Chapter 14 shuru! 🎉 Chapter 13 mein number ka square nikaala: 97 → 9409. Aaj reverse: square diya hai, number dhoondho. 49 kis number ka square hai? 7, kyunki 7 × 7 = 49. Ise square root kehte hain, aur sign √. Aaj √ ka matlab, perfect squares ki list (jo aap Chapter 13 ki trick se khud bana sakte ho), aur round numbers ka root.
⏱ Before Timer: Pehle Purane Method Se
Pehle “try karke” method (ek doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). Sawaal: kaunsa number apne aap se multiply hokar 169 deta hai? Estimates se try karo:
- 10 × 10 = 100: chhota.
- 15 × 15 = 225: bada. To number 10 aur 15 ke beech.
- 12 × 12 = 144: abhi chhota. 13 × 13 = 169: mil gaya! To √169 = 13.
Kai baar multiply karna pada. 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
√196 = 14, kyunki 14 × 14 = 196
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: square ka matlab (Chapter 13), aur 11² se 19² (Post 13.2 ki trick). Naya: √ ka sign aur “reverse square”. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Square root (varg-mool): woh number jiska square diya hua number hai. 49 ka square root 7.
- √ (root ka sign): √49 padho “root 49” ya “49 ka square root”.
- Perfect square: woh number jo kisi poore number ka square ho: 1, 4, 9, 16, 25 … 50 perfect square nahi.
- Whole number: 0, 1, 2, 3 … bina point ya pieces wala number.
- Bhuja: square ka ek kinaara. Square ka area = bhuja × bhuja, to bhuja = √area.
- Area: kisi cheez ki sataha (upar ki jagah) kitni hai. Square ka area = bhuja × bhuja.
- Metre, cm: length measure karne ki ikaai. 1 metre lagbhag ek bade man ke ek long step jitna; 1 cm (centimetre) ungli ki chaudai jitna. 100 cm = 1 metre.
- Square metre, square cm: area ki ikaai: 1 metre × 1 metre ke chhote square ki jagah = 1 square metre. Isi tarah 1 cm × 1 cm = 1 square cm.
- Taaq number: jo 2 se poora na bante (2 ke jodon mein na bante): 1, 3, 5, 7 …
📏 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 kya hai? Chhota ² ka matlab (Chapter 6 yaad karo)
Kisi number ko usi number se multiply karna us number ka square (Hindi mein “varg”) kehlata hai. Ise number ke upar-right mein chhota ² likh kar dikhate hain. 7² padho “7 ka square” ya “7 square”, matlab 7 × 7 = 49. Dhyaan: 7² ka matlab 7 × 2 nahi hai (woh 14 hota)!
Name “square” kyun? Square ka ek matlab square bhi hai (jiski all four bhujaayein, yaani kinaare, equal hon). 7 line banao, har line mein 7 dots: dots ka ek square ban jaata hai, aur total dots 7 × 7 = 49. To 7² = 49 dots ka square.
| Square | Matlab | Jawab |
|---|---|---|
| 3² | 3 × 3 | 9 |
| 7² | 7 × 7 | 49 |
| 10² | 10 × 10 | 100 |
| 35² | 35 × 35 | 1225 (Post 6.1 ki trick) |
| 97² | 97 × 97 | 9409 (Post 13.1 ki trick) |
Chapter 7 se relation: 97² bas 97 × 97 hai, yaani base method ka woh sawaal jisme dono numbers ek jaise hain. Isliye dono numbers base se utne hi door hote hain, aur trick aur bhi aasaan ho jaati hai.
🔄 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.
0️⃣ Round number ka square root: zero half
Post 13.3 mein seekha: round number ka square = digit ka square, phir zero double: 60² = 3600 (6 × 6 = 36, ek 0 → do 0). Ab reverse: square root mein zero half ho jaate hain.
- √3600: zero remove karo (do 0), bacha 36. √36 = 6. Do 0 ka half, ek 0: 60. Check: 60 × 60 = 3600 ✅.
- √40000: chaar 0, bacha 4. √4 = 2. Chaar ka half do 0: 200. Check: 200 × 200 = 40000 ✅.
Dhyaan: zero add kiye mein (2, 4, 6 …) hone chahiye. 900 (do 0) ✅ → 30. Par 90 (ek 0) ya 4000 (teen 0) perfect square nahi hain, kyunki kisi round number ka square hamesha do-do zero badhaata hai.
🔁 Chhota sa warm-up
- 8² = ?
- Kaunsa number × khud = 81?
- 13² = ?
