✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- (a + m)(b + n) = ?
- 40² = ?
- 3² + 4² = ?
Answers dekho
- ab + an + mb + mn
- 1600
- 25 (par 7² = 49!)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 2 — Guna Mein Badlaav: Increments dekh lijiye.
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🎬 Kahani: Rangoli Ka Bada Square
Diwali pe Anya ne 40 cm ka rangoli square banaaya (1600 varg cm). Didi boli: “Har taraf 3 cm aur badha do!” Ab side 43 cm. 🪔
Anya ne socha: “40² = 1600, 3² = 9, to 43² = 1609!” Rang ki thaili 1609 varg cm ke hisaab se li… aur rangoli adhoori reh gayi. 😟
Kitna rang chahiye tha asal mein? Aur Anya ki soch mein kya chhoota? Chitra mein jawab hai. 🎨
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📖 Concept: Square Ke Chaar Tukde: Do Square, Do Rectangle
💡 Is post ki ek badi baat: (a + b) side ka square kaato: ek a × a square, ek b × b square, aur do a × b rectangle. Isliye (a + b)² = a² + 2ab + b². “2ab” woh do pattiyaan hain jo log bhool jaate.
Step 1: Chitra
Anya: a = 40, b = 3: 1600 + 2 × 120 + 9 = 1849. Usne do pattiyaan (240) bhoola!
Check: 43 × 43 = 1849 ✅
Step 2: Algebra (distributive se)
(a + b)(a + b) = a·a + a·b + b·a + b·b = a² + ab + ab + b² = a² + 2ab + b².
WHY 2ab? ab aur ba like terms hain (guna ka kram nahi badalta), to jud ke 2ab.
Step 3: Mann mein squares
71² = (70 + 1)² = 4900 + 140 + 1 = 5041.
205² = (200 + 5)² = 40000 + 2000 + 25 = 42025.
Kaise todo? Ek gol number (10, 100 ka multiple) + chhota hissa. Gol number ka square aasaan.
Step 4: Letters ke saath
(4x + 3)² = (4x)² + 2 × 4x × 3 + 3² = 16x² + 24x + 9.
(2p + 5q)² = 4p² + 20pq + 25q².
Do tareeke: identity (a = 4x, b = 3) ya seedha distributive (4x + 3)(4x + 3). Kab kaunsa? Identity tez; agar a, b pehchaanne mein confusion ho (jaise (4x)² ko 4x² likhna) to distributive surakshit.
Step 5: 5 pe khatam numbers ki trick
(10n + 5)² = 100n² + 100n + 25 = 100 × n(n + 1) + 25.
85²: n = 8 → 8 × 9 = 72 → 7225. 115²: 11 × 12 = 132 → 13225.
WHY? 2ab = 2 × 10n × 5 = 100n: n² ke saath mil ke n(n + 1) banaata.
🪜 Atak Gaye? Aise Socho
- (… + …)² dekho: pehla term a, doosra b.
- a² likho (poore term ka square: (4x)² = 16x²).
- 2 × a × b likho.
- b² likho.
- x = 1 daal ke dono taraf check.
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🔐 Deep Concept: Kyun (a + b)² ≠ a² + b²? (Square “jodne” pe nahi baant-ta)
Ch 6 ka pehla sabak: guna jod pe baant-ta hai. Par square = khud se guna. (a + b)² = (a + b)(a + b): yahan do brackets hain, isliye chaar tukde, do nahi!
Kitna farak? (a + b)² − (a² + b²) = 2ab. a, b dono positive → (a + b)² hamesha bada. Ek positive, ek negative → chhota! (3 + (−2))² = 1, par 9 + 4 = 13.
Aage ka pattern: (a + b)³ = (a + b)(a² + 2ab + b²) = a³ + 3a²b + 3ab² + b³. Coefficients 1, 2, 1 → 1, 3, 3, 1: Pascal ka triangle (Bharat mein Pingala ka Meru-prastara)!
Unifying idea: identity = distributive ka shortcut, aur chitra uska saboot. 🎯
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✏️ Solved Examples
Example 1 (Easy): 53² mann mein.
(50 + 3)² = 2500 + 300 + 9 = 2809.
Check: 53 × 53 = 2650 + 159 = 2809 ✅
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Example 2 (Medium): (3m + 2n)² kholo.
9m² + 12mn + 4n².
Check: m = n = 1: 5² = 25; 9 + 12 + 4 = 25 ✅
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Example 3 (NCERT style): a + b = 9, ab = 14. a² + b²?
(a + b)² = a² + b² + 2ab → 81 = a² + b² + 28 → 53.
