Post 15.2 ka rule yaad hai? Teen digit alag, ikai ka add kiya, left se dahai. Aaj usi ko bade numbers pe chalaayenge: peeche zero wale cubes (∛64000, ∛1728000), 100 ke paas wale bade cubes (∛912673 = 97), jawab ko teen tarah se check karna, aur sabse zaroori: kab yeh trick nahi lagani. Aakhir mein ek saaf list: kaunsi trick kab.
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 729000 ek perfect cube hai. Kaunsa number teen baar multiply pe 729000?
- 80 × 80 × 80: 80 × 80 = 6400, 6400 × 80 = 512000. Chhota.
- 90 × 90 × 90: 90 × 90 = 8100, 8100 × 80 = 648000 aur 8100 × 10 = 81000; 648000 + 81000 = 729000. Mil gaya! Root 90.
Bade multiply, aur zero ginne mein mistake ka darr. 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. 912673 ek perfect cube hai. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
∛912673 = 97, kyunki 97 × 97 × 97 = 912673
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 ki list aur “zero teen times”, Post 15.2 ka rule. Chapter 13 (97² = 9409) aur Chapter 6 (95² = 9025) check mein kaam aayenge. Yaad ke liye boxes neeche. Naye word:
📖 Word jo is post mein aayenge
- Check: jawab sahi hai ya nahi, doosre method se pakka karna.
- Ikai check: root ki ikai ka cube, number ki ikai deta hai ya nahi.
- Beech check: root 90-kuch hai to number 90³ aur 100³ ke beech hona chahiye.
- Teen-teen zero: zero 3, 6, 9 … ki count mein: 64000 (teen 0), 8000000 (chhah 0).
- Perfect cube nahi: kisi poore number ka cube nahi, jaise 2000 (12³ = 1728 aur 13³ = 2197 ke beech).
📏 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.
🔁 Chhota sa warm-up
- ∛64 = ?
- 40³ = ?
- ∛2197 = ?
Answers dekho
- 4
- 64000 (64, teen 0)
- 13 (Post 15.2: 2 | 197)
Doosra aur pehla saath: ∛64000 = 40. Aaj ka pehla step!
— 📌 charumam.com
🎬 Story: Robotics Club Ka Bada Cube
School ke robotics club ne 912673 chhote LEGO jaise blocks se ek bada cube banaya tha. Principal ma’am ne poocha: “Har kinaare pe kitne block?”
Aditya ne Post 15.2 lagaya: “Ikai 3, to root ki ikai 7. Left 912 mein 9³ = 729 baitha. 97!” Tanvi boli: “Ruko, check karte hain. 97 to 90 aur 100 ke beech hai, aur 90³ = 729000, 100³ = 1000000. 912673 inke beech hai ✅. Aur 7³ = 343 ki ikai 3 ✅.”
Principal ma’am: “Jawab bhi, check bhi. Shabaash!” 🏗️ Aaj yahi: bade cube, zero wale cube, aur check.
— 📌 charumam.com
🪄 Trick Step by Step: Zero Remove karo, Rule Lagao, Teen Tarah Se Check
💡 Is post ki ek badi baat: Kaunsi trick kab: (1) 1000 tak → list (Post 15.1). (2) Peeche teen-teen zero → zero remove karo, bache ka root, aur har teen zero ka ek zero waapas. (3) Baaki perfect cube (6 digit tak) → Post 15.2 ka rule. (4) Pata nahi perfect cube hai ya nahi → trick lagao, phir check; check fail = perfect cube nahi.
Step 1: Peeche ke zero count karo
∛64000: peeche teen 0. Teen-teen ke jhund mein hain, to aage badho. (Agar 1, 2, 4 ya 5 zero hon, to round number ka cube ho hi nahi sakta: Post 15.1 ka box.)
Step 2: Zero remove karo, bache ka root, zero waapas
∛64000: teen 0 remove karo → 64. ∛64 = 4 (list). Teen 0 ka ek 0 waapas: 40.
