✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- (k + 2)² − k² = ?
- a(b + c) = ?
- Step 1, 2, 3 = 8, 12, 16: agla?
Answers dekho
- 4k + 4
- ab + ac
- 20
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Distributive Property Kya Hai? se Post 6 — Algebra Ki Galti Pakdo dekh lijiye.
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🎬 Kahani: Chaar Dost, Chaar Jawab… Ya Ek?
Teacher ne board pe dots ka pattern banaaya aur poochha: “Step k mein kitne dots?” Chaar dost, chaar formule: 🤔
Aarav: 4(k + 2) − 4. Bela: 2(k + 2) + 2k. Chirag: 4(k + 1). Diya: (k + 2)² − k².
Sab ladne lage: “Mera sahi!” Teacher muskurayi: “Shayad sab sahi hain.” Kya yeh ho sakta hai? Aaj dekhte hain. 🛤️
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📖 Concept: Alag Raaste, Ek Manzil: Distributive Se Saboot
💡 Is post ki ek badi baat: Ek hi cheez ko alag tareekon se ginne pe expressions alag dikhte, par sahi hon to distributive se saral karne pe ek hi ban jaate. Saral karke milaana = saboot ki sab tareeke sahi. Na mile = kahin galti.
Step 1: Pattern
Step k: (k + 2) × (k + 2) square ke sirf kinaare ke dots.
Step 2: Chaar soch, chaar formula
- Aarav: chaar sides × (k + 2), par kone do baar gine → 4(k + 2) − 4.
- Bela: upar-neeche poori (k + 2), baayein-daayein beech ke k → 2(k + 2) + 2k.
- Chirag: har side se ek kona chhod ke (k + 1) dots ke 4 group → 4(k + 1).
- Diya: poora square − andar ka square → (k + 2)² − k².
Step 3: Saral karo: sab ek!
4k + 8 − 4 = 4k + 4. 2k + 4 + 2k = 4k + 4. 4k + 4 = 4k + 4. k² + 4k + 4 − k² = 4k + 4.
Check: Step 3: 16 ✅ (4 × 3 + 4). Step 10: 44.
WHY sab same? Dots wahi hain; bas ginne ka kram badla. Distributive ne dikhaya ki kram badalne se ginti nahi badalti.
Step 4: Area: do sadke ek khet mein
20 m × 15 m ka khet, beech mein ek lambi aur ek chaudi sadak, dono 2 m chaudi. Sadak ka area?
Tareeka A: 20 × 2 + 15 × 2 − 2 × 2 (chauraha do baar gina) = 66.
Tareeka B: poora − bache 4 tukde (jo milke 18 × 13): 300 − 234 = 66.
General: aw + bw − w² = ab − (a − w)(b − w) ✅ (distributive se kholo).
Step 5: Kaunsa tareeka kab?
Jo chitra mein sabse saaf dikhe, wahi. Kinaare jaise patterns mein Chirag (barabar group) sabse saaf. Andar khaali ho to Diya (bada − chhota). Overlap (chauraha) ho to “jodo phir double gina hatao”.
Do tareekon se karna = khud ka check! Dono same aaye to bharosa.
🪜 Atak Gaye? Aise Socho
- Chhote steps (1, 2, 3) gino aur table banao.
- Ek tareeka chuno: groups dhundho jo har step mein dohraate.
- Formula likho, Step 1–3 pe check.
- Doosra tareeka socho (bada − chhota, ya overlap hatao).
- Dono saral karo: same aaye to pakka.
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🔐 Deep Concept: Kyun Kai Tareeke Ek Hi Jawab Dete? (Ginti Ka Niyam)
Ek tokri ke aam ko kisi bhi kram mein gino, ginti wahi. Maths mein yeh “invariance” hai: jo cheez kram badalne se na badle. Har sahi tareeka usi ginti ka alag “raasta” hai.
