Post 14.1 mein 400 tak ke perfect squares ka root list se nikaala. Par √2209? List itni long nahi hoti! Aaj ek tez method: last digit se root ki ikai ke do option, left part se dahai, aur Chapter 6 ke 5 wale square (45² = 2025) se decision. Yeh tab chalta hai jab pata ho ki number perfect square hai.
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). √1849: kaunsa number × khud = 1849?
- 40 × 40 = 1600: chhota. 50 × 50 = 2500: bada. To root 40 aur 50 ke beech.
- 45 × 45 = 2025 (Chapter 6): bada. 44 × 44 = 1936: bada. 43 × 43 = 1849: mil gaya! To √1849 = 43.
Teen-chaar 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, 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
√2209 = 47, kyunki 47 × 47 = 2209
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 14.1 ka √, 1² se 9² (1 se 81), aur Chapter 6 ke 5 wale square (25², 35², …, 95²). Comparison ke word (chhota, bada, beech) aur 25 wala reverse neeche boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Last digit (ikai): number ka sabse right wala digit. 2209 ki ikai 9.
- Left part: last do digit hata kar jo bacha: 2209 → 22.
- Do option: ikai ke rule se root ki ikai ke do sambhav digit, jaise 3 ya 7.
- Beech wala (5 wala) square: dono option ke beech ka number 5 pe khatam: 43 aur 47 ke beech 45.
- Perfect square hai ya nahi: yeh trick tabhi jab pata ho number perfect square hai (sawaal mein diya ho).
- Square metre: area ki ikaai: 1 metre × 1 metre (metre = length ki ikaai, lagbhag ek long step) ke square ki jagah.
📏 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.
🔄 Square root = “reverse square” (√ ka matlab)
Square mein hum number dete hain aur poochte hain “iska square kya?” 7 → 7 × 7 = 49. Square root mein reverse poochte hain: “Kis number ka square 49 hai?” Jawab 7, kyunki 7 × 7 = 49. Isliye square root ko reverse square samjho (Hindi mein “varg-mool”: mool matlab jad, yaani woh number jisse square uga).
Sign √ (square root ka sign): square root dikhane ke liye number ke aage √ lagate hain. Yeh ek sahi-ka-tick (✓) jaisa dikhta hai, jiski long roof number ke upar faili hoti hai. √49 padho “49 ka square root” ya “root 49”. √49 = 7 ka matlab: 7 × 7 = 49.
| Square (aage) | Square root (reverse) |
|---|---|
| 3 × 3 = 9, yaani 3² = 9 | √9 = 3 |
| 7 × 7 = 49, yaani 7² = 49 | √49 = 7 |
| 10 × 10 = 100, yaani 10² = 100 | √100 = 10 |
| 12 × 12 = 144, yaani 12² = 144 | √144 = 12 |
Check hamesha aasaan: jo root mila, use usi se multiply karo. Wahi number wapas aaye to root sahi. √144 = 12? 12 × 12 = 144 ✅.
Dhyaan: hum sirf 0, 1, 2, 3 … jaise numbers (0 ya usse bade) ki baat kar rahe hain. √ ka sign hamesha aisa hi root batata hai: √49 = 7.
↔️ Comparison ke word: chhota, bada, beech mein
Is chapter mein numbers ki comparison hum shabdon mein likhenge, koi sign nahi:
- “49, 50 se chhota hai”: 49 kam hai.
- “64, 50 se bada hai”: 64 zyada hai.
- “16, 22 se chhota ya equal hai”: 16 ya to 22 se kam hai, ya bilkul 22.
- “50, 49 aur 64 ke beech hai”: 50, 49 se bada aur 64 se chhota.
🟠 25 pe khatam perfect square: Chapter 6 reverse
Chapter 6: 65² = 4225, kyunki 6 × 7 = 42, peeche 25. Ab reverse: √4225. Peeche ka 25 remove karo → 42. Kaunsa number × agla number = 42? 6 × 7 = 42. To root = 6 aur peeche 5: 65.
| Square | 25 remove karo | Number × agla | Root |
|---|---|---|---|
| 1225 | 12 | 3 × 4 = 12 | 35 |
| 2025 | 20 | 4 × 5 = 20 | 45 |
| 5625 | 56 | 7 × 8 = 56 | 75 |
| 9025 | 90 | 9 × 10 = 90 | 95 |
🔁 Chhota sa warm-up
- 4² aur 5² kya hain?