∛1728000: teen 0 remove karo → 1728. ∛1728 Post 15.2 se: 1 | 728, ikai 8 → 2, left 1 → 1 → 12. Ek 0 waapas: 120.
Step 3: Bade cube: wahi rule
∛912673: 912 | 673. Ikai 3 → 7. Left 912: 9³ = 729 baitha, 10³ = 1000 bada (912 se bada) → dahai 9. Jawab 97.
Step 4: Check 1 aur 2: ikai aur beech
Ikai check: 97 ki ikai 7, aur 7³ = 343 ki ikai 3; 912673 ki ikai bhi 3 ✅.
Beech check: 97, 90 aur 100 ke beech hai. 90³ = 729000, 100³ = 1000000 (zero teen times). 912673, 729000 aur 1000000 ke beech hai ✅. Do second mein do check!
Step 5: Check 3: poora multiply (jab pakka karna ho)
97 × 97 = 9409 (Chapter 13: 3 kam → 94 | 09). Phir 9409 × 97: 9409 × 90 = 846810 (9409 × 9 = 84681, peeche 0), 9409 × 7 = 65863; 846810 + 65863 = 912673. To 97 × 97 × 97 = 912673 ✅.
Kab yeh trick NAHI (ya pehle check)
| Number | Trick kya kehti | Check | Sach |
|---|---|---|---|
| 2000 | Teen 0 hain, to Step 1: zero remove karo → 2 bacha. 2 cube list mein nahi (1³ = 1, 2³ = 8) | 10³ = 1000, 2000 nahi ❌ | Perfect cube nahi (12³ = 1728, 13³ = 2197) |
| 117650 | 117 | 650: dahai 4, ikai 0 → “40” | 40³ = 64000 ❌ (bahut door) | Perfect cube nahi (49³ = 117649, sirf 1 kam!) |
| 4000 | teen 0 remove karo → 4; ∛4 poora nahi | 1³ = 1, 2³ = 8 ❌ | Perfect cube nahi |
| 1030301 | left 1030: list 9³ = 729 tak hi hai | root 100 se bada | 101³ hai, par yeh is post ke bahar |
Seekh: trick hamesha koi na koi jawab de deti hai. Agar sawaal mein “perfect cube” nahi likha, to check zaroor karo.
🆘 Atak gaye? Aise socho
- Zero: teen-teen ke jhund mein hon to remove karo; har jhund ka ek 0 waapas.
- Bada cube: wahi Post 15.2: teen alag, ikai ka add kiya, left se dahai.
- Ikai check: root ki ikai ka cube → number ki ikai?
- Beech check: root 90-kuch → number 90³ aur 100³ ke beech?
- Check fail: perfect cube nahi hai.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Zero remove karna kyun chalta hai? 40³ = 40 × 40 × 40. Post 15.1: 4 × 4 × 4 = 64, aur teen 10 ka multiply 10 × 10 × 10 = 1000. To 40³ = 64 × 1000 = 64000. Reverse chalo: 64000 = 64 × 1000, to root = (64 ka root 4) aur (1000 ka root 10) ka multiply = 40. Har teen zero ek 10 hain.
Beech check kyun? Bada number, bada cube. Agar root 90 aur 100 ke beech hai, to cube bhi 90³ aur 100³ ke beech hoga. Agar number wahan nahi, to root galat (ya number perfect cube nahi).
7-digit pe kya hota hai?
100³ = 1000000 (7 digit). To 7 digit ya zyada ke cube ka root 100 ya bada hoga. Tab left part 1000 ya zyada, aur hamari 1³ se 9³ ki list chhoti pad jaati hai. Sirf zero wale (jaise 1728000) chal jaate hain, kyunki zero remove karne ke baad number chhota ho jaata hai.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: ∛343000 = ?
Peeche teen 0 → remove karo → 343. ∛343 = 7 (list). Ek 0 waapas: 70.
