Chapter 15 ka last post! 🏆 Aaj 15.1–15.3 ka mix, soch wale sawaal: perfect cube hai ya nahi (aur yahan Chapter 14 wala ikai test kaam nahi karta!), cube box, kitna add karein ya subtract karein, reverse sawaal (□³ = 9261), point wale root jaise ∛0.027, aur ek bada dhokha: ∛8 + ∛27. Har sawaal pe pehle identify karo, phir sahi method.
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). Sawaal: kya 4920 perfect cube hai?
- 10³ = 1000, 20³ = 8000: root 10 aur 20 ke beech.
- 17 × 17 × 17: 17 × 17 = 289, 289 × 17 = 4913. Thoda chhota.
- 18 × 18 × 18: 18 × 18 = 324, 324 × 18 = 5832. Bada. 4920, 4913 aur 5832 ke beech: perfect cube nahi.
Do bade multiply! Ab timer wala sawaal isi tarah try karke kijiye.
Ab phone ya watch mein stopwatch (time measure karne wali watch) chalu kijiye. Neeche wala sawaal isi “try karke” method se kijiye (upar jaisa: estimate lagao, teen baar multiply karke dekho), copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
Nahi
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki compare karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Chahiye: Post 15.1–15.3 ke method. Yaad ke liye ∛, zero teen times, comparison ke word aur point wale cube ke boxes neeche. Naye word:
📖 Word jo is post mein aayenge
- Perfect cube test: bina root nikaale pehchanna ki number perfect cube hai ya nahi.
- Dhokha (trap): aisa sawaal jo dekhne mein aasaan, par jaldi mein mistake karwa de.
- Kinaara: cube boxes ki ek seedhi line. Boxes mein total chhote cube = kinaara × kinaara × kinaara.
- Agla cube, pichla cube: kisi number ke theek baad aur theek pehle wale perfect cube. 100 ke liye 125 aur 64.
- Reverse sawaal: jawab (cube) diya hai, number (root) dhoondhna hai: □³ = 9261.
- Square bhi, cube bhi: aisa number jo kisi ka square bhi ho aur kisi ka cube bhi: 64 = 8 × 8 = 4 × 4 × 4.
📏 Bade numbers padhna aur round number mein add karna
Har digit ki ek jagah (place) hoti hai. Right se count karo: pehli jagah Ikai (ones, 1 wale), uske left mein Dahai (tens, 10 wale), phir Saikda (hundreds, 100 wale), phir Hazaar (thousands, 1000 wale).
10 ka bundle: 10 ikai = 1 dahai (10). 10 dahai = 1 saikda (100). 10 saikde = 1 hazaar (1000). Har jagah apne right wali jagah se 10 times badi hai.
| Number | Kaise padhein | Kis cheez se bana |
|---|---|---|
| 98 | attaanve | 9 dahai (90) + 8 ikai |
| 165 | ek sau painsath | 1 saikda (100) + 6 dahai (60) + 5 ikai |
| 999 | nau sau ninyaanve | 9 saikde (900) + 9 dahai (90) + 9 ikai |
| 1011 | ek hazaar gyaarah | 1 hazaar (1000) + 0 saikda + 1 dahai (10) + 1 ikai |
Round number mein chhota number add karna (place box rule): chhote number ko right se likho, yaani ikai ke neeche ikai, dahai ke neeche dahai. Phir har boxes mein jo hai, woh jawab mein utaar do. Ek digit wala number sirf ikai ka zero badalta hai, do digit wala ikai aur dahai dono.
- 100 + 65: 65 mein do digit, to 6 dahai mein aur 5 ikai mein: 1|6|5 = 165.
- 100 + 2: 2 sirf ikai mein, dahai ka 0 waise hi rehta hai: 1|0|2 = 102 (12 nahi, 120 nahi!).
- 200 + 6: 6 sirf ikai mein: 2|0|6 = 206.
- 1000 + 11: 11 mein do digit, to 1 dahai mein aur 1 ikai mein: 1|0|1|1 = 1011.
🔄 Cube root = “reverse cube” (∛ ka matlab)
Cube mein number dete hain: 4 → 4 × 4 × 4 = 64. Cube root mein reverse poochte hain: “Kaunsa number teen baar multiply karne pe 64 aaya?” Jawab 4. Hindi mein ise ghan-mool kehte hain (mool = jad, jahan se ghan uga).
