Class 9 Mathematics Chapter 10 'The Other Side of Zero' Important Questions with Complete Solutions

Chapter 10 is a gateway to high school mathematics. The CBSE Class 9 board exam typically dedicates 6–8 marks to integers and their operations—appearing in MCQs, short-answer, and application-based questions. The rationalized 2024-25 syllabus emphasizes: (1) **Conceptual understanding of the number line and integer representation**, (2) **Comparison of integers using < and >**, (3) **Addition and subtraction rules** without memorizing formulas—by understanding direction and magnitude, (4) **Real-life modeling**: temperature changes (e.g., −5°C to +3°C), financial contexts (debt = negative, credit = positive), and depth (below sea level = negative). Examiners test whether you can *reason* about integers, not just compute. A student who struggles here often faces difficulty in linear equations, coordinate geometry, and algebra later. This guide targets all three cognitive levels: **Knowledge** (recall definitions), **Application** (solve word problems), and **Analysis** (compare strategies, explain why −7 < −2).

**Q1. On a number line, which integer is 5 steps to the left of +2?** (a) +7 (b) −3 (c) +3 (d) −7 **Answer: (b) −3** Explanation: Moving left decreases value. +2 − 5 = −3. --- **Q2. Which statement is true?** (a) −8 > −3 (b) −8 < −3 (c) −8 = −3 (d) None **Answer: (b) −8 < −3** Explanation: On the number line, −8 is further left than −3, so it is smaller. --- **Q3. The additive inverse of −15 is:** (a) +15 (b) −15 (c) 0 (d) 1/15 **Answer: (a) +15** Explanation: −15 + 15 = 0. The additive inverse undoes a number. --- **Q4. If the temperature drops by 7°C from +5°C, the new temperature is:** (a) +12°C (b) −2°C (c) +2°C (d) −12°C **Answer: (b) −2°C** Explanation: +5 − 7 = −2. --- **Q5. Which pair of integers are opposites?** (a) −6 and 6 (b) −6 and −6 (c) 0 and 0 (d) 6 and 12 **Answer: (a) −6 and 6** Explanation: Opposites lie equidistant from zero on opposite sides of the number line.

**Q1. Compare −45 and −54. Which is greater? Show on a number line.** **Answer:** −45 > −54 On the number line: …−54 … −45 … 0. Since −45 is to the right of −54, it is greater. Alternatively: −45 is "less deep in debt" than −54, so it's greater. --- **Q2. Simplify: (−12) + 8 − (−5). Show your steps.** **Answer:** (−12) + 8 − (−5) = (−12) + 8 + 5 [subtracting a negative = adding its opposite] = −4 + 5 = +1 --- **Q3. A bank account shows −₹250 (overdraft). The account holder deposits ₹400. What is the new balance?** **Answer:** New balance = −250 + 400 = +150 The account now has a credit of ₹150. --- **Q4. Arrange in ascending order: 0, −8, +3, −2, +1** **Answer:** −8 < −2 < 0 < +1 < +3 (From left to right on the number line.) --- **Q5. The sum of two integers is −10. If one integer is −6, find the other.** **Answer:** Let the other integer be x. −6 + x = −10 x = −10 − (−6) = −10 + 6 = −4 The other integer is −4.

**Q1. Over 4 days, the temperature changes as follows: Day 1: drops 3°C, Day 2: rises 5°C, Day 3: drops 2°C, Day 4: rises 1°C. If the starting temperature is 8°C, what is the final temperature?** **Answer:** Starting: +8°C After Day 1: 8 − 3 = +5°C After Day 2: 5 + 5 = +10°C After Day 3: 10 − 2 = +8°C After Day 4: 8 + 1 = +9°C Final temperature: **+9°C** --- **Q2. Prove that (−a) + (−b) = −(a + b) using a number line. Take a = 3 and b = 2.** **Answer:** With a = 3, b = 2: LHS: (−3) + (−2) = −5 RHS: −(3 + 2) = −5 Both sides equal −5. On the number line: start at 0, move 3 steps left (−3), then 2 steps left (−2), land at −5. The net effect is moving 5 steps left, which represents −5. ✓ --- **Q3. A diver is 40 metres below sea level. She ascends 15 metres, then descends 8 metres. Represent her movements using integers and find her final position relative to sea level.** **Answer:** Initial position: −40 m (below sea level) After ascending 15 m: −40 + 15 = −25 m After descending 8 m: −25 − 8 = −33 m Final position: **33 metres below sea level** (or −33 m). --- **Q4. If x is an integer and −5 < x < 2, list all possible values of x and find their sum.** **Answer:** Integers between −5 and 2: −4, −3, −2, −1, 0, +1 Sum = (−4) + (−3) + (−2) + (−1) + 0 + 1 = −9 [Grouping: (−4 − 3 − 2 − 1) + (0 + 1) = −10 + 1 = −9]