- 7 × 7 ki ikai?
- 45² = ?
Answers dekho
- 16 aur 25
- 9 (49)
- 2025 (4 × 5 = 20, peeche 25)
Teeno aa gaye? Yahi teen cheezein √2209 ka jawab hain!
— 📌 charumam.com
🎬 Story: Square Parade Ground
Republic Day ke liye 2209 cadets ek square mein khade hone the: jitni line, har line mein utne cadet. Coach Gill sir ne poocha: “Har line mein kitne?”
Ishaan bola: “Sir, 2209 ki ikai 9. Square ki ikai 9 tab aati hai jab root 3 ya 7 pe khatam ho. Aur 22 mein 4² = 16 baitha, to root 40-kuch. 43 ya 47. 45² = 2025 se 2209 bada, to 47!”
Coach Gill sir ne 47 line banwaayi: har line mein 47, ek bhi cadet nahi bacha. 🎖️ Aaj yahi trick! 🔚
— 📌 charumam.com
🪄 Trick Step by Step: Do Digit Alag, Dahai, Ikai Ke Option, 5 Se Decision
💡 Is post ki ek badi baat: Perfect square N ke liye: (1) last do digit alag karo. (2) Left part: sabse bada square jo usme baith jaaye → root ki dahai. (3) Last digit se root ki ikai ke do option. (4) Beech wale 5 wale square se decision: N bada → bada option, N chhota → chhota option.
Step 1: Do digit alag karo
√2209: last do digit alag: 22 | 09.
Step 2: Left se dahai
Left 22. Kaunsa sabse bada square 22 ke andar baithta hai? 4² = 16, 22 se chhota hai, par 5² = 25, 22 se bada. To root ki dahai 4 (root 40-kuch, kyunki 40² = 1600 aur 50² = 2500 ke beech).
Step 3: Last digit se ikai ke do option
2209 ki ikai 9. Neeche ki table: square ki ikai 9 tab, jab root ki ikai 3 (3 × 3 = 9) ya 7 (7 × 7 = 49). To root 43 ya 47.
| Number ki ikai | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Square ki ikai | 0 | 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 |
| Kaise | 0 × 0 = 0 | 1 × 1 = 1 | 2 × 2 = 4 | 3 × 3 = 9 | 4 × 4 = 16 | 5 × 5 = 25 | 6 × 6 = 36 | 7 × 7 = 49 | 8 × 8 = 64 | 9 × 9 = 81 |
Step 4: 5 wale square se decision
43 aur 47 ke beech 45. 45² = 2025 (Chapter 6). 2209, 2025 se bada hai, to root bhi 45 se bada: 47. (Agar N, 2025 se chhota hota, to 43.)
Step 5: Check karo
47 × 47: 47 × 7 = 329 (40 × 7 = 280, 7 × 7 = 49, 280 + 49 = 329), 47 × 40 = 1880 (47 × 4 = 188, peeche 0), 329 + 1880 = 2209 ✅.
Aur examples, ek ke baad ek
√7744: last do digit alag: 77 | 44. Left 77: 8² = 64, 77 se chhota ya equal; par 9² = 81, 77 se bada. To root ki dahai 8 (root 80 aur 90 ke beech). Last digit 4 → root ki ikai 2 ya 8, yaani 82 ya 88. Beech wala 85² = 7225 (Chapter 6: 8 × 9 = 72, peeche 25). 7744, 7225 se bada hai, to bada option: 88. Jawab: √7744 = 88.
√1849: last do digit alag: 18 | 49. Left 18: 4² = 16, 18 se chhota ya equal; par 5² = 25, 18 se bada. To root ki dahai 4 (root 40 aur 50 ke beech). Last digit 9 → root ki ikai 3 ya 7, yaani 43 ya 47. Beech wala 45² = 2025 (Chapter 6: 4 × 5 = 20, peeche 25). 1849, 2025 se chhota hai, to chhota option: 43. Jawab: √1849 = 43.