Sign ∛: Chapter 14 wala √ hi, bas uske corners (baayein kandhe) pe chhota 3. ∛64 padho “64 ka cube root”. ∛64 = 4 ka matlab: 4 × 4 × 4 = 64. (Bina 3 ke √ hamesha square root hota hai.)
| Cube (aage) | Cube root (reverse) |
|---|---|
| 2 × 2 × 2 = 8, yaani 2³ = 8 | ∛8 = 2 |
| 3 × 3 × 3 = 27, yaani 3³ = 27 | ∛27 = 3 |
| 5 × 5 × 5 = 125, yaani 5³ = 125 | ∛125 = 5 |
| 10 × 10 × 10 = 1000, yaani 10³ = 1000 | ∛1000 = 10 |
Perfect cube: woh number jo kisi poore number (0, 1, 2, 3 …) ka cube ho: 1, 8, 27, 64, 125 … 100 perfect cube nahi, kyunki 4³ = 64 (chhota) aur 5³ = 125 (bada), beech mein koi whole number nahi.
Check hamesha aasaan: jo root mila, use teen baar likh kar multiply karo. Wahi number wapas aaye to sahi. ∛125 = 5? 5 × 5 = 25, 25 × 5 = 125 ✅.
0️⃣ Round number ka cube aur cube root: zero teen times, zero ek-tihai
Cube mein zero teen times: 20³ = 20 × 20 × 20. Pehle digit: 2 × 2 × 2 = 8. Har 20 ke saath ek 0 tha, aur teen 20 hain, to teen 0: 8000. Isi tarah 30³: 3 × 3 × 3 = 27, peeche teen 0, yaani 27000.
Kyun? 20 = 2 × 10. To 20 × 20 × 20 mein teen 2 aur teen 10 hain. Multiply mein jagah badal sakte hain (Chapter 7 ka rule), to pehle 2 × 2 × 2 = 8, phir 10 × 10 × 10 = 1000. Aur 8 × 1000 = 8000 (1000 se multiply = peeche teen 0).
Cube root mein reverse, zero ek-tihai (teen parts mein ek): ∛8000: peeche ke teen 0 remove karo, bacha 8. ∛8 = 2. Teen 0 ka ek-tihai, ek 0: 20. Check: 20 × 20 × 20 = 8000 ✅.
Dhyaan: zero teen-teen ke jhund mein (3, 6, 9 …) hone chahiye. 27000 (teen 0) ✅ → 30. Par 2700 (do 0) ya 270 (ek 0) kisi round number ka cube nahi, kyunki round number ka cube hamesha teen-teen zero badhaata hai.
↔️ Comparison ke word: chhota, bada, “baith gaya”
Is chapter mein comparison hum shabdon mein likhenge, koi sign nahi:
- “64, 117 se chhota hai”: 64 kam hai.
- “125, 117 se bada hai”: 125 zyada hai.
- “64, 117 mein baith gaya” matlab 64, 117 se chhota ya equal hai (117 ke andar aa jaata hai).
- “117, 64 aur 125 ke beech hai”: 117, 64 se bada aur 125 se chhota.
🔸 Point (decimal) wale number ka cube
0.3 padho “zero point teen”: ek ke 10 equal pieces mein se 3 pieces (jaise ek roti ke 10 pieces, unme se 3). Chapter 14 (Post 14.3) mein point wale numbers dekhe the.
Point wale number ka cube: point hata kar multiply karo, phir jawab mein point waapas lagao. Kitni jagah? Teeno numbers mein point ke baad jitne digit, utne total, right se gin kar. 0.3 × 0.3 × 0.3: point remove karo → 3 × 3 × 3 = 27. Har 0.3 mein point ke baad 1 digit, teen number, total 3. 27 ko 3 digit ka banao: 027, aur right se 3 gin kar point: 0.027.
Reverse (cube root): ∛0.027 = 0.3. Point ke baad 3 digit the, root mein uska ek-tihai, 1 digit.
🔁 Chhota sa warm-up
- ∛27000 = ?
- 5³ = ?
- 0.3 × 0.3 × 0.3 = ?
