitted steps, not wrong final answers. Redo any question where your working differed. **Step 3: Attempt 5-Mark Questions Under Exam Conditions (40 mins)** Set a timer for 10 minutes per question. Don't pause to check working mid-question. After the timer ends, check your solution against the full breakdown. Look for *where* you lost marks: setup, calculation, or verification? **Step 4: Mistake Log** Create a simple table: | Question | My Error | Correct Method | Practise Count | |----------|----------|----------------|----------------| | 3-mark Q2 | Didn't extract common factor first | Always check for GCD before identity | | Fill this after each session. Redo that specific question type twice more in the next week. **Step 5: Timed Mixed Tests** After 1 week of step-by-step practice, take a mixed 20-minute test (4 × 1-mark, 2 × 3-mark, 1 × 5-mark). Your score should be ≥ 16/20. If not, identify which *method* (common factor, regrouping, identity, division) is weakest and target it. **Why This Works:** Past papers are a record of what examiners actually ask. By solving them deeply—not just reading solutions—you train your brain to recognise patterns and execute under pressure. Theory alone leaves gaps; past papers fill them.