Ikai 5 ya 0 ho to?
Ikai 5 → root ki ikai sirf 5 (1 option): 25 wale box ka method (√4225 = 65). Ikai 0 → peeche 00 hona chahiye: zero half (√3600 = 60, Post 14.1).
Kabhi nahi: ikai 2, 3, 7, 8
Table mein square ki ikai sirf 0, 1, 4, 5, 6, 9 aati hai. To 2207, 1233, 5678 jaise number perfect square ho hi nahi sakte!
🆘 Atak gaye? Aise socho
- Left part: last do digit remove karo.
- Dahai: 1² se 9² mein sabse bada jo left mein baith jaaye.
- Ikai: 1 → 1/9, 4 → 2/8, 9 → 3/7, 6 → 4/6, 5 → 5, 0 → 0.
- Decision: beech wale 5 wale square (jaise 45², Chapter 6) se comparison.
- 3-digit (576): left ek digit (5).
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Ikai wala rule kyun? 47 × 47 ko pieces mein break karo (jaise Chapter 6 mein 35 × 35 ko pieces mein kiya: 35 ko 30 baar, phir 35 ko 5 baar): 40 × 40, 40 × 7, 7 × 40, aur 7 × 7. Pehle teen mein 40 hai, to unki ikai 0. Ikai sirf 7 × 7 = 49 se aati hai: 9. Matlab square ki ikai sirf root ki ikai pe tikti hai.
Isliye har root-ikai se square ki ikai pakki hai. Table dekho: 5 aur 0 ko chhod kar, har ikai do digit se aati hai:
- 1 × 1 = 1 aur 9 × 9 = 81: dono ki ikai 1.
- 2 × 2 = 4 aur 8 × 8 = 64: dono ki ikai 4.
- 3 × 3 = 9 aur 7 × 7 = 49: dono ki ikai 9.
- 4 × 4 = 16 aur 6 × 6 = 36: dono ki ikai 6.
- Sirf 5 × 5 = 25 (ikai 5) aur 0 × 0 = 0 (ikai 0) akele hain.
Isiliye ikai ke do option aate hain, aur 5 wale square se decision karna padta hai.
Dahai wala rule kyun? 40² = 1600 aur 50² = 2500. 2209 inke beech, to root 40 aur 50 ke beech. 22 (left part) aur 16, 25 ki comparison yahi hai, bas do zero remove karke.
5 wala square decision kyun karta hai?
Bada number, bada square. 45, 43 aur 47 ke beech hai, to 45² bhi 43² aur 47² ke beech. 45² = 2025 hum jaante hain (Chapter 6). To N agar 2025 se bada, root 45 se bada: 47.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: √5329 = ?
√5329: last do digit alag: 53 | 29. Left 53: 7² = 49, 53 se chhota ya equal; par 8² = 64, 53 se bada. To root ki dahai 7 (root 70 aur 80 ke beech). Last digit 9 → root ki ikai 3 ya 7, yaani 73 ya 77. Beech wala 75² = 5625 (Chapter 6: 7 × 8 = 56, peeche 25). 5329, 5625 se chhota hai, to chhota option: 73. Jawab: √5329 = 73.
✔️ Normal method se check: 73 × 73: 73 × 3 = 219, 73 × 70 = 5110, 219 + 5110 = 5329 ✅
Example 2: √576 = ? (3-digit)
√576: last do digit alag: 5 | 76. Left 5: 2² = 4, 5 se chhota ya equal; par 3² = 9, 5 se bada. To root ki dahai 2 (root 20 aur 30 ke beech). Last digit 6 → root ki ikai 4 ya 6, yaani 24 ya 26. Beech wala 25² = 625 (Chapter 6: 2 × 3 = 6, peeche 25). 576, 625 se chhota hai, to chhota option: 24. Jawab: √576 = 24.
✔️ Normal method se check: 24 × 24: 24 × 4 = 96, 24 × 20 = 480, 96 + 480 = 576 ✅
Example 3: √9604 = ?