Answers dekho
- 30 (teen 0 ka ek)
- 125
- 0.027 (3 × 3 × 3 = 27, point ke baad 3 digit)
Teesra sawaal reverse karo: ∛0.027 = 0.3. Aaj ka ek tricky sawaal yahi hai!
— 📌 charumam.com
🎬 Story: Quiz Ka Dhokha
Inter-school quiz mein sawaal aaya: “∛8 + ∛27 = ?” Kabir ne jhat se add kar diya: “8 + 27 = 35, to ∛35!” Phir atak gaya: 35 to kisi ka cube hi nahi.
Zoya ne aaraam se kaha: “Pehle har ek ka root alag-alag: ∛8 = 2, ∛27 = 3. Phir add karo: 2 + 3 = 5.” Quiz master Iyer ma’am ne poocha: “Aur ∛35 kyun galat?” Zoya: “5³ = 125, 35 nahi. Root aur sum ki jagah nahi badal sakte!”
Team ko point mila. 🧩 Aaj aise hi dhokhe pakadna seekhenge.
— 📌 charumam.com
🪄 Trick Step by Step: Identify karo, Test Karo, Phir Root Ya Decision
💡 Is post ki ek badi baat: Pehle identify karo: root poocha (jaise ∛9261), “perfect hai ya nahi”, “kitna add karein”, box, ya point wala? Test: zero teen-teen mein nahi → perfect nahi. Ikai se kuch pakka nahi hota (har ikai mumkin hai). To beech check: diya hua number kin do agle cubes ke beech hai (3000: 14³ = 2744 aur 15³ = 3375)? Beech mein ho → perfect nahi. Perfect ho to Post 15.1 (list), 15.2 (rule), 15.3 (zero). Aur hamesha check.
Step 1: Identify karo
Sawaal ki type dekho: root poocha (∛9261), “perfect hai ya nahi”, “kitna add karein/subtract karein”, cube box, reverse sawaal (□³ = 9261), ya point wala (∛0.027).
Step 2: Zero test
Peeche zero count karo. Teen-teen mein nahi (1, 2, 4, 5 zero) → perfect cube nahi. Teen-teen mein hain → remove karo aur bache ko dekho (5000 → 5 → nahi).
Step 3: Ikai se dhokha mat khao
Chapter 14 mein square ki ikai 2, 3, 7, 8 kabhi nahi hoti thi, to test aasaan tha. Cube mein har ikai aati hai (Post 15.2 ki table: 8³ = 512, ikai 2!). To ikai dekh kar “perfect cube nahi” mat kaho.
Step 4: Beech check
Diye hue number ke paas ke do agle cubes dhoondho. Number unme se kisi ke equal → perfect cube. Beech mein → nahi. 3000: 14³ = 2744 (14 × 14 = 196, 196 × 14 = 2744) aur 15³ = 3375 (15 × 15 = 225, 225 × 15 = 3375); 3000 beech mein → nahi. (3000 pe zero test bhi yahi kehta hai: teen 0 remove karo → 3, jo perfect cube nahi.)
Step 5: Root ya decision, phir check
Perfect hai to list, Post 15.2 ya zero wala method. “Kitna add karein” → agla cube − diya hua number (100: 125 − 100 = 25). “Kitna subtract karein” → diya hua number − pichla cube (100 − 64 = 36).
Type 1: Perfect cube hai ya nahi?
| Number | Test | Decision |
|---|---|---|
| 5000 | teen 0 remove karo → 5, perfect cube nahi | ❌ Nahi |
| 2700 | sirf do 0 | ❌ Nahi |
| 3000 | 14³ = 2744, 15³ = 3375, beech mein | ❌ Nahi |
| 5832 | 5 | 832: ikai 2 → 8; left 5 mein 1³ = 1 baitha (2³ = 8 bada) → dahai 1 → 18. Check 18 × 18 = 324, 324 × 18 = 5832 | ✅ Haan (18³) |
| 2008 | ikai 8 (chal sakti hai), par 12³ = 1728, 13³ = 2197, beech mein | ❌ Nahi |
Dhyaan: 2008 ki ikai 8 dekh kar “2 ka root” mat socho. Ikai sahi ho sakti hai aur number phir bhi perfect cube na ho.