✔️ Normal method se check: 70 × 70 = 4900, 4900 × 70 = 343000 (49 × 7 = 343, peeche teen 0) ✅
Example 2: ∛857375 = ? (perfect cube)
∛857375: last teen digit alag: 857 | 375. Last digit 5 → root ki ikai 5 (5 apne jaisa). Left 857: 9³ = 729, 857 mein baith gaya; par 10³ = 1000, 857 se bada. To root ki dahai 9. Jawab: ∛857375 = 95.
✔️ Normal method se check: 95 × 95 = 9025 (Chapter 6: 9 × 10 = 90, peeche 25). Phir 9025 × 95: 9025 × 90 = 812250 (9025 × 9 = 81225, peeche 0), 9025 × 5 = 45125; 812250 + 45125 = 857375 ✅
Example 3: ∛1728000 = ?
Teen 0 remove karo → 1728. 1 | 728: ikai 8 → 2 (2 ↔ 8), left 1 → 1³ = 1 baitha → 1. ∛1728 = 12. Ek 0 waapas: 120.
✔️ Normal method se check: 12 × 12 = 144, 144 × 12 = 1728 (144 × 10 = 1440, 144 × 2 = 288, 1440 + 288 = 1728); peeche teen 0 → 1728000 ✅
Example 4: Kya 13000 perfect cube hai?
Peeche teen 0, theek. Remove karo → 13. Par 2³ = 8 aur 3³ = 27; 13 inke beech hai, perfect cube nahi. To 13000 perfect cube nahi (∛13000, 20 aur 30 ke beech).
✔️ Normal method se check: 20³ = 8000 aur 30³ = 27000; 13000 inke beech, aur 20 aur 30 ke beech ke number (jaise 23³ = 12167, 24³ = 13824) bhi 13000 nahi dete ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki har trick ka ek “check” wala part hota hai: har tez method ke saath ek tez check. Aaj ke do check (ikai aur beech) isi soch ke hain. Aage Chapter 18 mein ek aur mashhoor check aayega: digit sum (9 wala check).
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Peeche teen-teen zero (64000, 1728000) | ✅ Zero remove karo | Phir list ya Post 15.2. |
| 100 ke paas bada cube (912673) | ✅ Post 15.2 | Left 9-sau ke aas-paas → dahai 9. |
| Zero 1, 2, 4 ya 5 (2700) | ❌ Round cube nahi | Zero teen-teen mein nahi. |
| “Perfect cube” nahi likha (117650) | ⚠️ Trick + check | Check fail = perfect nahi. |
| 7-digit, zero nahi (1030301) | ❌ Is chapter mein nahi | Root 100 se bada. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Zero waapas lagana bhool gaye
Sawaal: ∛64000
❌ Galat: “zero hataye, ∛64 = 4. Jawab 4.”
✅ Sahi: teen 0 ka ek 0 waapas: 40. Check: 4 × 4 × 4 to sirf 64 hai.
Mistake 2: Saare zero waapas laga diye
Sawaal: ∛27000
❌ Galat: “∛27 = 3, teeno 0 waapas: 3000.”
✅ Sahi: teen 0 ka ek 0: 30. (3000 ka cube to bahut bada hota.)
Mistake 3: Perfect cube nahi, phir bhi trick
Sawaal: ∛2000
❌ Galat: “left 2 → 1, ikai 0 → 0, jawab 10.”
✅ Sahi: check: 10³ = 1000, 2000 nahi. 2000, 12³ = 1728 aur 13³ = 2197 ke beech: perfect cube nahi.
Mistake 4: Zero teen-teen mein nahi
Sawaal: ∛2700
❌ Galat: “zero remove karo, 27, ∛27 = 3, to 30.”
✅ Sahi: 2700 mein sirf do 0. 30³ = 27000 hota hai (teen 0). 2700 perfect cube nahi.