√9604: last do digit alag: 96 | 04. Left 96: 9² = 81, 96 se chhota ya equal; par 10² = 100, 96 se bada. To root ki dahai 9 (root 90 aur 100 ke beech). Last digit 4 → root ki ikai 2 ya 8, yaani 92 ya 98. Beech wala 95² = 9025 (Chapter 6: 9 × 10 = 90, peeche 25). 9604, 9025 se bada hai, to bada option: 98. Jawab: √9604 = 98.
(Post 13.1 se bhi: 98² → 96 | 04!)
✔️ Normal method se check: 98² → 96 | 04 → 9604 (base 100) ✅
Example 4: √4225 = ? (25 pe khatam)
25 remove karo → 42. 6 × 7 = 42. Root 65.
✔️ Normal method se check: 65²: Chapter 6: 6 × 7 = 42, peeche 25 → 4225 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Perfect square ke root ke liye CharuMam ki trick bilkul yahi “last digit + pehle digit” wali soch hai, jise hum Pehle Dekho (dekh kar) kehte hain. Ikai ka 3 aur 7 wala pair (dono ka sum 10) Chapter 3 ke “10 se poora” jaisa hai: 3 aur 7, 4 aur 6, 2 aur 8, 1 aur 9.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 4-digit perfect square (2209) | ✅ Haan | Do option, 5 se decision. |
| 3-digit perfect square (576) | ✅ Haan | Left ek digit. |
| 25 pe khatam (4225) | ✅ Chapter 6 reverse | Number × agla. |
| Ikai 2, 3, 7, 8 (2207) | ❌ Perfect nahi | Root poora nahi. |
| Pata nahi perfect hai ya nahi (2219) | ⚠️ Pehle check | 2219 ke last do digit alag: 22 aur 19. Dahai 4, ikai 9 → 43 ya 47; 2219, 45² = 2025 se bada → trick “47” degi. Par 47² = 2209 hai, 2219 nahi. To 2219 perfect square nahi (47² = 2209 aur 48² = 2304 ke beech). Trick tabhi lagao jab pata ho ki number perfect square hai; warna Post 14.3. |
| 5-digit (15129) | ✅ Haan | Left 151: 12² = 144, to 12-kuch. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Ikai ka ek hi option socha
Sawaal: √2209
❌ Galat: “ikai 9 → root ki ikai 3 → 43.”
✅ Sahi: 9 ke do option: 3 ya 7. 45² = 2025 se decision → 47.
Mistake 2: Left parts mein galat square
Sawaal: √7744
❌ Galat: “77 mein 9² = 81 baitha → 9-kuch.”
✅ Sahi: 81, 77 se bada hai, to nahi baithta. 8² = 64, 77 se chhota → dahai 8 → 82 ya 88 → 85² = 7225 se bada → 88.
Mistake 3: Decision opposite
Sawaal: √1849
❌ Galat: 1849, 2025 se chhota tha, phir bhi bada option 47.
✅ Sahi: chhota hai to chhota option: 43. Check: 43 × 43 = 1849 ✅.
Mistake 4: Perfect square nahi, phir bhi trick
Sawaal: √2207
❌ Galat: “ikai 7… root 4-kuch.”
✅ Sahi: ikai 7 kisi square ki nahi hoti: 2207 perfect square nahi.
🧠 Rule of thumb: Do digit alag, left se dahai, ikai ke do option, beech wale 5 wale square (jaise 45²) se decision, aur check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 perfect squares ka root. Target: 8 minute.
- √529 = ?
- √676 = ?
- √784 = ?
- √841 = ?
- √961 = ?
- √1024 = ?
- √1296 = ?
- √1444 = ?
- √1681 = ?
- √2116 = ?
- √2809 = ?
- √3481 = ?
- √4096 = ?
- √6241 = ?
- √9216 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 23
- 26
- 28
- 29
- 31
- 32
- 36
- 38
- 41
- 46
- 53
- 59
- 64
- 79
- 96
Kaise kiya (kuch):
√1296: last do digit alag: 12 | 96. Left 12: 3² = 9, 12 se chhota ya equal; par 4² = 16, 12 se bada. To root ki dahai 3 (root 30 aur 40 ke beech). Last digit 6 → root ki ikai 4 ya 6, yaani 34 ya 36. Beech wala 35² = 1225 (Chapter 6: 3 × 4 = 12, peeche 25). 1296, 1225 se bada hai, to bada option: 36. Jawab: √1296 = 36.