Type 2: Cube box (kinaara dhoondho)
∛2197: last teen digit alag: 2 | 197. Last digit 7 → root ki ikai 3 (7 aur 3 jagah badalte hain). Left 2: 1³ = 1, 2 mein baith gaya; par 2³ = 8, 2 se bada. To root ki dahai 1. Jawab: ∛2197 = 13.
2197 chhote cube se bana box: har kinaare pe 13.
Type 3: Kitna add karein / subtract karein?
100 mein kam se kam kitna add karein ki perfect cube bane? 100 ke baad agla cube 125 (5³). 125 − 100 = 25.
100 mein se kam se kam kitna subtract karein? 100 se pehle cube 64 (4³). 100 − 64 = 36.
Type 4: Point wale root
∛0.027: point hata kar 27 → ∛27 = 3. Point ke baad 3 digit the, root mein uska ek-tihai, 1 digit: 0.3. Isi tarah ∛0.008 = 0.2 aur ∛1.331 = 1.1 (1331 → 11, point ke baad 3 digit → 1 digit).
Type 5: Sum ka dhokha
∛8 + ∛27: pehle har root alag: 2 aur 3. Phir add karo: 2 + 3 = 5. ∛35 likhna galat, kyunki 5³ = 125 hai, 35 nahi.
🆘 Atak gaye? Aise socho
- Zero teen-teen mein nahi → perfect cube nahi.
- Ikai se “nahi” mat kaho: cube mein har ikai aati hai.
- Beech check: do agle cubes ke beech → perfect nahi.
- Kitna add karein: agla cube − diya hua number (125 − 100 = 25). Subtract karein: diya hua number − pichla cube (100 − 64 = 36).
- Point: point ke baad digit ka ek-tihai (∛0.027 = 0.3).
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Cube mein ikai test kyun nahi chalta? Square ki ikai sirf 0, 1, 4, 5, 6, 9 thi, kyunki do-do digit ek hi ikai dete the (3 aur 7 dono 9). Cube mein dasoon digit dasoon alag ikai dete hain, to har ikai kisi na kisi cube ki hoti hai. Isliye ikai se “nahi” keh hi nahi sakte.
Beech check kyun pakka hai? Agle do cubes ke beech koi whole number nahi bachta (14 aur 15 ke beech koi whole number nahi). To 2744 aur 3375 ke beech ka koi number kisi poore number ka cube ho hi nahi sakta.
∛8 + ∛27, ∛35 kyun nahi?
Root pehle, sum baad mein. 2 + 3 = 5, aur 5 × 5 × 5 = 125. Agar ∛35 sahi hota to 5 × 5 × 5 ka jawab 35 aana chahiye tha, par woh 125 hai. Isliye “sum ka root” aur “root ka sum” alag cheezein hain.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: □³ = 9261. □ = ?
∛9261: last teen digit alag: 9 | 261. Last digit 1 → root ki ikai 1 (1 apne jaisa). Left 9: 2³ = 8, 9 mein baith gaya; par 3³ = 27, 9 se bada. To root ki dahai 2. Jawab: ∛9261 = 21.
✔️ Normal method se check: 21 × 21 = 441, 441 × 21: 441 × 20 = 8820, 441 × 1 = 441; 8820 + 441 = 9261 ✅
Example 2: 200 mein kam se kam kitna add karein ki perfect cube bane?
5³ = 125 (200 se chhota), 6³ = 216 (200 se bada). Agla cube 216. 216 − 200 = 16.
✔️ Normal method se check: 200 + 16 = 216 = 6 × 6 × 6 (6 × 6 = 36, 36 × 6 = 216) ✅
Example 3: ∛0.008 = ?
Point hata kar 8 → ∛8 = 2. Point ke baad 3 digit (008), root mein 1 digit: 0.2.
✔️ Normal method se check: 0.2 × 0.2 × 0.2: 2 × 2 × 2 = 8, point ke baad 1 + 1 + 1 = 3 digit → 0.008 ✅
Example 4: Kaunsa sabse chhota number (1 se bada) square bhi hai aur cube bhi?
Cubes: 8, 27, 64 … Squares: 4, 9, 16, 25, 36, 49, 64 … Pehla jo dono mein: 64 (8 × 8 = 64 aur 4 × 4 × 4 = 64).