Overlap ka niyam: Aarav ne kone do baar gine aur −4 kiya; khet mein chauraha −w². Yahi (a − b)² mein kone ka +b² (Post 5 — (a − b)² Aur (a + b)(a − b)) tha. Ek hi idea: jo do baar gina, use ek baar hatao.
Galti pakadna: agar kisi ka formula saral karke alag aaye, to woh galat hai (ya woh koi aur pattern gin raha). Isliye kai tareeke = muft ka check (Post 6 — Algebra Ki Galti Pakdo).
Unifying idea (poora chapter): distributive property alag dikhne wale expressions ko ek saath laati hai: “We distribute, yet things multiply.” 🎯
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✏️ Solved Examples
Example 1 (Easy): Step k mein (k + 1) × (k + 1) poora square + ek line mein k dots. Do formulas aur Step 5?
(k + 1)² + k = k² + 3k + 1; ya k(k + 3) + 1.
Check: k = 5: 36 + 5 = 41; 25 + 15 + 1 = 41 ✅
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Example 2 (Medium): 27 × 33 ya 28 × 32: bada kaun (bina pura guna)?
30² − 3² = 891 aur 30² − 2² = 896 → 28 × 32.
Check: 27 × 33 = 891 ✅, 28 × 32 = 896 ✅
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Example 3 (NCERT style): Square bageecha side s, chaaron taraf andar w chauda raasta. Raaste ka area do tareeke se.
A: s² − (s − 2w)². B: 4 pattiyaan w × (s − w): 4w(s − w). Dono = 4ws − 4w².
Check: s = 10, w = 1: 100 − 64 = 36; 4 × 9 = 36 ✅
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Chaaron dost: saral kiya to sabka 4k + 4! Teacher: “Maths mein ek hi sahi raasta nahi hota. Apna pasand ka raasta chuno, par doosron ke raaste bhi samjho.” 🤝
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🏠 Kai Tareeke Roz Ki Zindagi Mein
- 🗺️ Google Maps: ghar se school ke kai raaste, manzil ek.
- 🧮 Bill jodna: upar se neeche ya neeche se upar: kul same, aur yahi check.
- 🏗️ Mistri: L-shape farsh ko do tarah kaat ke area nikaalte, milaa ke check.
- 🍰 Cake baantna: pehle aadha phir chauthaai, ya seedha 4 tukde: har tukda same.
- 🎨 Rangoli ke dots: design ko lines mein gino ya group mein: ginti ek.
Bill ko do tarah jodna sabse aasaan galti-check hai, aur yeh is chapter ka hi sabak hai!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Kinaare ke patterns | ✅ Barabar groups | 4(k + 1) |
| Andar khaali | ✅ Bada − chhota | (k + 2)² − k² |
| Overlap/chauraha | ✅ Jodo − double | aw + bw − w² |
| Check chahiye | ✅ Doosra tareeka | Saral karke milao |
| Formula galat lage | ✅ Step 1–3 pe test | Value test |
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❌→✅ Worked Mistake
Sawaal: 20 × 15 khet, 2 m ki do sadke (cross). Sadak ka area?
❌ Galat: 20 × 2 + 15 × 2 = 70. Chauraha (2 × 2) do baar gina!
✅ Sahi: 70 − 4 = 66. Check: 300 − 18 × 13 = 300 − 234 = 66 ✅
Doosri common galti: sirf Step 1 pe formula check karna. 4k + 4 aur 8k dono Step 1 pe 8 dete! Step 2 pe 12 vs 16: farak pakdo.
🧠 Rule of thumb: Jo do baar gina, use ek baar hatao; formula ko kam se kam do steps pe check karo.
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📝 10 MCQs
Q1. 4(k + 2) − 4 = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 4k + 8 − 4 = 4k + 4 ✅
- (b) galat: −4 bhool gaye, kone dobara gine.
- (c) galat: 8 − 4 = +4, minus nahi.
- (d) galat: Constant hissa bhool gaye.
Q2. (k + 2)² − k² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: k² + 4k + 4 − k² ✅
- (b) galat: 4k wala tukda bhool gaye.