√6241: last do digit alag: 62 | 41. Left 62: 7² = 49, 62 se chhota ya equal; par 8² = 64, 62 se bada. To root ki dahai 7 (root 70 aur 80 ke beech). Last digit 1 → root ki ikai 1 ya 9, yaani 71 ya 79. Beech wala 75² = 5625 (Chapter 6: 7 × 8 = 56, peeche 25). 6241, 5625 se bada hai, to bada option: 79. Jawab: √6241 = 79.
— 📌 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.
- √625 = ?
- √1225 = ?
- √2025 = ?
- √3025 = ?
- √4225 = ?
- √5625 = ?
- √7225 = ?
- √9025 = ?
- √8100 = ?
- √6400 = ?
| 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
- 25
- 35
- 45
- 55
- 65
- 75
- 85
- 95
- 90
- 80
Dhyaan: 25 wale: 25 remove karo, number × agla dhoondho (3025: 30 = 5 × 6 → 55). 8100, 6400: zero half → 90, 80.
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. √2209 = ?
Jawab aur reason dekho
Sahi jawab: (b) 47
- (a) galat ❌: 49: ikai ke do option ki jagah 2209 ki last digit 9 hi root ki ikai maan li.
- (b) sahi ✅: Dahai 4, ikai 3 ya 7; 2209, 2025 se bada → 47.
- (c) galat ❌: 43: decision opposite kiya: 2209, 45² = 2025 se bada tha, phir bhi chhota option liya.
- (d) galat ❌: 53: dahai galat: 22 mein 5² = 25 baithta nahi (25, 22 se bada), phir bhi 5 liya. Phir 55² = 3025 se 2209 chhota → 53.
Q2. √1849 = ?
Jawab aur reason dekho
Sahi jawab: (a) 43
- (a) sahi ✅: Dahai 4, ikai 3 ya 7; 1849, 2025 se chhota → 43.
- (b) galat ❌: 47: decision opposite kiya: 1849, 2025 se chhota tha, phir bhi bada option liya.
- (c) galat ❌: 49: ikai ke do option ki jagah last digit 9 hi root ki ikai maan li.
- (d) galat ❌: 37: dahai galat: 18 mein sabse bada square 16 (4²) baithta hai, par 9 (3²) le liya. Phir 35² = 1225 se 1849 bada → 37.
Q3. Square ki ikai 6 hai. Root ki ikai kya ho sakti hai?
Jawab aur reason dekho
Sahi jawab: (d) 4 ya 6
- (a) galat ❌: 6: Mistake: sirf ek option socha; 4 bhi chalta hai.
- (b) galat ❌: Mistake: ikai 9 wala rule laga diya (3 aur 7 se ikai 9 aati hai).
- (c) galat ❌: Mistake: ikai 4 wala rule laga diya (2 aur 8 se ikai 4 aati hai).
- (d) sahi ✅: 4 × 4 = 16, 6 × 6 = 36: dono ki ikai 6.
Q4. Kaunsa number perfect square ho hi nahi sakta?
Jawab aur reason dekho
Sahi jawab: (c) 2207
- (a) galat ❌: Mistake: 4 ko woh ikai samjha jo kisi square ke aakhir mein nahi aati (woh sirf 2, 3, 7, 8 hain). 2304 = 48².
- (b) galat ❌: Mistake: 1 ko woh ikai samjha jo kisi square ke aakhir mein nahi aati (woh sirf 2, 3, 7, 8 hain). 2401 = 49².
- (c) sahi ✅: Ikai 7: kisi square ki ikai 7 nahi hoti.
- (d) galat ❌: Mistake: 9 ko woh ikai samjha jo kisi square ke aakhir mein nahi aati (woh sirf 2, 3, 7, 8 hain). 2209 = 47².
Q5. √7744 = ?