Answers dekho
- 64
- 9 (9 × 9 = 81)
- 169 (16 | 9, Post 13.2)
Doosra sawaal hi square root hai: √81 = 9!
— 📌 charumam.com
🎬 Story: Square Rangoli
Diwali pe Meher ko 196 diye mile. Usne socha: “Inhe ek square mein lagaungi: utni hi line, har line mein utne hi diye.” Par har line mein kitne?
Bhai Aarav ne 10, 12, 15 try kiye… Meher ne Chapter 13 ki trick se bani list dekhi: “14² = 196. To 14 line, har line mein 14 diye!”
Rangoli bilkul square bani. ✨ Aarav: “Tune to square reverse kar diya!” Meher: “Haan, isi ko square root kehte hain: √196 = 14.” 🌱
— 📌 charumam.com
🪄 Trick Step by Step: Reverse Socho, List Dekho, Zero Half, Check Karo
💡 Is post ki ek badi baat: √N ka matlab: “kaunsa number × khud = N?” Perfect squares ki list (1² se 20²) yaad ya taiyaar rakho. Round number ho to zero half (√3600 = 60). Aur hamesha check: root × root = N.
Step 1: Reverse sawaal bolo
√144 ko bolo: “Kaunsa number apne aap se multiply hokar 144 deta hai?”
Step 2: List mein dhoondho
Neeche ki list mein 144 kahan hai? 12² = 144. To √144 = 12.
Step 3: List khud banao (Chapter 13 se)
11 se 19 ke square Post 13.2 ki trick se: 12 → 2 zyada → 14 | 4 → 144. 17 → 7 zyada → 24 | 49 → 49 mein 4 dahai, left mein: 28 | 9 → 289.
Step 4: Round number: zero half
√3600: do 0 remove karo → 36. √36 = 6. Ek 0 waapas → 60.
Step 5: Check karo
12 × 12 = 144 ✅. 60 × 60 = 3600 ✅. Root ko usi se multiply karo: wahi number aaya, to sahi.
Perfect squares ki list: 1² se 20²
| Square | Matlab | Perfect square |
|---|---|---|
| 1² | 1 × 1 | 1 |
| 2² | 2 × 2 | 4 |
| 3² | 3 × 3 | 9 |
| 4² | 4 × 4 | 16 |
| 5² | 5 × 5 | 25 |
| 6² | 6 × 6 | 36 |
| 7² | 7 × 7 | 49 |
| 8² | 8 × 8 | 64 |
| 9² | 9 × 9 | 81 |
| 10² | 10 × 10 | 100 |
| 11² | 11 × 11 | 121 |
| 12² | 12 × 12 | 144 |
| 13² | 13 × 13 | 169 |
| 14² | 14 × 14 | 196 |
| 15² | 15 × 15 | 225 |
| 16² | 16 × 16 | 256 |
| 17² | 17 × 17 | 289 |
| 18² | 18 × 18 | 324 |
| 19² | 19 × 19 | 361 |
| 20² | 20 × 20 | 400 |
(1² se 10² chhote multiply hain; 11² se 19² Post 13.2 ki trick se; 20² = 400 round number ka rule: 2 × 2 = 4, phir 00.)
Round squares (zero wale)
| Square | Kaise | To square root |
|---|---|---|
| 30² = 900 | 3 × 3 = 9, do 0 | √900 = 30 |
| 40² = 1600 | 4 × 4 = 16, do 0 | √1600 = 40 |
| 50² = 2500 | 5 × 5 = 25, do 0 | √2500 = 50 |
| 70² = 4900 | 7 × 7 = 49, do 0 | √4900 = 70 |
| 100² = 10000 | 1 × 1 = 1, chaar 0 | √10000 = 100 |
List mein ek pattern (Post 13.4 yaad karo)
1, 4, 9, 16, 25, 36 … ka difference dekho: 3, 5, 7, 9, 11 … har baar agla taaq number. Kyunki agla square = square + number + agla number (4 + 5 = 9, to 16 + 9 = 25). List bhool jaao to isi se bana lo.
🆘 Atak gaye? Aise socho
- Bolo: “kaunsa number × khud = N?”
- List: 1 se 400 tak ke perfect squares list mein.
- Round: zero add kiye mein hon to half karo: √4900 = 70.
- Half mat karo: √16 = 4, 8 nahi (8 to 16 ka half hai).
- Check: root × root.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Chitra se: 49 dots ko square mein lagao: 7 line, har line mein 7. Square ki bhuja (ek kinaara) mein 7 dots hain. To square root = square ki bhuja.