Check: a = 7, b = 2: 49 + 4 = 53 ✅
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Anya ki rangoli: 1849 varg cm chahiye tha: 1609 + do pattiyaan (240). Agli baar usne pehle chitra banaya, phir rang khareeda. Rangoli poori, aur sabse sundar! 🪔✨
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🏠 (a + b)² Roz Ki Zindagi Mein
- 🏠 Kamre ka farsh: 12 ft square kamre mein 2 ft badhaao: 144 + 48 + 4 = 196.
- 🖼️ Photo frame: photo ke chaaron taraf border: bahar ka square − andar ka.
- 🌱 Kyaari: square kyaari ki side badhaao to mitti kitni zyada: 2ab + b².
- 🧮 Mental maths: 85², 95², 105² jaise squares turant.
- 📐 Tiles: 30 × 30 tiles ka farsh 32 × 32 karo: 4 × 30 + 4 = 124 tiles aur.
Tile wala hisaab mistri roz karte: “do pattiyaan + kona”. Wahi 2ab + b²!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| (gol + chhota)² | ✅ a² + 2ab + b² | Mann mein |
| 5 pe khatam | ✅ n(n + 1) | 25 | 85² = 7225 |
| Letters | ✅ Identity ya distributive | (4x)² dhyaan |
| a + b, ab diye | ✅ a² + b² = (a + b)² − 2ab | Bina a, b nikaale |
| (a + b)² vs a² + b² | ✅ Farak 2ab | Sign pe nirbhar |
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❌→✅ Worked Mistake
Sawaal: (3p + 7q)² kholo.
❌ Galat: 9p² + 49q². Beech wala 2ab gayab!
✅ Sahi: 9p² + 2 × 3p × 7q + 49q² = 9p² + 42pq + 49q². Check: p = q = 1: 10² = 100; 9 + 42 + 49 = 100 ✅ (galat wala 58 deta)
Doosri common galti: (4x)² = 4x². Galat: (4x)² = 16x².
🧠 Rule of thumb: Square = do brackets = chaar tukde. Beech ka 2ab kabhi mat chhodo.
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📝 10 MCQs
Q1. (a + b)² = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2ab wale do rectangle bhool gaye.
- (b) sahi: Chaar tukde: a², ab, ab, b² ✅
- (c) galat: Yeh to perimeter jaisa hai.
- (d) galat: ab do baar aata hai, ek baar nahi.
Q2. 61² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 3600 + 120 + 1 = 3721 ✅
- (b) galat: 2ab = 120 bhool gaye, isliye yeh galat.
- (c) galat: 2ab = 120 hai, 60 nahi.
- (d) galat: Jod mein galti ho gayi.
Q3. (2x + 1)² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: (2x)² + 2·2x·1 + 1 ✅
- (b) galat: (2x)² = 4x², aur beech ka term?
- (c) galat: Beech ka 4x bhool gaye.
- (d) galat: 2 × 2x × 1 = 4x hota hai.
Q4. 75² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 7 × 8 = 56 | 25 ✅
- (b) galat: 49 to 7² hai, 7 × 8 chahiye.
- (c) galat: 55 nahi, 7 × 8 = 56, isliye yeh galat.
- (d) galat: Aakhir mein hamesha 25 aata.
Q5. a + b = 7, ab = 10: a² + b²?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 49 − 20 = 29 ✅
- (b) galat: 49 to (a + b)² hai, isliye yeh galat.
- (c) galat: 2ab = 20 ghatana tha, 10 nahi.
- (d) galat: 20 to 2ab hai, isliye yeh galat.
Q6. (3 + (−5))² vs 3² + 5²?
Solution dekho
Sahi jawab: (a)
- (a) sahi: ab negative → (a + b)² chhota ✅
- (b) galat: 2ab = −30, barabar nahi.
- (c) galat: 4 to 34 se chhota hai.
- (d) galat: Seedha hisaab se pata chalta.
Q7. (a + b)³ mein a²b ka coefficient?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 1, 3, 3, 1 pattern ✅
- (b) galat: 2 to (a + b)² mein tha.
- (c) galat: 1 sirf a³ aur b³ ka, isliye yeh galat.
- (d) galat: 6 kahin nahi aata, isliye yeh galat.
Q8. (4x)² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 4² × x² = 16x² ✅
- (b) galat: 4 ka square bhool gaye.
- (c) galat: Square, double nahi, isliye yeh galat.
- (d) galat: x ka square bhi chahiye.
Q9. (10n + 5)² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 100n² + 100n + 25 ✅
- (b) galat: Beech ka 100n bhool gaye.
- (c) galat: (10n)² = 100n² hota, isliye yeh galat.
- (d) galat: n² wala term bhool gaye.
Q10. 12 ft square kamra 2 ft badha: kitna zyada?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2 × 24 + 4 = 52 ✅
- (b) galat: Sirf kona gina, pattiyaan nahi.
- (c) galat: Kona bhool gaye, isliye yeh galat.