Jawab aur reason dekho
Sahi jawab: (b) 88
- (a) galat ❌: 84: ikai ke do option ki jagah last digit 4 hi root ki ikai maan li.
- (b) sahi ✅: Dahai 8 (64, 77 se chhota), ikai 2 ya 8; 7744, 85² = 7225 se bada → 88.
- (c) galat ❌: 82: decision opposite: 7744, 7225 se bada tha, phir bhi chhota option.
- (d) galat ❌: 92: dahai galat: 77 mein 9² = 81 baithta nahi (81, 77 se bada), phir bhi 9 liya. Phir 95² = 9025 se 7744 chhota → 92.
Q6. √5329 = ?
Jawab aur reason dekho
Sahi jawab: (a) 73
- (a) sahi ✅: Dahai 7 (49, 53 se chhota), ikai 3 ya 7; 5329, 75² = 5625 se chhota → 73.
- (b) galat ❌: 77: decision opposite: 5329, 5625 se chhota tha, phir bhi bada option.
- (c) galat ❌: 79: ikai ke do option ki jagah last digit 9 hi root ki ikai maan li.
- (d) galat ❌: 67: dahai galat: 53 mein sabse bada square 49 (7²) baithta hai, par 36 (6²) le liya. Phir 65² = 4225 se 5329 bada → 67.
Q7. √4225 = ?
Jawab aur reason dekho
Sahi jawab: (d) 65
- (a) galat ❌: 42: 25 remove karke bacha 42 hi jawab maan liya.
- (b) galat ❌: 6: 42 = 6 × 7 se 6 mila, par peeche 5 lagana bhool gaye.
- (c) galat ❌: 75: number × agla (6 × 7) mein agla wala 7 le liya, phir 5 lagaya.
- (d) sahi ✅: 25 remove karo → 42 = 6 × 7 → 6 ke baad 5: 65.
Q8. 47² ki ikai kya hogi?
Jawab aur reason dekho
Sahi jawab: (c) 9
- (a) galat ❌: 4: 7 × 7 = 49 ka dahai wala digit (4) le liya, ikai nahi.
- (b) galat ❌: 6: dahai digit 4 ka square (4 × 4 = 16) dekh ke uski ikai 6 le li.
- (c) sahi ✅: 7 × 7 = 49 → ikai 9.
- (d) galat ❌: 7: root ki ikai hi square ki ikai likh di.
Q9. √576 = ?
Jawab aur reason dekho
Sahi jawab: (b) 24
- (a) galat ❌: 16: dahai galat: 5 mein sabse bada square 4 (2²) baithta hai, par 1 (1²) le liya. Phir 15² = 225 se 576 bada → 16.
- (b) sahi ✅: Left 5: 2² = 4 baithta hai → dahai 2; ikai 4 ya 6; 576, 25² = 625 se chhota → 24.
- (c) galat ❌: 26: decision opposite: 576, 625 se chhota tha, phir bhi bada option.
- (d) galat ❌: 34: dahai galat: 5 mein 3² = 9 baithta nahi (9, 5 se bada), phir bhi 3 liya. Phir 35² = 1225 se 576 chhota → 34.
Q10. Is trick mein 45² = 2025 kis kaam aata hai?
Jawab aur reason dekho
Sahi jawab: (a) Do options (43, 47) mein se chunne ke liye
- (a) sahi ✅: Bada ya chhota option tay karta hai.
- (b) galat ❌: Mistake: dahai left part (22) se aati hai, 45² se nahi.
- (c) galat ❌: Mistake: ikai last digit (9) se aati hai.
- (d) galat ❌: Mistake: bina iske 43 aur 47 mein decision nahi hota.
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🧠 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. √9801 = ? (do methods se)
Solution dekho
Left 98: 9² = 81, 98 se chhota → dahai 9. Ikai 1 → 1 ya 9 → 91 ya 99. 95² = 9025, aur 9801, 9025 se bada → 99. Post 13.1 se bhi: 99² → 98 | 01 → 9801.
Check: 99 × 99 = 9900 − 99 = 9801 ✅
Q2. Ek square land ka area 3136 square metre. Bhuja?