🧠 Rule of thumb: Pehle zero count karo (teen-teen), phir list ya Post 15.2, aur hamesha do check: ikai aur beech.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: cube root, ya “perfect cube hai?” (Haan/Nahi). Target: 5 minute.
- ∛8000 = ?
- ∛216000 = ?
- ∛512000 = ?
- ∛729000 = ?
- ∛1331000 = ?
- ∛857375 = ?
- ∛912673 = ?
- ∛804357 = ?
- Kya 2000 perfect cube hai?
- Kya 27000 perfect cube hai?
- Kya 2700 perfect cube hai?
- Kya 117650 perfect cube hai?
- ∛970299 = ?
- ∛2744000 = ?
- ∛175616 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 20
- 60
- 80
- 90
- 110
- 95
- 97
- 93
- Nahi
- Haan
- Nahi
- Nahi
- 99
- 140
- 56
Kaise kiya (kuch): ∛1331000: teen 0 remove karo → 1331 → 1 | 331, ikai 1 → 1, left 1 → 1 → 11 → 110. ∛804357: 804 | 357, ikai 7 → 3, 804 mein 9³ = 729 baitha → 93. 2700: sirf do 0, perfect cube nahi.
— 📌 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.
- ∛1000 = ?
- ∛27000 = ?
- ∛64000 = ?
- ∛125000 = ?
- ∛343000 = ?
- ∛1000000 = ?
- ∛6859000 = ?
- ∛91125 = ?
- ∛941192 = ?
- ∛884736 = ?
| 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
- 10
- 30
- 40
- 50
- 70
- 100
- 190
- 45
- 98
- 96
Dhyaan: ∛1000000: chhah 0 → do jhund → do 0 waapas, ∛1 = 1 → 100. ∛6859000: teen 0 remove karo → 6859 → 19 → 190. ∛884736: 884 | 736, ikai 6 → 6, left 884 mein 9³ = 729 → 96.
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. ∛64000 = ?
Jawab aur reason dekho
Sahi jawab: (c) 40
- (a) galat ❌: 400: teen 0 ki jagah do 0 waapas laga diye.
- (b) galat ❌: 4000: teeno 0 waapas laga diye (4 ke peeche 000); teen 0 ke jhund ka sirf ek 0 waapas aata hai.
- (c) sahi ✅: Teen 0 remove karo → 64 → 4, ek 0 waapas → 40.
- (d) galat ❌: 4: zero waapas lagana bhool gaye.
Q2. ∛912673 = ? (perfect cube)
Jawab aur reason dekho
Sahi jawab: (b) 97
- (a) galat ❌: 79: dahai aur ikai reverse likh di.
- (b) sahi ✅: 912 | 673: ikai 3 → 7; 912 mein 9³ = 729 → 97.
- (c) galat ❌: 93: ikai 3 ko 3 hi rakha; 3 aur 7 jagah badalte hain.
- (d) galat ❌: 87: 912 mein sabse bada 9³ = 729 baithta hai, par 8³ = 512 le liya.
Q3. Kaunsa number perfect cube NAHI hai?
Jawab aur reason dekho
Sahi jawab: (a) 2700
- (a) sahi ✅: 2700 mein sirf do 0: round number ke cube mein zero teen-teen increase hote hain.
- (b) galat ❌: Mistake: 27000 = 30³ (teen 0, 27 = 3³).
- (c) galat ❌: Mistake: 8000 = 20³.
- (d) galat ❌: Mistake: 1000 = 10³.
Q4. ∛1728000 = ?
Jawab aur reason dekho
Sahi jawab: (d) 120
- (a) galat ❌: 12: zero waapas lagana bhool gaye.
- (b) galat ❌: 1200: ek ki jagah do 0 waapas laga diye.
- (c) galat ❌: 12000: teeno 0 waapas laga diye (12 ke peeche 000); teen 0 ke jhund ka sirf ek 0 waapas aata hai.
- (d) sahi ✅: Teen 0 remove karo → 1728 → 12, ek 0 waapas → 120.