✔️ Normal method se check: 8 × 8 = 64 ✅ aur 4 × 4 = 16, 16 × 4 = 64 ✅; 8 aur 27 squares ki list mein nahi ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki soch hai: “pehle identify karo, phir method chuno”. Har sawaal ke liye ek hi method nahi; zero wala, list wala, last digit wala, aur check. Aaj ke tricky sawaal isi pehchaan ki practice hain. Zero remove karne ka idea Chapter 14 (Post 14.1) ke round number wale square root jaisa hai (√3600 = 60), bas wahan zero half hote the aur yahan ek-tihai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| “Perfect cube hai?” (peeche zero) | ✅ Zero test | Teen-teen mein nahi → nahi. |
| “Perfect cube hai?” (zero nahi) | ✅ Beech check | Ikai se decision nahi. |
| Cube box, total cube diye | ✅ ∛ = kinaara | Post 15.2. |
| Kitna add karein / subtract karein | ✅ Agla / pichla cube | Difference. |
| Point wala (0.027) | ✅ Point remove karo, root, ek-tihai digit | Upar “point wale number ka cube” box. |
| ∛8 + ∛27 jaisa sawaal | ⚠️ Root pehle, sum baad mein | 2 + 3 = 5; ∛35 likhna galat. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Ikai dekh ke “perfect cube nahi” kaha
Sawaal: Kya 512 perfect cube hai?
❌ Galat: “ikai 2, square ki tarah, to nahi.”
✅ Sahi: cube mein ikai 2 aati hai: 8³ = 512. Haan, perfect cube.
Mistake 2: Sum ko root ke andar daala
Sawaal: ∛8 + ∛27
❌ Galat: “∛35.”
✅ Sahi: alag-alag root: 2 + 3 = 5.
Mistake 3: Point galat jagah
Sawaal: ∛0.027
❌ Galat: “0.03” (point ke baad 2 digit rakhe).
✅ Sahi: 3 digit ka ek-tihai, 1 digit: 0.3.
Mistake 4: Kitna add karein mein beech wala cube chhoda
Sawaal: 100 mein kitna add karein?
❌ Galat: “216 tak: 116.”
✅ Sahi: sabse paas ka agla cube 125: 25.
🧠 Rule of thumb: Identify karo, zero test, ikai se “nahi” kabhi nahi, beech check, phir root, aur check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 mix sawaal. Target: 6 minute.
- Kya 5832 perfect cube hai?
- Kya 3000 perfect cube hai?
- Kya 512 perfect cube hai?
- Kya 2700 perfect cube hai?
- Kya 4096 perfect cube hai?
- Kya 9000 perfect cube hai?
- □³ = 9261. □ = ?
- □³ = 4913. □ = ?
- □³ = 103823. □ = ?
- ∛27000000 = ?
- ∛0.008 = ?
- ∛0.125 = ?
- ∛1.331 = ?
- ∛0.064 = ?
- ∛0.001 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Haan
- Nahi
- Haan
- Nahi
- Haan
- Nahi
- 21
- 17
- 47
- 300
- 0.2
- 0.5
- 1.1
- 0.4
- 0.1
Kaise kiya (kuch): 4096: 4 | 096, ikai 6 → 6, left 4 mein 1³ baitha → 16; check 16 × 16 = 256, 256 × 16 = 4096 → Haan. 9000: teen 0 remove karo → 9, perfect cube nahi → Nahi. ∛27000000: chhah 0 → do jhund → ∛27 = 3 → 300.
— 📌 charumam.com
⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- ∛8000 = ?
- ∛343 = ?
- ∛1000000 = ?
- ∛1728 = ?
- ∛2744 = ?
- ∛216000 = ?
- ∛0.027 = ?
- ∛0.216 = ?
- 6³ = ?
- 0.2³ = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya start! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 20
- 7
- 100
- 12
- 14
- 60
- 0.3
- 0.6
- 216
- 0.008
Dhyaan: ∛1000000: chhah 0 → do 0 waapas → 100. ∛1728 = 12 (1 | 728: ikai 8 → 2). ∛0.216 = 0.6 (6³ = 216, point ke baad 3 digit → 1). 0.2³: 2 × 2 × 2 = 8, point ke baad 3 digit → 0.008. (1728 se ek zyada, 1729, ki ek mashhoor story neeche Challenge mein!)