Isiliye har number perfect square nahi hota: 50 dots se poora square nahi banta. 7 × 7 = 49 mein 1 bach jaata hai, aur 8 × 8 = 64 ke liye 14 kam padte hain. (50 ka root 7 aur 8 ke beech hai: Post 14.3.)
Zero half kyun?
60² = 60 × 60 = 6 × 6 × 10 × 10 = 36 × 100 = 3600. Root mein ek 10 hai, square mein do (100). To square se root pe jaate hue 100 (do zero) ka ek 10 (ek zero) bachta hai: zero half.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: √144 = ?
List: 12² = 144. Ya Post 13.2: 12 → 14 | 4 → 144. Jawab 12.
✔️ Normal method se check: 12 × 12 = 120 + 24 = 144 ✅ (12 ko 10 baar = 120, 12 ko 2 baar = 24)
Example 2: √361 = ?
361, 18² = 324 aur 20² = 400 ke beech. 19²: 9 zyada → 28 | 81 → 81 mein 8 dahai, left mein: 36 | 1 → 361. Jawab 19.
✔️ Normal method se check: 19 × 19 = 190 + 171 = 361 ✅ (19 ko 10 baar = 190; 19 ko 9 baar = 190 − 19 = 171; 190 + 171 = 361)
Example 3: √2500 = ?
Do 0 remove karo → 25. √25 = 5. Ek 0 waapas → 50.
✔️ Normal method se check: 50 × 50 = 5 × 5 × 100 = 2500 ✅
Example 4: Ek square room ka area 225 square metre hai. Bhuja?
Bhuja × bhuja = 225, to bhuja = √225. List: 15² = 225 (Chapter 6: 1 × 2 = 2, peeche 25). Bhuja 15 metre (metre: length ki ikaai, upar word-list dekho).
✔️ Normal method se check: 15 × 15 = 150 + 75 = 225 ✅ (15 ko 10 baar = 150, 5 baar = 75)
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Square root ke liye CharuMam ka pehla rule “Pehle Dekho” (dekh kar identify karo) hai: chhote perfect squares ko dekhte hi root pehchanna. Aur reverse sochna (square → root) “Poora Aur Parts” wali soch hai: poore ko parts se, parts ko poore se samjho. Agle post mein ek aur tez method: last digit se root.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 1 se 400 tak perfect square | ✅ List | 1² se 20², turant. |
| Round, zero add kiye mein (3600) | ✅ Zero half | √36 = 6 → 60. |
| 4-digit perfect square (2209) | ➡️ Post 14.2 | Last digit ka rule. |
| Perfect square nahi (50) | ➡️ Post 14.3 | Kin do squares ke beech, estimate. |
| 90, 4000 (zero taaq) | ❌ Perfect nahi | Zero do-do mein nahi. |
| Root ka matlab half? | ❌ Nahi | √16 = 4, half 8 hai. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Root ko half samajh liya
Sawaal: √16
❌ Galat: 8 (16 ka half).
✅ Sahi: 4 × 4 = 16, to 4. Check: 8 × 8 = 64, 16 nahi.
Mistake 2: Zero galat
Sawaal: √900
❌ Galat: 300 ya 3.
✅ Sahi: do 0 ka half ek 0: √9 = 3 → 30. Check: 30 × 30 = 900 ✅.
Mistake 3: √ aur ² ulta-pulta
Sawaal: √49
❌ Galat: 49 ka square kar diya: 49 × 49 = 2401 (49 ko 50 baar = 2450, ek 49 kam = 2401).
✅ Sahi: √ reverse square hai: 7, kyunki 7 × 7 = 49. (2401 to 49² hai.)
Mistake 4: Har number ka poora root maan liya
Sawaal: √50
❌ Galat: “√50 = 25” ya “√50 = 5”.
✅ Sahi: 50 perfect square nahi. 7 × 7 = 49, 8 × 8 = 64, to √50 7 aur 8 ke beech (Post 14.3).
🧠 Rule of thumb: √N = woh number jo × khud = N. List dekho, zero half, aur root × root se check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 square root. Target: 4 minute. List chhupa ke karo!
- √4 = ?
- √9 = ?
- √25 = ?
- √36 = ?
- √49 = ?
- √64 = ?
- √81 = ?
- √100 = ?
- √121 = ?
- √144 = ?
- √169 = ?
- √196 = ?
- √225 = ?
- √400 = ?
- √900 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 2
- 3
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
- 13
- 14
- 15
- 20
- 30
Kaise kiya (kuch): √169: 13² → 16 | 9 → 169, to 13. √900: √9 = 3, ek 0 → 30.
— 📌 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.
- √16 = ?
- √1 = ?
- √256 = ?