Solution dekho
√3136: last do digit alag: 31 | 36. Left 31: 5² = 25, 31 se chhota ya equal; par 6² = 36, 31 se bada. To root ki dahai 5 (root 50 aur 60 ke beech). Last digit 6 → root ki ikai 4 ya 6, yaani 54 ya 56. Beech wala 55² = 3025 (Chapter 6: 5 × 6 = 30, peeche 25). 3136, 3025 se bada hai, to bada option: 56. Jawab: √3136 = 56.
Check: 56 × 56 = 56 × 6 + 56 × 50 = 336 + 2800 = 3136 ✅
Q3. Kya 1234 perfect square hai?
Solution dekho
Ikai 4 chal sakti hai, to aur dekho. 35² = 1225 (Chapter 6), 36² = 1225 + 35 + 36 = 1296 (agla square, Post 13.4). 1234, 1225 aur 1296 ke beech hai: beech mein koi poore number ka square nahi. To nahi.
Check: 1234 − 1225 = 9, square se 9 bachte hain ✅
Q4. 5-digit: √15129 = ?
Solution dekho
√15129: last do digit alag: 151 | 29. Left 151: 12² = 144, 151 se chhota ya equal; par 13² = 169, 151 se bada. To root ka ikai se pehle wala part 12 (root 120 aur 130 ke beech). Last digit 9 → root ki ikai 3 ya 7, yaani 123 ya 127. Beech wala 125² = 15625 (Chapter 6: 12 × 13 = 156, peeche 25). 15129, 15625 se chhota hai, to chhota option: 123. Jawab: √15129 = 123.
Check: 123 × 123 = 12300 + 2829 = 15129 ✅ (123 ko 100 baar = 12300; 123 ko 23 baar = 123 × 20 + 123 × 3 = 2460 + 369 = 2829)
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Perfect square ka root, “kya yeh perfect square hai?”. |
| Maths Olympiad | Ikai ke pattern, 5-digit root, square area. |
| NTSE / MAT | Tez root, number series. |
| SSC / Bank (aage) | √ wale simplification, 5 second mein. |
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. Class 6–8 = school ki chhathi se aathvi kaksha.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “do digit alag, dahai, do option, paanch se decision.” Ikai table (1 → 1/9, 4 → 2/8, 9 → 3/7, 6 → 4/6) ek baar khud bana lo. Agle post mein woh numbers jo perfect square nahi hain: √50 ka estimate! 🔚
— 📌 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)
22 | 09. Left 22: 4² = 16, 22 se chhota; 5² = 25, 22 se bada → dahai 4. Last digit 9 → ikai 3 ya 7 → 43 ya 47. 45² = 2025, aur 2209, 2025 se bada → bada option: 47 ✅
| 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. 🚀
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🤖 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:
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❓ FAQs (Students Ke Asli Sawaal)
1. Yeh trick har number pe?
Sirf perfect squares pe. Pehle ikai dekho: 2, 3, 7, 8 ho to perfect square nahi.
2. Ikai ke do option kyun?
Do digit ek hi ikai dete hain: 3 × 3 = 9 aur 7 × 7 = 49.
3. Ikai 5 ho to?
Sirf 5: 25 remove karo, number × agla dhoondho.
4. Chapter 13 se kya relation?
Base ke paas wale (9604, 10816) Chapter 13 ka reverse (Post 13.4).
5. Agle post mein kya hai?
Post 14.3: jo perfect square nahi, unka estimate (√50, √200).
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Do digit alag: 22 | 09.
- Left → dahai (4² = 16, 22 se chhota).
- Ikai: 9 → 3/7, 4 → 2/8, 6 → 4/6, 1 → 1/9.
- Decision: 45² = 2025 se comparison → 47.
- 25 wale: Chapter 6 reverse.
- Ikai 2, 3, 7, 8: perfect square nahi.
— charumam.com 🎯
👉 Aage Kya?
Ab woh numbers jo perfect square nahi: Post 14.3 mein √50, √200 ka estimate, “kin do squares ke beech” aur ek decimal tak (decimal: point wale numbers, jaise 7.1)!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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- 🌐 Website: charumam.com
- 🤖 AI Tutor (doubt poochho, 24×7): tutor.charumam.com
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