- (c) galat: k² to kat jaata hai, isliye yeh galat.
- (d) galat: 2ab = 4k hota, 2k nahi.
Q3. Step 10 (4k + 4)?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 4 × 10 + 4 = 44 ✅
- (b) galat: +4 bhool gaye, isliye yeh galat.
- (c) galat: Guna ki jagah jod kiya.
- (d) galat: 4 × 11 = 44, 48 nahi, isliye yeh galat.
Q4. Sadak area 20 × 15, w = 2?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 70 − 4 (chauraha) = 66 ✅
- (b) galat: Chauraha do baar gina.
- (c) galat: Chauraha ghataana tha, jodna nahi.
- (d) galat: Do baar ghata diya, isliye yeh galat.
Q5. 27 × 33 vs 28 × 32?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 900 − 4 > 900 − 9 ✅
- (b) galat: 891 < 896 hai, isliye yeh galat.
- (c) galat: 891 aur 896 alag hain.
- (d) galat: a² − b² se turant pata.
Q6. Do formula Step 1 pe same: pakka same?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 4k + 4 aur 8k: Step 1 pe same ✅
- (b) galat: Ek value kaafi nahi hoti.
- (c) galat: Hamesha nahi, test karo.
- (d) galat: Kabhi-kabhi sach bhi hote.
Q7. s² − (s − 2w)² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: s² − s² + 4ws − 4w² ✅
- (b) galat: 4ws wala tukda bhool gaye.
- (c) galat: Galat expansion hai yeh.
- (d) galat: Beech ka tukda bhool gaye.
Q8. Overlap ka niyam?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Chauraha, kone sab yahi ✅
- (b) galat: Pehle se jud chuka hai.
- (c) galat: Ek baar to ginna hai, isliye yeh galat.
- (d) galat: Double se aur galat, isliye yeh galat.
Q9. Kai tareeke kyun achhe?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Same aaye to bharosa ✅
- (b) galat: Check se time bachta hai.
- (c) galat: Kai sahi ho sakte hain.
- (d) galat: Saral karke sab saaf, isliye yeh galat.
Q10. (k + 1)² + k = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: k² + 2k + 1 + k ✅
- (b) galat: Aakhri k bhool gaye, isliye yeh galat.
- (c) galat: 2k bhool gaye, isliye yeh galat.
- (d) galat: 1 bhool gaye, isliye yeh galat.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya har pattern ke kai tareeke hote?
Aksar haan. Jitna chitra ko alag nazar se dekho, utne tareeke.
2. Kaunsa tareeka “best”?
Jo aapko sabse saaf dikhe aur jisme galti kam ho. Doosre check ke liye.
3. Agar do formule alag aayein?
Saral karo. Phir bhi alag to ek galat hai; Step 1–3 pe test karke pakdo.
4. Yeh Ch 1 ke patterns se kaise juda?
Wahan patterns dekhe; yahan algebra se saabit kiya ki alag soch ek hi jawab de.
5. Overlap har jagah?
Haan: kone, chauraha, (a − b)² ka b², Venn diagram. “Do baar gina, ek baar hatao.”
6. Aage kya?
Book 1 Ch 7 (proportional reasoning) aur aage algebra ke bade saboot.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Step k: 3k + 2 dots. Step 7?
Solution dekho
23. Check: k = 1 → 5 ✅
Q2. Easy 2(k + 3) + 2k saral karo.
Solution dekho
4k + 6. Check: k = 1 → 10 ✅
Q3. Easy (k + 3)² − k² saral karo.
Solution dekho
6k + 9. Check: k = 1 → 15 ✅
Q4. Easy 36 × 44 ya 38 × 42?
Solution dekho
40² − 16 vs 40² − 4 → 38 × 42. Check: 1584 < 1596 ✅
Q5. Medium Step k: (k + 3) × (k + 3) square ke kinaare. Formula?