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har day thoda better, bas yahi chahiye. 💪
— 📌 charumam.com
📝 10 MCQs (Har Option Ka Reason)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. Kaunsa perfect cube hai?
Jawab aur reason dekho
Sahi jawab: (d) 27000
- (a) galat ❌: Mistake: teen 0 remove karo → 3; 3 perfect cube nahi.
- (b) galat ❌: Mistake: sirf do 0; zero teen-teen mein nahi.
- (c) galat ❌: Mistake: chaar 0; zero teen-teen mein nahi.
- (d) sahi ✅: 30³ = 27000 (teen 0, 27 = 3³).
Q2. □³ = 9261. □ = ?
Jawab aur reason dekho
Sahi jawab: (c) 21
- (a) galat ❌: 29: ikai 1 ka square root wala rule (1 → 1 ya 9) laga ke 9 chuna.
- (b) galat ❌: 12: dahai aur ikai reverse likh di.
- (c) sahi ✅: 9 | 261: ikai 1 → 1; 9 mein 2³ = 8 baitha → 21.
- (d) galat ❌: 31: left 9 mein 3³ = 27 baitha samjha, par 27, 9 se bada hai.
Q3. 100 mein kam se kam kitna add karein ki perfect cube bane?
Jawab aur reason dekho
Sahi jawab: (b) 25
- (a) galat ❌: 116: 216 (6³) tak add kiya, beech ka 125 chhod diya.
- (b) sahi ✅: Agla cube 125: 125 − 100 = 25.
- (c) galat ❌: 36: pichle cube 64 tak ka difference (100 − 64 = 36) le liya; woh subtract karna hota.
- (d) galat ❌: 21: 121 tak add kiya (11 × 11 = 121), par woh square hai, cube nahi.
Q4. ∛0.027 = ?
Jawab aur reason dekho
Sahi jawab: (a) 0.3
- (a) sahi ✅: ∛27 = 3, point ke baad 3 digit ka ek-tihai 1 digit → 0.3.
- (b) galat ❌: 3: point lagana bhool gaye.
- (c) galat ❌: 0.03: point ke baad 2 digit rakh diye; teen ka ek-tihai 1 hota hai.
- (d) galat ❌: 0.009: 0.027 ke 3 equal parts kar diye (cube root parts nahi).
Q5. Ek cube box 2197 chhote cubes se bana. Har kinaare pe kitne?
Jawab aur reason dekho
Sahi jawab: (d) 13
- (a) galat ❌: 17: ikai 7 ko 7 hi rakha; 3 aur 7 jagah badalte hain.
- (b) galat ❌: 169: sirf ek parat gin li (parat = ek tah/layer, jaise 13 × 13 ka ek floor: 13 × 13 = 169).
- (c) galat ❌: 23: left 2 mein 2³ = 8 baitha samjha, par 8, 2 se bada hai.
- (d) sahi ✅: 2 | 197: ikai 7 → 3; 2 mein 1³ baitha → 13.
Q6. Kaunsa number square bhi hai aur cube bhi?
Jawab aur reason dekho
Sahi jawab: (c) 64
- (a) galat ❌: Mistake: 27 = 3³ cube hai, par square nahi (5 × 5 = 25, 6 × 6 = 36).
- (b) galat ❌: Mistake: 8 = 2³ cube hai, par square nahi (2 × 2 = 4, 3 × 3 = 9).
- (c) sahi ✅: 8 × 8 = 64 aur 4 × 4 × 4 = 64.
- (d) galat ❌: Mistake: 16 = 4 × 4 square hai, par cube nahi (2³ = 8, 3³ = 27).
Q7. ∛8 + ∛27 = ?
Jawab aur reason dekho
Sahi jawab: (b) 5
- (a) galat ❌: 6: 2 aur 3 ko multiply kar diya (2 × 3 = 6), add karna tha.
- (b) sahi ✅: 2 + 3 = 5.
- (c) galat ❌: Mistake: sum ko root ke andar daal diya; pehle har root alag nikaalo.
