Chapter 14 ka last post! 🏆 Aaj 14.1–14.3 ka mix, soch wale sawaal: perfect square hai ya nahi (bina multiply kiye), √4000 ka dhokha (20 nahi!), kitna add karein ya subtract karein ki perfect square ban jaaye, area se kinaara, aur point wale root jaise √0.25. 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 1850 perfect square hai?
- 40² = 1600, 50² = 2500: root 40 aur 50 ke beech.
- 45² = 2025 (Chapter 6): bada. 43² = 1849: thoda chhota. 44² = 1936: bada.
- Kai multiply ke baad: 1850, 1849 aur 1936 ke beech aa gaya, to perfect square nahi.
Teen 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
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 14.1–14.3 ke method. Yaad ke liye √, zero half, comparison ke word (chhota, bada, beech) aur decimal ke boxes neeche. Naye word:
📖 Word jo is post mein aayenge
- Perfect square test: bina root nikaale pehchanna ki number perfect square hai ya nahi.
- Taaq (odd): jo 2 se poora na bante: 1, 3, 5 …
- m: metre (length ki ikai): 42 m = 42 metre.
- Jist (even): jo 2 se poora bante: 0, 2, 4, 6 …
- Dhokha (trap): aisa sawaal jo dekhne mein aasaan, par jaldi mein mistake karwa de.
- Kinaara (bhuja): square ki ek side. Area = kinaara × kinaara.
- Metre, cm: length ki ikaai. 1 metre lagbhag ek long step; 1 cm ungli ki chaudai jitna; 100 cm = 1 metre.
- Square metre, square cm: area ki ikaai: 1 metre × 1 metre (ya 1 cm × 1 cm) ke square ki jagah.
- Negative number: 0 se chhote numbers, jaise −1, −4 (thermometer pe 0 se neeche). Class 8 tak inka root nahi lete.
📏 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.
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.
↔️ 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.
🔸 Decimal (point) wale numbers: 7.5 ka matlab
Kabhi jawab do whole numbers ke beech mein hota hai. Use likhne ke liye point (.) lagate hain, jise decimal point kehte hain. Point ke left wala part poore number, aur right wala part “ek ke pieces”.
- 7.5 padho “saat point paanch” ya “saadhe saat”: 7 aur half. Jaise ₹7.50 = 7 rupees 50 money (50 money = half rupaya).
- 0.1 = ek ka 10 mein se 1 part (tenth part). Jaise ek roti ke 10 equal pieces karo, unme se 1. To 7.1 = 7 aur ek tenth part, 7 se thoda sa zyada.
- 0.25 = ek ka chautha part (one-fourth), jaise 25 money = ₹1 ka chautha part.
Kaun bada? Point ke left wala part pehle dekho: 8.1 bada hai 7.9 se. Left same ho to right dekho: 7.5 bada hai 7.1 se (5 pieces, 1 pieces se zyada).
Point wale number ka square kaise? Point hata kar multiply karo, phir jawab mein point waapas lagao, dono numbers ke point ke baad jitne digit the, utne right se gin kar. 7.5 × 7.5: point remove karo → 75 × 75 = 5625 (Post 6.1: 7 × 8 = 56, peeche 25). Dono 7.5 mein point ke baad 1-1 digit, total 2. To 5625 mein right se 2 digit gin kar point: 56.25.
Yeh kyun sahi hai? 7.5 asal mein 75 ka tenth part hai (75 ÷ 10; ÷ matlab equal parts mein divide karna, Chapter 5). Multiply mein do baar tenth part liya, to jawab 5625 ka hundredth part: 5625 ÷ 100 = 56.25. (100 se divide karna = point do jagah left.)
Point wale numbers ka subtract karna: money ki tarah socho, point ke neeche point rakh kar. ₹50.41 − ₹50 = ₹0.41 (sirf 41 money). ₹10 − ₹9.61 = ₹0.39 (39 money), kyunki ₹9.61 + ₹0.39 = ₹10 (61 money aur 39 money milkar 100 money, yaani ₹1).
🔁 Chhota sa warm-up
- √1600 = ?
- Square ki ikai kabhi kya nahi hoti?
- √50 kin do whole numbers ke beech?