**Q1. 'Addition of integers on a number line is a visual journey.' Explain this statement with two examples: (a) (+3) + (−5), and (b) (−2) + (−4). Draw number lines and describe each step.** **Answer:** The statement means that integer addition can be understood as movement on a number line: - Positive integers = move right. - Negative integers = move left. - The final position is the sum. **(a) (+3) + (−5):** Start at 0. Move 3 steps right → land at +3. Now move 5 steps left → land at −2. So (+3) + (−5) = −2. [Number line: 0 →(+3)→ +3 →(−5)→ −2] **(b) (−2) + (−4):** Start at 0. Move 2 steps left → land at −2. Now move 4 steps left → land at −6. So (−2) + (−4) = −6. [Number line: 0 ←(−2)← −2 ←(−4)← −6] Conclusion: The visual journey clarifies *why* addition works and removes reliance on memorized rules. Students who understand movement naturally generalize to algebraic equations later. --- **Q2. 'Subtraction of integers is addition of the opposite.' Verify this rule using three examples and explain how it simplifies computation.** **Answer:** *Example 1:* 7 − 5 = 7 + (−5) = 2 ✓ *Example 2:* −3 − 4 = −3 + (−4) = −7 ✓ *Example 3:* −6 − (−2) = −6 + 2 = −4 ✓ Why this rule works: Subtracting means 'taking away.' Taking away 5 is the same as adding its opposite, −5. This converts subtraction into addition, which uses a single consistent rule. Simplification: - No need to memorize separate subtraction rules. - Negative × negative = positive follows naturally from the definition of opposites. - Reduces errors: students work with one operation (addition) and one concept (opposites). - Extends to algebra: solving x − 5 = 3 becomes x + (−5) = 3, then x = 3 − (−5) = 8. --- **Q3. A financial advisor explains a client's account using integers over 6 months. Month 1: Deposit ₹5000 (+5000). Month 2: Withdrawal ₹8000 (−8000). Month 3: Deposit ₹3000 (+3000). Month 4: Withdrawal ₹1000 (−1000). Month 5: Deposit ₹6000 (+6000). Month 6: Withdrawal ₹2000 (−2000). (a) Calculate the net change. (b) If the account started at ₹10,000, find the final balance. (c) In which month did the account first go into overdraft (if at all)?** **Answer:** **(a) Net change:** Sum of all transactions = 5000 − 8000 + 3000 − 1000 + 6000 − 2000 = (5000 + 3000 + 6000) − (8000 + 1000 + 2000) = 14,000 − 11,000 = +3000 (net gain) **(b) Final balance:** Starting balance: ₹10,000 Final balance = 10,000 + 3000 = **₹13,000** **(c) Month-by-month balance:** - Start: +10,000 - Month 1: 10,000 + 5000 = +15,000 - Month 2: 15,000 − 8000 = +7000 - Month 3: 7000 + 3000 = +10,000 - Month 4: 10,000 − 1000 = +9000 - Month 5: 9000 + 6000 = +15,000 - Month 6: 15,000 − 2000 = +13,000 The account **never went into overdraft** (never became negative). The lowest balance was ₹7000 in Month 2.

**Case Study: 'Climate Monitoring at a Himalayan Weather Station'** A weather station at an altitude of 3500 m records temperature hourly. The station is at the edge of a glacier, and scientists monitor seasonal trends. Here's a 24-hour snapshot (in °C): | Hour | 12 AM | 3 AM | 6 AM | 9 AM | 12 PM | 3 PM | 6 PM | 9 PM | |------|-------|------|------|------|-------|------|------|------| | Temp | −8 | −12 | −15 | −9 | +2 | +5 | +1 | −4 | **(a) Arrange temperatures in ascending order.** **Answer:** −15, −12, −9, −8, −4, +1, +2, +5 **(b) Calculate the total temperature change from 12 AM to 9 PM.** **Answer:** Change = Final − Initial = −4 − (−8) = −4 + 8 = +4°C The temperature rose by 4°C over 21 hours (net effect). **(c) The coldest temperature was −15°C (6 AM). By 9 AM, the temperature rose to −9°C. How much did it rise?** **Answer:** Change = −9 − (−15) = −9 + 15 = +6°C Temperature rose by 6°C from 6 AM to 9 AM. **(d) If the glacier melts when daily average temperature exceeds 0°C, did this glacier risk melting on this day?** **Answer:** Average = (−8 − 12 − 15 − 9 + 2 + 5 + 1 − 4) ÷ 8 = −40 ÷ 8 = −5°C Since −5°C < 0°C, the glacier did **not** risk melting on this day. The 24-hour average is well below the melting threshold. **(e) Challenge: Scientists predict tomorrow's average will be +3°C. What does this mean for the glacier? Explain in terms of integer change.** **Answer:** Change in daily average = +3 − (−5) = +8°C If the average rises by 8°C and reaches +3°C, the glacier would be at risk of melting. This represents a significant warming trend. Continued warming above 0°C accelerates melt rates, threatening the glacier's stability and downstream water supplies.