- (d) galat ❌: 35: 8 + 27 add kar diya, root lena hi bhool gaye.
Q8. Ek perfect cube ki ikai 8 hai. Uske cube root ki ikai?
Jawab aur reason dekho
Sahi jawab: (a) 2
- (a) sahi ✅: 2³ = 8 (2 ↔ 8).
- (b) galat ❌: 8: jagah change karna bhool gaye; 8³ = 512 ki ikai 2 hai.
- (c) galat ❌: Mistake: yeh square root ka rule jaisa soch (do option); cube mein ek hi.
- (d) galat ❌: Mistake: 2³ = 8, to ikai 8 wale cube hote hain.
Q9. 125 mein se kam se kam kitna subtract karein ki ek aur (chhota) perfect cube bane?
Jawab aur reason dekho
Sahi jawab: (d) 61
- (a) galat ❌: 25: 100 tak subtract kiya; 100 round hai, par perfect cube nahi.
- (b) galat ❌: 117: 8 (2³) tak subtract kiya (125 − 8 = 117), beech ka 64 chhod diya.
- (c) galat ❌: 64: difference ki jagah pichla cube hi likh diya.
- (d) sahi ✅: Pichla cube 64: 125 − 64 = 61.
Q10. ∛1.331 = ?
Jawab aur reason dekho
Sahi jawab: (c) 1.1
- (a) galat ❌: 0.11: point ke baad 2 digit rakh diye.
- (b) galat ❌: 1.21: 1.1 ka square (1.1 × 1.1 = 1.21) likh diya, cube root nahi.
- (c) sahi ✅: ∛1331 = 11, point ke baad 3 digit ka ek-tihai 1 → 1.1.
- (d) galat ❌: 11: point lagana bhool gaye.
— 📌 charumam.com
🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda mind bhi lagana padta hai. Olympiad ek maths competition (pratiyogita) hota hai.
Q1. 4 × 4 × 4 ka box (64 chhote cube) bahar se colour diya. Kitne cube pe bilkul colour nahi? Aur kitne pe sirf ek taraf colour?
Solution dekho
Bina colour: bahar ki parat (ek tah/layer) har taraf se remove karo: dono taraf se 1-1 remove karo, to 2 kam: 4 − 2 = 2. Andar 2 × 2 × 2 = 8.
Sirf ek taraf: har chehre (face) ke beech wale, edges ko chhod kar: 2 × 2 = 4. Boxes ke 6 chehre: 4 × 6 = 24.
Q2. 500 ko kam se kam kis number se multiply karein ki perfect cube ban jaaye?
Solution dekho
500 ko pieces mein socho: 500 = 125 × 4 (125 × 4 = 500, kyunki chaar 125 milkar 500). 125 = 5 × 5 × 5 pehle se cube hai. 4 = 2 × 2. 10 × 10 × 10 = (2 × 5) × (2 × 5) × (2 × 5), yaani teen 2 aur teen 5; isliye har piece teen-teen baar chahiye. Cube ke liye teen 2 chahiye, ek kam hai. To 2 se multiply: 500 × 2 = 1000 = 10³ ✅.
Q3. 1729 ko do alag methods se do cubes ke sum mein likho.
Solution dekho
12³ = 1728, to 1³ + 12³ = 1 + 1728 = 1729. Aur 9³ = 729, 10³ = 1000, to 9³ + 10³ = 729 + 1000 = 1729. 🤩
Yeh mashhoor “Ramanujan number” hai: Bharat ke mahaan mathematician Srinivasa Ramanujan ne bataya tha ki yeh sabse chhota number hai jo do methods se do cubes ka sum hai.
Q4. Series: 1, 8, 27, 64, … Agle do number? Aur do agle numbers ka difference kaise increase hota hai?
Solution dekho
Yeh cubes hain: agle 125 (5³) aur 216 (6³).
Difference: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37, 125 − 64 = 61, 216 − 125 = 91. Difference bhi har baar zyada increase hota hai: 19 − 7 = 12, 37 − 19 = 18, 61 − 37 = 24, 91 − 61 = 30 (har baar 6 zyada!).