Answers dekho
- 40 (zero half)
- 2, 3, 7, 8
- 7 aur 8
Doosra jawab hi aaj ka sabse tez test hai!
— 📌 charumam.com
🎬 Story: Quiz Ka Dhokha
School quiz mein sawaal aaya: “√4000 = ?” Riya ne turant buzzer dabaya: “20! Kyunki √400 = 20, aur ek 0 extra.”
Quiz master Verma sir muskuraaye: “20 × 20 = 400 hai, 4000 nahi.” Riya ruk gayi. Dost Ishaan bola: “4000 mein teen zero hain. Zero half karne ke liye zero add kiye mein hone chahiye. Teen ka half nahi hota, to √4000 whole number hai hi nahi!”
Ishaan ne Post 14.3 lagaya: 63² = 3969, 64² = 4096. “√4000, 63 aur 64 ke beech!” Team ko point mila. 🧩
— 📌 charumam.com
🪄 Trick Step by Step: Identify karo, Test Karo, Phir Root Ya Estimate
💡 Is post ki ek badi baat: Pehle identify karo: perfect square hai ya nahi? Test: ikai 2, 3, 7, 8 → nahi; aakhir mein zero taaq count mein → nahi. Perfect ho to Post 14.1 (list), 14.2 (last digit). Nahi ho to Post 14.3 (beech, paas, half, ek decimal). Aur hamesha check: root × root.
Step 1: Identify karo
Sawaal mein root poocha hai (jaise √4000), ya “perfect hai ya nahi”, ya “kitna add karein”, ya area? Type identify karo.
Step 2: Ikai test
Diye hue number ki ikai 2, 3, 7 ya 8 → perfect square nahi (Post 14.2 ki table). 2207, 4003, 998 sab nahi.
Step 3: Zero test
Aakhir mein zero ginlo. 1 ya 3 (taaq) → perfect square nahi. 4000 mein 3 zero, 90 mein 1 zero: dono nahi. 3600 mein 2 zero: ho sakta hai (36 perfect hai).
Step 4: Perfect: root nikaalo
Chhota → list (Post 14.1). 4-digit → last do digit alag, dahai, ikai ke do option, 5 wale square se decision (Post 14.2).
Step 5: Perfect nahi: beech aur estimate
Kin do squares ke beech (Post 14.3). “Kitna add karein” → agla square − diya hua number (50: 64 − 50 = 14). “Kitna subtract karein” → diya hua number − pichla square (150: 150 − 144 = 6). Ek decimal chahiye → half check, phir Post 13.4 ka “agla square” rule (jaise Example 2 mein √5000).
Type 1: Perfect square hai ya nahi?
| Number | Test | Decision |
|---|---|---|
| 2207 | Ikai 7 | ❌ Nahi |
| 4000 | 3 zero (taaq) | ❌ Nahi |
| 1600 | 2 zero, 16 = 4² | ✅ Haan (40²) |
| 2025 | 25 pe khatam, 20 = 4 × 5 | ✅ Haan (45²) |
| 2024 | Ikai 4 (test paas), par 44² = 1936, 45² = 2025 | ❌ Nahi (beech mein) |
Dhyaan: ikai test sirf “nahi” pakka karta hai. Ikai 1, 4, 5, 6, 9 ho to bhi number perfect na ho sakta hai (2024). Tab beech check karo.
Type 2: √4000 ka dhokha
Step 1 (beech): 63² = 3969 aur 64² = 4096. 4000, 3969 aur 4096 ke beech hai, to √4000, 63 aur 64 ke beech. Distance: 4000 − 3969 = 31, aur 4096 − 4000 = 96. 4000, 3969 ke zyada paas, to sabse paas whole number 63.
Step 2 (half check): 63.5² nikaalo. Point remove karo: 635² (Chapter 6: 63 × 64 = 4032, peeche 25) = 403225. Point do jagah: 63.5² = 4032.25. 4000, 4032.25 se chhota hai, to √4000, 63.5 se chhota: 63 aur 63.5 ke beech.
Step 3 aur 4 (ek decimal): 63 se 0.1-0.1 increase karo, point hata kar Post 13.4 ka “agla square” rule (square + number + agla number), 630² = 396900 se: 631² = 396900 + 630 + 631 = 398161, 632² = 398161 + 631 + 632 = 399424, 633² = 399424 + 632 + 633 = 400689. Point do jagah: 63.2² = 3994.24 aur 63.3² = 4006.89. Distance: 4000 − 3994.24 = 5.76, aur 4006.89 − 4000 = 6.89. 4000, 3994.24 ke zyada paas.