- (d) galat: Hisaab galat ho gaya, isliye yeh galat.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. (a + b)² ko yaad kaise rakhein?
Chitra yaad rakho: bada square, chhota square, do rectangle. Formula khud ban jaata.
2. Kya negative b pe bhi?
Haan: b negative lo to 2ab negative, wahi (a − b)²: Post 5 — (a − b)² Aur (a + b)(a − b).
3. (a + b + c)²?
a² + b² + c² + 2ab + 2bc + 2ca (3 × 3 grid, 9 tukde).
4. Kab (a + b)² > a² + b²?
Jab ab > 0 (dono same sign). Ab < 0 ho to chhota.
5. 5 wali trick 3-digit pe?
Haan: 125² → 12 × 13 = 156 → 15625.
6. Aage kya?
(a − b)² aur (a + b)(a − b): Post 5 — (a − b)² Aur (a + b)(a − b).
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy (x + 4)² kholo.
Solution dekho
x² + 8x + 16. Check: x = 1 → 25 ✅
Q2. Easy 32²?
Solution dekho
900 + 120 + 4 = 1024. Check ✅
Q3. Easy 65 ke jaise 45²?
Solution dekho
4 × 5 = 20 → 2025. Check ✅
Q4. Easy (m + 6)² kholo.
Solution dekho
m² + 12m + 36. Check: m = 1 → 49 ✅
Q5. Medium (5y + 2)² kholo.
Solution dekho
25y² + 20y + 4. Check: y = 1 → 49 ✅
Q6. Medium 102²?
Solution dekho
10000 + 400 + 4 = 10404. Check ✅
Q7. Medium a + b = 11, ab = 24: a² + b²?
Solution dekho
121 − 48 = 73. Check: 8, 3 → 64 + 9 ✅
Q8. Medium (3a + 4b)² kholo.
Solution dekho
9a² + 24ab + 16b². Check: a = b = 1 → 49 ✅
Q9. NCERT style (x + 1/x)² agar x + 1/x = 5: x² + 1/x²?
Solution dekho
25 − 2 = 23. Check: 2ab = 2 × x × 1/x = 2 ✅
Q10. NCERT style (a + b + 1)² kholo.
Solution dekho
a² + b² + 1 + 2ab + 2a + 2b. Check: a = b = 1 → 9 ✅
Q11. NCERT style 995²?
Solution dekho
99 × 100 = 9900 → 990025. Check ✅
Q12. NCERT style (a + b)³ = ? a = 2, b = 1 pe check.
Solution dekho
a³ + 3a²b + 3ab² + b³ → 8 + 12 + 6 + 1 = 27 = 3³ ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. 2(a² + b²) = (a + b)² + (a − b)²: 7 aur 4 pe check, aur kyun?
Solution dekho
2 × 65 = 130; 121 + 9 = 130 ✅. 2ab aur −2ab kat jaate.
Q2. Kya do lagaataar numbers ke squares ka farak hamesha odd?
Solution dekho
(n + 1)² − n² = 2n + 1 → haan. Check: 25 − 16 = 9 ✅
Q3. Ek square ki side 10% badhaai: area kitna % badha?
Solution dekho
(1.1)² = 1.21 → 21%. Check: 10² → 11² = 121 ✅
Q4. (a + b)² hamesha ≥ 4ab? Kyun?
Solution dekho
(a + b)² − 4ab = (a − b)² ≥ 0 → haan. Check: 3, 5 → 64 ≥ 60 ✅
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🏆 Olympiad Corner
Q1. 1001²?
Solution dekho
1002001.
Q2. x + y = 12, xy = 32: x² + y²?
Solution dekho
144 − 64 = 80.
Q3. (a + b)² − (a − b)²?
Solution dekho
4ab.
Q4. 11² , 111² , 1111²: pattern?
Solution dekho
121, 12321, 1234321.
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💗 CharuMam Motivation
Anya ki galti bahut aam hai: bade-bade log bhi (a + b)² = a² + b² likh dete! Aapke paas ab chitra hai, to yeh galti kabhi nahi. 🌟
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🟢 Master Recap
- 💡 Chaabi: (a + b) side ka square kaato: ek a × a square, ek b × b square, aur do a × b rectangle. Isliye (a + b)² = a² + 2ab + b². “2ab” woh do pattiyaan hain jo log bhool jaate.
- (a + b)² = a² + 2ab + b².
- Chitra: do square + do rectangle.
- 2ab = like terms ab + ba.
- Mann mein: gol + chhota.
- 5 pe khatam: n(n + 1) | 25.
- (a + b)² − (a² + b²) = 2ab.
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👉 Aage Kya?
Ab ghataav wala square aur “jod × farak”: Post 5 — (a − b)² Aur (a + b)(a − b).
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