Q5. Trick ne ∛117650 ke liye “40” diya. Check (40³ = 64000) se kya pata chala?
Jawab aur reason dekho
Sahi jawab: (c) 117650 perfect cube nahi hai
- (a) galat ❌: Mistake: 49³ = 117649 hai, 117650 nahi (1 ka difference).
- (b) galat ❌: Mistake: 50³ = 125000, bahut bada.
- (c) sahi ✅: Check fail: 64000, 117650 se bahut door. Aur 49³ = 117649, 50³ = 125000: beech mein koi cube nahi.
- (d) galat ❌: Mistake: check fail hone pe jawab nahi maana jaata.
Q6. ∛857375 = ? (perfect cube)
Jawab aur reason dekho
Sahi jawab: (b) 95
- (a) galat ❌: 59: dahai aur ikai reverse likh di.
- (b) sahi ✅: 857 | 375: ikai 5 → 5; 857 mein 9³ = 729 baitha → 95.
- (c) galat ❌: 85: 857 mein sabse bada 9³ = 729 baithta hai, par 8³ = 512 le liya.
- (d) galat ❌: 75: 857 mein 7³ = 343 le liya; 8³ aur 9³ bhi baithte hain, sabse bada lena tha.
Q7. 97 ka “beech check”: 912673 kin do cubes ke beech hona chahiye?
Jawab aur reason dekho
Sahi jawab: (a) 729000 aur 1000000
- (a) sahi ✅: 90³ = 729000, 100³ = 1000000.
- (b) galat ❌: Mistake: yeh 20³ aur 30³ hain.
- (c) galat ❌: Mistake: yeh 97² aur 100² (square) hain, cube nahi.
- (d) galat ❌: Mistake: yeh root ke beech hai, cube ke nahi.
Q8. ∛343000 = ?
Jawab aur reason dekho
Sahi jawab: (d) 70
- (a) galat ❌: 7: zero waapas lagana bhool gaye.
- (b) galat ❌: 700: ek ki jagah do 0 waapas laga diye.
- (c) galat ❌: 7000: teeno 0 waapas laga diye (7 ke peeche 000); teen 0 ke jhund ka sirf ek 0 waapas aata hai.
- (d) sahi ✅: Teen 0 remove karo → 343 → 7, ek 0 waapas → 70.
Q9. Kis number pe Post 15.2 wala rule (zero ke bina) seedha nahi chalega?
Jawab aur reason dekho
Sahi jawab: (c) 1030301
- (a) galat ❌: Mistake: 2197 = 13³, rule chalta hai.
- (b) galat ❌: Mistake: 54872 = 38³, rule chalta hai.
- (c) sahi ✅: 7 digit, left 1030: root 100 se bada, list 9³ tak hi.
- (d) galat ❌: Mistake: 912673 = 97³, rule chalta hai.
Q10. Ek bade cube ke peeche chhah 0 hain: 8000000. ∛ = ?
Jawab aur reason dekho
Sahi jawab: (b) 200
- (a) galat ❌: 2: zero waapas lagana bhool gaye.
- (b) sahi ✅: Chhah 0 = do jhund → do 0 waapas, ∛8 = 2 → 200.
- (c) galat ❌: 2000: square root (Chapter 14) wala rule laga diya: zero half kar diye (chhah ka half teen), 2 ke peeche 000. Cube root mein zero ka ek-tihai aata hai: chhah ka ek-tihai do.
- (d) galat ❌: 20: do jhund ki jagah ek hi 0 waapas lagaya.
— 📌 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. ∛ 474552000 = ? (Yeh perfect cube hai.)
Solution dekho
Teen 0 remove karo → 474552. Post 15.2: 474 | 552, ikai 2 → 8, left 474 mein 7³ = 343 baitha (8³ = 512 bada) → 78. Ek 0 waapas: 780.