Matlab √4000 ≈ 63.2, 20 bilkul nahi. (√400 = 20 aur √40000 = 200: dono mein zero add kiye mein hain.)
Type 3: Kitna add karein / subtract karein?
50 mein kam se kam kitna add karein ki perfect square bane? 50 ke baad agla square 64. 64 − 50 = 14.
150 mein se kam se kam kitna subtract karein? 150 se pehle square 144 (12²). 150 − 144 = 6.
Type 4: Area se kinaara
√1764: last do digit alag: 17 | 64. Left 17: 4² = 16, 17 se chhota ya equal; par 5² = 25, 17 se bada. To root ki dahai 4 (root 40 aur 50 ke beech). Last digit 4 → root ki ikai 2 ya 8, yaani 42 ya 48. Beech wala 45² = 2025 (Chapter 6: 4 × 5 = 20, peeche 25). 1764, 2025 se chhota hai, to chhota option: 42. Jawab: √1764 = 42.
Square ka area 1764 square metre ho to kinaara √1764 = 42 metre.
Type 5: Point wale root
√0.25: 0.5 × 0.5 = 0.25 (5 × 5 = 25, point ke baad total 2 digit). To √0.25 = 0.5. Isi tarah √0.49 = 0.7, √1.44 = 1.2.
🆘 Atak gaye? Aise socho
- Ikai 2, 3, 7, 8 → perfect nahi.
- Zero taaq → perfect nahi (4000).
- Kitna add karein: agla square − diya hua number (64 − 50 = 14).
- Kitna subtract karein: diya hua number − pichla square (150 − 144 = 6).
- Point: root mein point ke baad half digit (√0.25 = 0.5).
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Zero taaq ho to perfect kyun nahi? Kisi number ke peeche jitne zero, uske square ke peeche do times zero (60² = 3600: ek zero → do zero). Do multiply hamesha jist (2, 4, 6 …). To agar kisi number ke peeche 1 ya 3 zero (taaq) hain (jaise 90 ya 4000), woh kisi poore number ka square ho hi nahi sakta.
Chitra se √4000: 4000, 63² aur 64² ke beech hai, 3969 ke zyada paas.
√0.25 = 0.5 kyun?
0.25 = ek ka chautha part (25 money). 0.5 = half. Half ka half = chautha part: 0.5 × 0.5 = 0.25. Isliye root 0.5. (Dhokha: √0.25 ko 0.05 mat likho: 0.05 × 0.05 = 0.0025, kyunki 5 × 5 = 25 aur point ke baad total 4 digit.)
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: Kya 4489 perfect square hai? Ho to root?
Ikai 9: test paas. √4489: last do digit alag: 44 | 89. Left 44: 6² = 36, 44 se chhota ya equal; par 7² = 49, 44 se bada. To root ki dahai 6 (root 60 aur 70 ke beech). Last digit 9 → root ki ikai 3 ya 7, yaani 63 ya 67. Beech wala 65² = 4225 (Chapter 6: 6 × 7 = 42, peeche 25). 4489, 4225 se bada hai, to bada option: 67. Jawab: √4489 = 67.
✔️ Normal method se check: 67 × 67 = 4020 + 469 = 4489 ✅ (67 ko 60 baar = 4020, 7 baar = 469)
Example 2: √5000 ek decimal tak? (Dhyaan: 3 zero, perfect nahi)
Step 1 (beech): 70² = 4900 aur 71² = 5041. 5000, 4900 aur 5041 ke beech hai, to √5000, 70 aur 71 ke beech. Distance: 5000 − 4900 = 100, aur 5041 − 5000 = 41. 5000, 5041 ke zyada paas, to sabse paas whole number 71.
Step 2 (half check): 70.5² nikaalo. Point remove karo: 705² (Chapter 6: 70 × 71 = 4970, peeche 25) = 497025. Point do jagah: 70.5² = 4970.25. 5000, 4970.25 se bada hai, to √5000, 70.5 se bada: 70.5 aur 71 ke beech.