At **cbsetutor.ai**, we've built an AI tutor that replicates the exact question patterns your Class 9 examiner will set. Here's how our system works for Chapter 10: **1. Adaptive Question Bank:** Our AI analyzes your weak areas (e.g., comparing negative integers, or subtracting negatives) and serves questions targeting those gaps. If you miss 3-mark application questions on temperature, the system prioritizes similar scenarios: depth, debt, elevation. You get 10–15 fresh variations, not the same 5 questions recycled. **2. Real-Time Number-Line Visualization:** Instead of static textbook diagrams, our interactive number line *animates* your steps. When you enter (+3) + (−5), you see a cursor move 3 right, then 5 left, landing at −2. This visual feedback cements the concept in seconds, not hours of passive reading. **3. Timed Drills Matching Board Tempo:** We offer 15-minute timed MCQ sets (like the 5-question block above), 30-minute mixed papers (MCQs + shorts + mediums), and 90-minute full simulations with 2-mark, 3-mark, and 5-mark sections in board-exam ratio. You practice speed *and* accuracy simultaneously. **4. Instant, Detailed Feedback:** After each question, the AI explains *why* your answer was right or wrong. For Q2 above (−8 < −3 or −8 > −3?), if you choose wrongly, our tutor asks: 'On a number line, is −8 to the left or right of −3? Which side = smaller?' This Socratic method builds reasoning, not just recall. **5. Real-Life Context Drills:** Our system weaves integers into daily scenarios. You solve 3 temperature problems, 2 financial queries, and 2 depth/elevation problems in a single session—all with different numbers, so no memorization occurs. By day 5, application questions feel routine, not novel. **6. Parent-Friendly Progress Reports:** Your parent sees that you've mastered 1-mark MCQs (90% accuracy), are developing 3-mark skills (72% accuracy), and need work on 5-mark long-answer structure (58% accuracy). Targeted, data-backed goals replace vague 'study harder' advice. **Start a 3-day free trial at cbsetutor.ai** to experience how AI-guided drills transform Chapter 10 from confusing to crystal-clear, all at your own pace.

Use this checklist 24 hours before your exam to confirm mastery of Chapter 10: ✓ **Number Line Basics:** Can you mark −8, 0, +5, −2, +10 on a number line in correct positions? ✓ **Comparison:** Without a number line, can you instantly say −9 < −4 < 0 < +3 is correct? ✓ **Addition Rule:** Do you *understand* (not memorize) why (−a) + (−b) = −(a + b)? Can you explain it on a number line? ✓ **Subtraction Rule:** Can you convert subtraction to addition and simplify (−6) − (+3) = (−6) + (−3) = −9 instantly? ✓ **Mixed Operations:** Can you solve (−5) + 8 − (−3) without errors in under 30 seconds? ✓ **Real-Life Modeling:** Given a word problem (temperature, debt, depth), can you: (i) assign positive/negative correctly, (ii) set up the integer equation, (iii) solve, (iv) interpret the answer in context? ✓ **Additive Inverse:** Given any integer, can you state its additive inverse instantly? (e.g., inverse of −17 is +17) ✓ **Board-Style MCQs:** Can you solve 5 single-choice questions on integers in 5 minutes with 100% accuracy? ✓ **2-Mark Shorts:** Can you write a clear, step-by-step 2-mark solution in under 3 minutes? ✓ **3-Mark Applications:** Can you solve a scenario-based 3-mark question (e.g., diver, account, temperature) with both calculation and interpretation in 5 minutes? If you ticked ≥8 boxes, you're exam-ready. If not, focus your final 24 hours on the unchecked areas.