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Perfect cube hai ya nahi, kis se multiply karein ki cube bane, ∛ point wale. |
| Maths Olympiad | Colour wale boxes, 1729 jaise number, cube series. |
| NTSE / MAT | Number series (1, 8, 27, 64 …), tez cube root. |
| SSC / Bank (aage) | Simplification ke sawaalon mein ∛, ∛0.027 jaise point wale. |
Shabdon ka matlab: Olympiad = school ke students ki maths pratiyogita (competition). NTSE = Class 10 ke students ka ek national scholarship (vazifa) exam. MAT = Mental Ability Test, yaani sochne-samajhne ki pariksha (NTSE ka ek part). SSC / Bank exam = badon ke government naukri ke exam. Simplification = long calculation ko chhota karke jawab nikaalna. Number series = numbers ki ek line jo kisi rule se aage increase hoti hai, aur agla number batana hota hai. Jaise 5, 10, 15, 20, … mein har baar 5 jud raha hai, to agla 25. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “identify karo, zero, beech, phir root.” Aur yaad rakho: cube mein ikai se “nahi” kabhi nahi; square mein chalta tha, cube mein nahi. Chapter 15 poora! 🏆 Agla chapter: Quick Division, yaani 5, 25 aur 9 se tez division.
— 📌 charumam.com
⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
Peeche teen 0 hain, to remove karo: bacha 5. 1³ = 1 aur 2³ = 8; 5 inke beech hai, perfect cube nahi. To 5000 bhi perfect cube nahi ✅ (Post 15.3: zero remove karo, bache ko dekho. Ek bhi bada multiply nahi!)
| Before time | After time | Kitna tez? |
|---|---|---|
| ____ second | ____ second | Before − After = ____ second bache! |
Pehli baar mein time zyada bhi lag sakta hai, ghabraiye nahi! Nayi trick ko 3 se 4 days ki practice chahiye. Phir yeh purane method se kaafi tez ho jaati hai. 🚀
— 📌 charumam.com
🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy kijiye aur CharuMam AI Tutor (tutor.charumam.com) mein paste kar dijiye. Tutor aapko naye sawaal dega aur mistake hone pe samjhayega:
— 📌 charumam.com
❓ FAQs (Students Ke Asli Sawaal)
1. Cube mein ikai test kyun nahi?
Har digit ka cube alag ikai deta hai, to har ikai kisi cube ki ho sakti hai.
2. Perfect cube pakka kaise karein?
Beech check: diya hua number agle do cubes ke beech ho to nahi (3000: 2744 aur 3375 ke beech); kisi cube ke equal ho to haan.
3. ∛8 + ∛27 aur ∛35 alag kyun?
Root pehle, sum baad mein: 2 + 3 = 5. 5³ = 125, 35 nahi.
4. Point wale cube root?
Point remove karo, root nikaalo, phir point ke baad digit ka ek-tihai (∛0.027 = 0.3).
5. Agla chapter kya hai?
Chapter 16: Quick Division by 5, 9 aur 25: tez division, shesh aur point.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Zero teen-teen mein nahi → perfect cube nahi.
- Ikai se “nahi” kabhi nahi (cube mein har ikai aati hai).
- Beech check: 14³ = 2744, 15³ = 3375 → 3000 nahi.
- Kitna add karein: agla cube − diya hua number (125 − 100 = 25).
- ∛0.027 = 0.3; ∛8 + ∛27 = 5.
- 64: square bhi, cube bhi.
— charumam.com 🎯
👉 Aage Kya?
Chapter 15 poora! 🎉 Ab Chapter 16: Quick Division: 5, 25 aur 9 se tez division. Pehla post: ÷ 5, double karo aur 10 se divide (85 ÷ 5 = 17).
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
💛 Post Achha Laga? Like, Share, Subscribe, Follow!
Agar aaj kuch naya samajh aaya, to post ko like kijiye, apne doston aur class ke group mein share kijiye, aur daily nayi trick ke liye humein subscribe aur follow kijiye! 🙏
- 🌐 Website: charumam.com
- 🤖 AI Tutor (doubt poochho, 24×7): tutor.charumam.com
- ▶️ YouTube (subscribe): @charumamofficial
- 📸 Instagram (follow): @charumamofficial
- ✈️ Telegram: t.me/charumam
- 💬 WhatsApp Channel: CharuMam WhatsApp Channel
— 📌 charumam.com