Step 3 (ek decimal): 70.5 se 0.1-0.1 increase karo. Har square “agla square” se (square + number + agla number), point hata kar: 706² = 497025 + 705 + 706 = 498436, to 70.6² = 4984.36; 707² = 498436 + 706 + 707 = 499849, to 70.7² = 4998.49; 708² = 499849 + 707 + 708 = 501264, to 70.8² = 5012.64. To √5000, 70.7 aur 70.8 ke beech.
Step 4 (kiske paas): 5000 − 4998.49 = 1.51, aur 5012.64 − 5000 = 12.64. 5000, 4998.49 ke zyada paas. To √5000 ≈ 70.7 (≈ matlab “lagbhag”).
✔️ Normal method se check: 70.7 × 70.7 = 4998.49, 5000 ke bahut paas ✅
Example 3: 200 mein kam se kam kitna add karein ki perfect square bane? Aur kitna subtract karein?
14² = 196 aur 15² = 225. Add karein: 225 − 200 = 25. Subtract karein: 200 − 196 = 4.
✔️ Normal method se check: 200 + 25 = 225 = 15 × 15 ✅; 200 − 4 = 196 = 14 × 14 ✅
Example 4: √4225 = ?
25 pe khatam: 25 remove karo → 42. 42 = 6 × 7 (number × agla). To root 6 ke baad 5: 65 (Chapter 6 reverse).
✔️ Normal method se check: 65 × 65 = 3900 + 325 = 4225 ✅ (65 ko 60 baar = 3900, 5 baar = 325)
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
Perfect square pehchanne ke liye CharuMam ke do test hain: ikai aur “digit ka sum” (saare digits add karte jao jab tak ek digit na bache). Aaj humne ikai aur zero wala test seekha. “Kitna add karein” pichle chapters ki “base se distance” (base yaani paas ka round number jaise 10, 100) wali soch jaisa hai: diya hua number agle square se kitna door (50, 64 se 14 door).
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Ikai 2, 3, 7, 8 | ✅ Turant “nahi” | Koi multiply nahi. |
| Zero taaq (90, 4000) | ✅ Turant “nahi” | Zero add kiye mein nahi. |
| Ikai 1, 4, 5, 6, 9, 0 | ⚠️ Pakka nahi | Beech check ya Post 14.2. |
| Kitna add karein / subtract karein | ✅ Beech wale squares | Agla square − number, number − pichla square. |
| Area se kinaara | ✅ √area | Perfect ho to 14.2, warna 14.3. |
| Point wala (0.25) | ✅ Point hata kar | √25 = 5 → 0.5. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Zero ka dhokha
Sawaal: √4000
❌ Galat: 20 ya 200.
✅ Sahi: 3 zero (taaq): perfect nahi. 63² = 3969, 64² = 4096 → 63 aur 64 ke beech.
Mistake 2: Ikai paas, to perfect maan liya
Sawaal: kya 2024 perfect square hai?
❌ Galat: haan (ikai 4 hai).
✅ Sahi: 44² = 1936, 45² = 2025: 2024 beech mein. Nahi.
Mistake 3: Add-subtract karne mein reverse
Sawaal: 150 mein se kitna subtract karein?
❌ Galat: 19 (169 − 150, yeh to add karna hai).
✅ Sahi: pichla square 144: 150 − 144 = 6.
Mistake 4: Point galat
Sawaal: √0.25
❌ Galat: 0.05.
✅ Sahi: 0.5 × 0.5 = 0.25, to 0.5.
🧠 Rule of thumb: Pehle test (ikai, zero), phir perfect ho to root (14.1/14.2), warna beech aur estimate (14.3). Hamesha root × root se check.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: pehle 7 “perfect square hai? (haan/nahi)”, phir 8 root. Target: 6 minute.
- 2207: perfect square?
- 1600: perfect square?
- 4000: perfect square?
- 2025: perfect square?
- 998: perfect square?
- 3481: perfect square?
- 2024: perfect square?
- √1764 = ?
- √4225 = ?
- √6889 = ?
- √3600 = ?
- √2116 = ?
- √7056 = ?
- √8464 = ?