Solution dekho
4(k + 2) = 4k + 8. Check: k = 1 → 16 − 4 = 12 ✅
Q6. Medium 30 × 20 khet, 3 m ki do cross sadke. Area?
Solution dekho
90 + 60 − 9 = 141. Check: 600 − 27 × 17 = 141 ✅
Q7. Medium Bageecha 12 m, andar 1.5 m raasta. Area?
Solution dekho
4 × 1.5 × 10.5 = 63. Check: 144 − 81 = 63 ✅
Q8. Medium k(k + 2) aur (k + 1)² − 1 same?
Solution dekho
Haan: dono k² + 2k. Check: k = 3 → 15 ✅
Q9. NCERT style Step k: k × (k + 1) rectangle ke dots. Teen tareeke se likho.
Solution dekho
k(k + 1) = k² + k = (k + 1)² − (k + 1). Check: k = 3 → 12 ✅
Q10. NCERT style Do formula 5k − 2 aur k² + k + 1: kis step pe same?
Solution dekho
k² − 4k + 3 = 0 → k = 1, 3. Check: k = 2: 8 vs 7 ✅
Q11. NCERT style Calendar 3 × 3 block ka beech a. Chaaron kone ka jod?
Solution dekho
(a − 8) + (a − 6) + (a + 6) + (a + 8) = 4a. Check: beech 16 → 8 + 10 + 22 + 24 = 64 ✅
Q12. NCERT style (1/2)(a + b) × (a + b) = (1/2)(a + b)²: kya sach?
Solution dekho
Haan: (a + b) × (a + b) = (a + b)². Check: a = 2, b = 4 → 18 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Triangle ke kinaare dots: side mein n dots. Kul? Do tareeke.
Solution dekho
3n − 3 (kone do baar) ya 3(n − 1). Check: n = 4 → 9 ✅
Q2. Cube ke kinaare (edges) pe n × n × n mein chhote cubes kitne?
Solution dekho
12(n − 2) + 8 = 12n − 16. Check: n = 3 → 20 ✅
Q3. 10 coins ka triangle ulta karne ke kam se kam kitne move?
Solution dekho
3 (teen kone ke coins). Check: 6 coins → 2 ✅
Q4. Square frame: bahar side a, andar side b. Area do tareeke?
Solution dekho
a² − b² = (a + b)(a − b). Check: 10, 6 → 64 = 16 × 4 ✅
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🏆 Olympiad Corner
Q1. Step k: 4k + 4. Step kab 100 dots?
Solution dekho
k = 24.
Q2. n × n grid ke andar (kinaare chhod ke) dots?
Solution dekho
(n − 2)².
Q3. 40 × 30 khet, 5 m ki cross sadke. Bacha khet?
Solution dekho
35 × 25 = 875.
Q4. 15 coins ka triangle ulta: kam se kam move?
Solution dekho
5.
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💗 CharuMam Motivation
Chaaron dost sahi the, aur sab ne ek doosre se naya dekhna seekha. Maths ek hi raaste ki race nahi; kai raaston ka mela hai! 🎉
Chapter 6 poora! Distributive se identities, fast guna, galti pakadna aur kai tareeke: sab ek hi property ki den.
— 📌 charumam.com
🟢 Master Recap
- 💡 Chaabi: Ek hi cheez ko alag tareekon se ginne pe expressions alag dikhte, par sahi hon to distributive se saral karne pe ek hi ban jaate. Saral karke milaana = saboot ki sab tareeke sahi. Na mile = kahin galti.
- Alag tareeke → alag expressions → saral karke same.
- Saral karke milaana = saboot/check.
- Overlap: do baar gina, ek baar hatao.
- Bada − chhota: andar khaali shapes.
- Formula ko kam se kam do steps pe test.
- Kai tareeke = muft ka check.
— charumam.com 🎯
👉 Aage Kya?
Agla safar: badlaav aur anupaat (proportional reasoning): Chapter 7 ka hub dekhiye. ⚖️
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