Check: 780 ki ikai 0, 0³ = 0, number ki ikai bhi 0 ✅. Beech: 700³ = 343000000 aur 800³ = 512000000; 474552000 beech mein ✅.
Q2. Riya kehti hai ∛24389 = 27. Bina poora multiply kiye, do check se batao sahi hai ya galat.
Solution dekho
Ikai check: 27 ki ikai 7, 7³ = 343 ki ikai 3. Par 24389 ki ikai 9 ❌. Galat!
Sahi: 24 | 389, ikai 9 → 9, left 24 mein 2³ = 8 baitha (3³ = 27 bada) → 29. Beech check: 20³ = 8000, 30³ = 27000; 24389 beech mein ✅.
Q3. Ek perfect cube 6 digit ka hai, uske peeche 000 hai. Sabse chhota aisa number kaunsa?
Solution dekho
Peeche 000, to root peeche 0 wala: 10, 20, 30 … 6 digit chahiye: 40³ = 64000 (5 digit), 50³ = 125000 (6 digit). To 125000.
Q4. 100 aur 10000 ke beech kitne perfect cube hain?
Solution dekho
5³ = 125 (pehla, 100 se bada). 21³ = 9261 (last, 10000 se chhota), kyunki 22³ = 10648 bada. To 5 se 21 tak ke numbers ke cube. Kitne numbers? 21 − 5 = 16, aur 5 khud bhi count mein hai, to 16 + 1 = 17 perfect cube.
Check: 4³ = 64 (100 se chhota) aur 22³ = 10648 (10000 se bada), isliye dono bahar ✅.
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🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Zero wale cubes ka root, “kya yeh perfect cube hai?”. |
| Maths Olympiad | Bade cube root, check ke sawaal, count (kitne perfect cube). |
| NTSE / MAT | Tez cube root aur check. |
| SSC / Bank (aage) | Simplification mein ∛, options mein se galat jawab remove karna (ikai check). |
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. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “zero count karo, rule lagao, do check.” Exam mein MCQ ho to ikai check se aksar 2-3 galat options seedhe kat jaate hain! Aur jab “perfect cube” na likha ho, check kabhi mat chhodo. Agla post: Chapter 15 ke tricky sawaal! 🏗️
— 📌 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)
912 | 673. Ikai 3 → root ki ikai 7 (3 ↔ 7). Left 912: 9³ = 729 baitha, 10³ = 1000 bada → dahai 9. Jawab 97 ✅
| 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. Zero remove karke root nikaalna kyun chalta hai?
Har teen zero ek 10 hain (10 × 10 × 10 = 1000). Unka root 10, yaani ek 0.
2. Ikai check kab kaafi hai?
Galat jawab pakadne ke liye bahut tez. Par ikai sahi hone se jawab pakka sahi nahi; beech check bhi karo.
3. Agar “perfect cube” nahi likha ho?
Trick lagao, phir check. Check fail ho to perfect cube nahi.
4. 7-digit cube ka root?
Root 100 ya zyada. Is chapter mein sirf zero wale (jaise 1728000 → 120).
5. Agle post mein kya hai?
Post 15.4: tricky sawaal: perfect cube hai ya nahi, cube box, ∛0.027 aur dhokhe.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Zero teen-teen mein: remove karo, root, har jhund ka ek 0 waapas (∛64000 = 40).
- Bade cube: Post 15.2 ka rule (∛912673 = 97).
- Ikai check: root ki ikai ka cube → number ki ikai.
- Beech check: 90³ = 729000 aur 100³ = 1000000.
- Poora multiply: 97 × 97 = 9409, 9409 × 97 = 912673.
- “Perfect cube” na likha ho to trick + check (2000, 117650 nahi).
— charumam.com 🎯
👉 Aage Kya?
Ab Chapter 15 ka last post: tricky sawaal! Perfect cube hai ya nahi (yahan ikai se pata nahi chalta!), cube box, ∛0.027 aur dhokhe, Post 15.4 mein.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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