- √9801 = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Nahi
- Haan
- Nahi
- Haan
- Nahi
- Haan
- Nahi
- 42
- 65
- 83
- 60
- 46
- 84
- 92
- 99
Kaise kiya (kuch):
√8464: last do digit alag: 84 | 64. Left 84: 9² = 81, 84 se chhota ya equal; par 10² = 100, 84 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). 8464, 9025 se chhota hai, to chhota option: 92. Jawab: √8464 = 92.
√9801: last do digit alag: 98 | 01. Left 98: 9² = 81, 98 se chhota ya equal; par 10² = 100, 98 se bada. To root ki dahai 9 (root 90 aur 100 ke beech). Last digit 1 → root ki ikai 1 ya 9, yaani 91 ya 99. Beech wala 95² = 9025 (Chapter 6: 9 × 10 = 90, peeche 25). 9801, 9025 se bada hai, to bada option: 99. Jawab: √9801 = 99.
— 📌 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.
- √0.25 = ?
- √0.49 = ?
- √1.44 = ?
- √0.09 = ?
- √0.64 = ?
- √2.25 = ?
- √0.81 = ?
- √1.21 = ?
- √0.16 = ?
- √1.69 = ?
| 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
- 0.5
- 0.7
- 1.2
- 0.3
- 0.8
- 1.5
- 0.9
- 1.1
- 0.4
- 1.3
Dhyaan: point hata kar root: √144 = 12, phir point ke baad digit half (2 → 1): 1.2. √0.09: √9 = 3 → 0.3.
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 square ho hi nahi sakta?
Jawab aur reason dekho
Sahi jawab: (d) 2207
- (a) galat ❌: Mistake: socha ki ikai 9 kisi square ki nahi hoti. Par hoti hai: 3 × 3 = 9, 7 × 7 = 49. Aur 47² = 2209.
- (b) galat ❌: Mistake: socha ki ikai 5 kisi square ki nahi hoti. Par 5 × 5 = 25, ikai 5. Aur 45² = 2025.
- (c) galat ❌: Mistake: zero dekh ke “nahi” bol diya, par yahan 2 zero (add kiye mein). 40² = 1600.
- (d) sahi ✅: Ikai 7: kabhi square nahi.
Q2. √4000 = ?
Jawab aur reason dekho
Sahi jawab: (c) 63 aur 64 ke beech
- (a) galat ❌: 200: 4000 mein ek 0 aur add karke 40000 bana liya, aur √40000 = 200 likh diya; par 200 × 200 = 40000.
- (b) galat ❌: 63: beech wale do numbers (63 aur 64) mein se chhota wala hi pakka root likh diya; par 63 × 63 = 3969, 4000 nahi.
- (c) sahi ✅: 4000, 3969 aur 4096 ke beech.
- (d) galat ❌: 20: Riya ki tarah 4000 ko 400 jaisa samjha aur √400 = 20 likh diya; par 20 × 20 = 400, 4000 nahi.
Q3. 50 mein kam se kam kitna add karein ki perfect square bane?
Jawab aur reason dekho
Sahi jawab: (b) 14
- (a) galat ❌: 31: agla square 64 chhod ke 81 tak add kar diya (81 − 50).
- (b) sahi ✅: 64 − 50 = 14.
- (c) galat ❌: 1: add karne ki jagah subtract kiya (50 − 49).
- (d) galat ❌: 50: paas wala square 64 chhod ke round square 100 tak add kar diya (100 − 50 = 50); 100 perfect hai, par kam se kam nahi.
Q4. 150 mein se kam se kam kitna subtract karein?
Jawab aur reason dekho
Sahi jawab: (a) 6
- (a) sahi ✅: 150 − 144 = 6.
- (b) galat ❌: 19: subtract karne ki jagah add karne wala nikaala (169 − 150).
- (c) galat ❌: 50: pichla square 144 chhod ke round square 100 tak subtract kiya (150 − 100 = 50); kam se kam nahi.
- (d) galat ❌: 29: pichla square 144 chhod ke 121 tak subtract kiya (150 − 121).
Q5. Square ka area 1764 square metre. Kinaara?
Jawab aur reason dekho
Sahi jawab: (d) 42 m
- (a) galat ❌: 1764: area ko hi kinaara maan liya, root nahi nikaala.