- √289 = ?
- √324 = ?
- √361 = ?
- √1600 = ?
- √2500 = ?
- √4900 = ?
- √10000 = ?
| 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
- 4
- 1
- 16
- 17
- 18
- 19
- 40
- 50
- 70
- 100
Dhyaan: √256: 16² (6 zyada: 22 | 36 → 36 mein 3 dahai, left mein: 22 + 3 = 25 → 25 | 6 → 256) = 256, to 16. √10000: chaar 0 → do 0, √1 = 1 → 100.
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. √49 = ?
Jawab aur reason dekho
Sahi jawab: (a) 7
- (a) sahi ✅: 7 × 7 = 49.
- (b) galat ❌: 98: √ ko “double” samjha, 49 × 2 kar diya.
- (c) galat ❌: 49: √ ka sign bhool gaye, number hi likh diya.
- (d) galat ❌: 2401: √ ki jagah square kar diya: 49 × 49 (49 ko 50 baar = 2450, ek 49 kam = 2401).
Q2. √ ka matlab kya hai?
Jawab aur reason dekho
Sahi jawab: (d) Kaunsa number × khud = yeh number
- (a) galat ❌: Mistake: root ko half samjha (49 ÷ 2); root half nahi.
- (b) galat ❌: Mistake: root ko double samjha; × 2 to double hai.
- (c) galat ❌: Mistake: reverse operation kiya; square se number bada hota hai, root se chhota.
- (d) sahi ✅: Haan: √49 → 7, kyunki 7 × 7 = 49.
Q3. √144 = ?
Jawab aur reason dekho
Sahi jawab: (c) 12
- (a) galat ❌: 288: root ko double samjha (144 × 2).
- (b) galat ❌: 24: root 12 sahi nikaala, phir bina wajah 2 se multiply kar diya (12 × 2).
- (c) sahi ✅: 12 × 12 = 144.
- (d) galat ❌: 72: root ko half samjha (144 ÷ 2).
Q4. Kaunsa perfect square hai?
Jawab aur reason dekho
Sahi jawab: (b) 196
- (a) galat ❌: Mistake: 9² = 81 yaad aaya to 90 bhi maan liya. 90 mein ek hi 0 hai, aur 90, 9² = 81 aur 10² = 100 ke beech.
- (b) sahi ✅: 14 × 14 = 196.
- (c) galat ❌: Mistake: round dikha to perfect maan liya. 200, 14² = 196 aur 15² = 225 ke beech hai: perfect nahi.
- (d) galat ❌: Mistake: 150 mein 15 dikha, 15² yaad aaya, to 150 ko bhi perfect square maan liya. Par 150, 12² = 144 aur 13² = 169 ke beech hai: perfect nahi.
Q5. √900 = ?
Jawab aur reason dekho
Sahi jawab: (a) 30
- (a) sahi ✅: √9 = 3, do 0 ka half ek 0: 30.
- (b) galat ❌: 300: √9 = 3 ke baad dono 0 waapas laga diye (zero half nahi kiye).
- (c) galat ❌: 3: √9 = 3 ke baad 0 waapas lagana bhool gaye.
- (d) galat ❌: 450: root ko half samjha (900 ÷ 2).
Q6. 17² = ?
Jawab aur reason dekho
Sahi jawab: (d) 289
- (a) galat ❌: 2449: 49 ke 4 dahai left mein bhejna (carry) bhool gaye, 24 aur 49 seedhe saath-saath likh diye.
- (b) galat ❌: 249: 49 ka sirf 9 likha, 4 dahai left mein add karna bhool gaye (24 | 9).
- (c) galat ❌: 34: square ki jagah double kiya (17 × 2).
- (d) sahi ✅: 7 zyada: 24 | 49 → 49 mein 4 dahai, left mein → 28 | 9 → 289.
Q7. √361 = ?
Jawab aur reason dekho
Sahi jawab: (c) 19
- (a) galat ❌: 91: root 19 ke digit reverse likh diye.
- (b) galat ❌: 722: root ko double samjha (361 × 2).
- (c) sahi ✅: 19 × 19 = 361.
- (d) galat ❌: 18: list mein ek line pehle wala padh liya (18² = 324).
Q8. √2500 = ?
Jawab aur reason dekho
Sahi jawab: (b) 50
- (a) galat ❌: 5: √25 = 5 ke baad 0 waapas lagana bhool gaye.
- (b) sahi ✅: √25 = 5, do 0 ka half ek 0: 50.
- (c) galat ❌: 500: √25 = 5 ke baad dono 0 waapas laga diye (zero half nahi kiye).