- (b) galat ❌: 441: area ko perimeter (all four edges ka sum) samjha, isliye 4 se baant diya: 1764 ÷ 4 = 441.
- (c) galat ❌: 882: root nikaalne ki jagah area ka half le liya: 1764 ÷ 2 = 882.
- (d) sahi ✅: 42 × 42 = 1764.
Q6. √0.25 = ?
Jawab aur reason dekho
Sahi jawab: (c) 0.5
- (a) galat ❌: 5: √25 = 5 ke baad point lagana bhool gaye.
- (b) galat ❌: 0.125: root nikaalne ki jagah 0.25 ka half le liya: 0.25 ÷ 2 = 0.125.
- (c) sahi ✅: 0.5 × 0.5 = 0.25.
- (d) galat ❌: 0.05: √25 = 5 sahi, par point ke baad 2 digit rakh diye, half (1) nahi kiye; 0.05 × 0.05 = 0.0025.
Q7. √4225 = ?
Jawab aur reason dekho
Sahi jawab: (b) 65
- (a) galat ❌: 75: 42 = 6 × 7 mein se chhota 6 ki jagah bada 7 le liya, phir peeche 5 lagaya: 75.
- (b) sahi ✅: 42 = 6 × 7, to 6 ke baad 5: 65.
- (c) galat ❌: 42: 25 remove karke bacha 42 hi jawab maan liya.
- (d) galat ❌: 6: 42 = 6 × 7 se 6 mila, par peeche 5 lagana bhool gaye.
Q8. 2024 ke baare mein sahi?
Jawab aur reason dekho
Sahi jawab: (a) Perfect nahi: 44² aur 45² ke beech
- (a) sahi ✅: 2024, 1936 aur 2025 ke beech.
- (b) galat ❌: Mistake: ikai test paas hone ko pakka “haan” maan liya; ikai 4 sirf opportunity deta hai.
- (c) galat ❌: Mistake: 2024 ko 2025 ke itna paas dekh kar equal maan liya; par 45 × 45 = 2025, 2024 nahi.
- (d) galat ❌: Mistake: root nikaalne ki jagah 2024 ka half le liya: 2024 ÷ 2 = 1012.
Q9. √5000 ≈ ?
Jawab aur reason dekho
Sahi jawab: (d) 70.7
- (a) galat ❌: 2500: root nikaalne ki jagah 5000 ka half le liya: 5000 ÷ 2 = 2500.
- (b) galat ❌: 7.07: 707 mein point right se 1 digit chhod ke lagana tha (70.7), par right se 2 digit chhod ke laga diya (7.07). 7.07 × 7.07 lagbhag 50 hai, 5000 nahi.
- (c) galat ❌: 707: 70.7 mein point lagana hi bhool gaye: 707 × 707 lagbhag 5 lakh (1,00,000 ka paanch times, yaani 5,00,000) hai.
- (d) sahi ✅: 70.7² = 4998.49, 5000 ke bahut paas.
Q10. Kaunse number ko sirf peeche ke zero dekh kar “perfect square nahi” keh sakte hain?
Jawab aur reason dekho
Sahi jawab: (c) 9000
- (a) galat ❌: Mistake: 4 zero (add kiye mein) bhi taaq count kiye. 40000 = 200².
- (b) galat ❌: Mistake: 2 zero (add kiye mein) bhi taaq count kiye. 3600 = 60².
- (c) sahi ✅: 9000 mein 3 zero (taaq): perfect nahi.
- (d) galat ❌: Mistake: 2 zero (add kiye mein) bhi taaq count kiye. 8100 = 90².
— 📌 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 50 tak kitne perfect squares?
Solution dekho
1, 4, 9, 16, 25, 36, 49: 1² se 7². 8² = 64, 50 se bada. Total 7.
Check: 7 × 7 = 49, 50 se chhota ✅
Q2. 1000 ke sabse paas perfect square kaun sa?
Solution dekho
31² = 961, 32² = 1024. 1000 − 961 = 39, 1024 − 1000 = 24. Sabse paas 1024.
Check: 32 × 32 = 960 + 64 = 1024 ✅ (32 ko 30 baar = 960, 2 baar = 64)
Q3. Shatranj (chess) ke board mein 64 boxes (chhote square boxes), aur board khud bhi square hai. Har box 5 cm × 5 cm. Board ka kinaara?
Solution dekho
√64 = 8 boxes ek line mein. Kinaara = 8 × 5 = 40 cm.
Check: 40 × 40 = 1600, aur 64 × 25 = 1600 ✅ (64 boxes × 25 square cm)
Q4. Ek number ka square 6889 hai, aur number 80 aur 90 ke beech hai. Number?
Solution dekho
√6889: last do digit alag: 68 | 89. Left 68: 8² = 64, 68 se chhota ya equal; par 9² = 81, 68 se bada. To root ki dahai 8 (root 80 aur 90 ke beech). Last digit 9 → root ki ikai 3 ya 7, yaani 83 ya 87. Beech wala 85² = 7225 (Chapter 6: 8 × 9 = 72, peeche 25). 6889, 7225 se chhota hai, to chhota option: 83. Jawab: √6889 = 83.
Check: 83 × 83 = 6640 + 249 = 6889 ✅ (83 ko 80 baar = 6640, 3 baar = 249)
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 8) | Perfect square test, kitna add karein/subtract karein, area se kinaara, decimal ka root. |
| Maths Olympiad | Perfect squares ginna, sabse paas square. |
| NTSE / MAT | “Kaunsa perfect square nahi” wale options. |
| SSC / Bank (aage) | √ wale long sawaal ko chhota karna aur estimate (√4000 ≈ 63.2) 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.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “pehle test, phir root.” Ikai 2, 3, 7, 8 ya taaq zero dikhe to ek second mein “nahi”. 🏆 Chapter 14 poora! Square aur square root dono ab aapke dost hain.
— 📌 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)
Last digit 7. Post 14.2 ki table: kisi bhi square ki ikai 2, 3, 7 ya 8 kabhi nahi hoti. To 2207 perfect square nahi ✅ (Ek bhi 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. Ikai 1, 4, 5, 6, 9 ho to perfect hoga?
Zaroori nahi (2024). Ikai test sirf “nahi” pakka karta hai.
2. Zero jist ho to perfect?
Zaroori nahi. 500 mein 2 zero (jist), par zero remove karke bacha 5 perfect square nahi; aur 22² = 484, 23² = 529, to 500 beech mein hai, perfect nahi. Zero remove karke bacha number bhi perfect hona chahiye: 3600 → 36 = 6², to 3600 = 60² ✅
3. √0.4 = 0.2?
Nahi! 0.2 × 0.2 = 0.04, 0.4 nahi. Point ke baad digit add kiye mein chahiye, to 0.4 ko 0.40 likho (40 money). Point hata kar 40: 6² = 36 aur 7² = 49, to √40, 6 aur 7 ke beech. Ek decimal: 6.3² = 39.69 aur 6.4² = 40.96 (63² = 3969, 64² = 4096); 40, 39.69 ke zyada paas (distance 0.31, aur 0.96), to √40 ≈ 6.3. 0.40, 40 ka hundredth part hai (40 ÷ 100), to uska root √40 ka tenth part, kyunki tenths × tenths = hundredths (0.1 × 0.1 = 0.01): 6.3 ÷ 10 = 0.63 (point ek jagah left). To √0.4 ≈ 0.63. Check: 0.63 × 0.63 = 0.3969, 0.4 ke bahut paas ✅
4. Negative number (0 se chhota, jaise −4) ka root?
Class 8 mein nahi; koi bhi number × khud, 0 se chhota number nahi deta.
5. Ab aage kya?
Chapter 14 poora! Agla chapter jald aa raha hai. Tab tak tricky sawaal dobara karo.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Ikai 2, 3, 7, 8 → perfect nahi.
- Zero taaq → perfect nahi (√4000, 20 nahi).
- Kitna add karein: agla square − diya hua number (64 − 50 = 14).
- Kitna subtract karein: diya hua number − pichla square (150 − 144 = 6).
- Area → kinaara = √area (√1764 = 42).
- √0.25 = 0.5.
— charumam.com 🎯
👉 Aage Kya?
🏆 Chapter 14 poora! Square aur square root ka safar khatam. Agla chapter jald aa raha hai. Tab tak Chapter 13 aur 14 ke drills dobara karo, aur charumam.com pe baaki chapters dekho!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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- 🌐 Website: charumam.com
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