- (d) galat ❌: 25: do 0 remove karke bacha 25 hi likh diya, uska root nikaala hi nahi.
Q9. Ek square ka area 64 square cm. Bhuja kitni?
Jawab aur reason dekho
Sahi jawab: (a) 8 cm
- (a) sahi ✅: 8 × 8 = 64.
- (b) galat ❌: 16: area ko perimeter (all four edges ka sum) samjha, 64 ÷ 4 kar diya.
- (c) galat ❌: 32: root ko half samjha (64 ÷ 2).
- (d) galat ❌: 64: area ko hi bhuja maan liya, root nahi nikaala.
Q10. 50 ke baare mein kya sahi hai?
Jawab aur reason dekho
Sahi jawab: (d) 50 perfect square nahi, √50 7 aur 8 ke beech
- (a) galat ❌: 25: root ko half samjha (50 ÷ 2); 25 × 25 = 625.
- (b) galat ❌: Mistake: round number dekh ke perfect maan liya. Koi whole number × khud = 50 nahi.
- (c) galat ❌: 100: root ko double samjha (50 × 2).
- (d) sahi ✅: 7 × 7 = 49, 8 × 8 = 64: 50 beech mein.
— 📌 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. 1 se 100 tak (100 bhi) kitne perfect squares hain?
Solution dekho
1², 2², …, 10², yaani 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Total 10. (11² = 121, 100 se bada.)
Check: 10 × 10 = 100 ✅
Q2. √(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) = ?
Solution dekho
Post 13.4: pehle 10 taaq numbers ka sum = 10² = 100. √100 = 10.
Check: sum 1 + 3 + 5 + … + 19: pair banao (1 + 19 = 20, 3 + 17 = 20 …), 5 pair × 20 = 100 ✅
Q3. Shatranj (chess) ke board mein 64 boxes hain, square mein. Har line mein kitne?
Solution dekho
√64 = 8, kyunki 8 × 8 = 64. Har line mein 8 boxes.
Check: 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64 ✅
Q4. Do agle perfect squares ka difference 41 hai. Dono kaun se?
Solution dekho
Post 13.4: agle squares ka difference = dono number add karo. Do agle numbers ka sum 41. Agla number 1 bada hai, to pehle 1 remove karo, phir half karo: 41 − 1 = 40, half 20. Numbers 20 aur 21. Squares 400 aur 441.
Check: 21² = 441 (13.4 ka “agla square” rule: 20² = 400, phir + 20 + 21 = 441), 441 − 400 = 41 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Square root, perfect square, square ki bhuja (area se). |
| Maths Olympiad | Perfect squares ginna, taaq numbers ka sum, patterns. |
| NTSE / MAT | Number series (1, 4, 9, 16 …). |
| SSC / Bank (aage) | Simplification mein √ turant. |
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.
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👩🏫 CharuMam Ka Tip
Mantra: “root bolo: kaunsa × khud?” 1² se 20² ki list ek card pe likho (Chapter 13 ki trick se khud banao) aur daily ek baar padho. Agle post mein 4-digit squares ka root, bina list ke! 🌱
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⏱ 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)
Perfect squares ki list mein dekho: 14² = 196 (Post 13.2: 14 → 4 zyada, 18 | 16 → 19 | 6 → 196). To √196 = 14 ✅
| 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. √ aur ² mein kya relation hai?
Ek doosre ke reverse: 7² = 49 aur √49 = 7.
2. √1 aur √0?
√1 = 1 (1 × 1 = 1), √0 = 0.
3. Har number ka square root hota hai?
Haan, par har number ka poora root nahi hota. √50 lagbhag 7.1 hai (7.1 padho “saat point ek”: 7 se thoda sa zyada; aise point wale, yaani decimal, numbers Post 14.3 mein samjhaayenge).
4. List ratni padegi?
1² se 10² pakke karo; 11² se 19² Chapter 13 ki trick se 5 second mein ban jaate hain.
5. Agle post mein kya hai?
Post 14.2: last digit se 4-digit perfect square ka root, jaise √2209 = 47.
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🟢 Recap: Aaj Kya Seekha
- √N: kaunsa number × khud = N.
- Square aur root reverse: 7² = 49, √49 = 7.
- Perfect squares: 1, 4, 9 … 400 (1² se 20²).
- Round: zero half (√3600 = 60).
- Root half nahi hai (√16 = 4).
- Check: root × root.
— charumam.com 🎯
👉 Aage Kya?
Ab bade perfect squares: Post 14.2 mein √2209, √7744 jaise root sirf last digit aur ek chhote estimates se!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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