NCERT Exemplar ke “Squares, Square Roots, Cubes and Cube Roots” unit ke topics pe poori practice, ek hi post mein: pehle chhota recap, phir 10 MCQs, fill in the blanks, true/false, short aur long answer, HOTS, aur aakhir mein answer key. Har sawaal ka hal Hinglish mein step by step hai. Yeh post aapke current chapter Book 1 Ch 1: A Square and a Cube ke saath padhiye: chapter se samjho, yahan se practice karo. 💪
🧠 Concept Recap: Pehle Yeh Yaad Kar Lo
📖 Pehle Book 1 Ch 1 padho: A Square and a Cube (hub)
📏 Pehle yeh shabd samjho:
- Natural number = ginti wale poore number 1, 2, 3, 4 …
- Odd / even = even = 2 se poora bant jaaye (2, 4, 6); odd = na bante (1, 3, 5)
- Factor = jo number poora-poora bhaag de: 12 ke factors 1, 2, 3, 4, 6, 12
- Prime = jiske sirf 2 factor hon, 1 aur khud: 2, 3, 5, 7, 11
- Prime factorisation = number ko sirf prime numbers ke guna mein todna (neeche 324 ka example)
- √ aur ∛ = √ = square root (√49 = 7 kyunki 7 × 7 = 49); ∛ = cube root (∛8 = 2 kyunki 2 × 2 × 2 = 8)
- > aur < = > = bada hai (5 > 3), < = chhota hai (3 < 5)
- Negative = zero se kam number, jaise −3 (thermometer mein zero se neeche)
- LCM = sabse chhota number jo dono se bant jaaye: 4 aur 6 ka LCM 12
- Perimeter aur m² = perimeter = chaaron taraf ki kul lambai (khet ki baad); m² = square metre, 1 m × 1 m ke tukde se naapa area
- Multiplier / divisor = jis number se guna karein / jis se bhaag dein
- Baayan / daayan = = ke left wala hissa / right wala hissa
- Pythagorean triplet = teen numbers jahan chhote do ke square ka jod = bade ka square: 3, 4, 5 (9 + 16 = 25)
- Decimal = point wala number, poore ke beech ka: 17.3 = 17 se thoda zyada
Prime factorisation ka worked example (324): 324 ÷ 2 = 162, 162 ÷ 2 = 81, 81 ÷ 3 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3. To 324 = 2 × 2 × 3 × 3 × 3 × 3 (sab prime).
1. Square (varg)
Kisi number ko usi number se guna karo to square milta hai: 7² = 7 × 7 = 49. Jo number kisi poore number (natural number) ka square ho, use perfect square kehte hain: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
- Aakhri digit ka niyam: perfect square ki aakhri (unit) digit sirf 0, 1, 4, 5, 6 ya 9 hoti hai. Jo number 2, 3, 7 ya 8 pe khatam ho, woh kabhi perfect square nahi hota.
- Zero ka niyam: perfect square ke end mein zero hamesha even (2, 4, 6…) ginti mein aate hain: 900 (do zero) theek, 9000 (teen zero) kabhi square nahi.
- Odd-even: odd number ka square odd, even number ka square even.
- Odd numbers ka jod: pehle 4 odd numbers 1 + 3 + 5 + 7 = 16 = 4 × 4. Yaani jitne odd numbers jode (yahan 4), us number ka square jawab hai. Isi ko chhota likhte hain: pehle n odd numbers ka jod = n².
- Beech ke numbers: n² aur (n + 1)² ke beech 2n aise numbers hote hain jo square nahi hain. Jaise 4² = 16 aur 5² = 25 ke beech 17 se 24 tak 8 numbers (2 × 4 = 8). Yaani: chhote wale ka root (4) double karo, utne non-square numbers beech mein. Chhota roop: n² aur (n + 1)² ke beech 2n.
- Pythagorean triplet: agar m > 1 ho to 2m, m² − 1, m² + 1 hamesha triplet banate hain (chhote do ke squares ka jod = bade ka square). m = 4 pe: 2 × 4 = 8, 4 × 4 − 1 = 15, 4 × 4 + 1 = 17. Check: 8 × 8 + 15 × 15 = 64 + 225 = 289 = 17 × 17. Yahan m bas ek chuna hua number hai (2, 3, 4 …).
2. Square root (vargmool)
Square ka ulta. √49 = 7, kyunki 7 × 7 = 49. Teen tareeke:
- Prime factorisation: number ke prime factors ki jodiyaan (pairs) banao, har jodi se ek factor lo. 324 = (2 × 2) × (3 × 3) × (3 × 3) → √324 = 2 × 3 × 3 = 18.
- Odd numbers ghatana: 1, 3, 5, … baar-baar ghatao; jitne step mein 0 aaye, utna root. 25 − 1 − 3 − 5 − 7 − 9 = 0, 5 step → √25 = 5.
- Andaaza (estimation): paas ke do squares dhoondo. √90: 81 = 9² aur 100 = 10², to √90 lagbhag 9 aur 10 ke beech, 9 ke zyada paas (9.5 se thoda kam).
3. Cube (ghan) aur cube root (ghanmool)
Cube = number ko teen baar guna: 4³ = 4 × 4 × 4 = 64. Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
| Number ki aakhri digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
|---|---|---|---|---|---|---|---|---|---|---|
| Cube ki aakhri digit | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 | 0 |
- Cube root prime factorisation se: factors ki teen-teen ki toli (triplet) banao, har toli se ek factor lo. 1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) → ∛1728 = 2 × 2 × 3 = 12.
- Chhota multiplier / divisor: jo factor jodi (square ke liye) ya toli (cube ke liye) mein poora na ho, usi se guna (ya bhaag) karo.
- Negative cube: negative number ka cube negative hota hai: (−3) × (−3) × (−3) = −27.
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📝 Section A: 10 MCQs
Har sawaal ke 4 options mein se ek sahi chuno. Pehle khud socho, phir “Hal dekho” kholo.
Q1. In mein se kaunsa number perfect square nahi ho sakta?
Hal (step by step) dekho
Sahi jawab: (c) 3127
Step 1: aakhri digit dekho. 1296 (6), 2025 (5), 1764 (4) teeno ki aakhri digit 0, 1, 4, 5, 6, 9 wali list mein hai, to yeh ho sakte hain (aur sach mein 36 × 36 = 1296, 45 × 45 = 2025, 42 × 42 = 1764).
Step 2: 3127 ki aakhri digit 7 hai. Koi bhi square 7 pe khatam nahi hota, isliye 3127 pakka perfect square nahi.
Q2. 87² ki aakhri (unit) digit kya hogi?
Hal (step by step) dekho
Sahi jawab: (d) 9
Square ki aakhri digit sirf number ki aakhri digit se tay hoti hai. 87 ki aakhri digit 7 → 7 × 7 = 49 → aakhri digit 9. (Poora: 87 × 87 = 7569.)
Q3. 15² aur 16² ke beech kitne aise numbers hain jo perfect square nahi hain?
Hal (step by step) dekho
Sahi jawab: (b) 30
Niyam: n² aur (n + 1)² ke beech 2n numbers. Yahan n = 15, to 2 × 15 = 30.
Check: 15² = 225, 16² = 256. Beech mein 226 se 255 tak: 255 − 226 + 1 = 30 (+1 kyunki 226 khud bhi ginti mein hai; jaise 3 se 5 tak 3, 4, 5 = 5 − 3 + 1 = 3 numbers) ✅
Q4. 1 + 3 + 5 + 7 + … + 19 ka jod kitna hai?
Hal (step by step) dekho
Sahi jawab: (a) 100
Step 1: kitne odd numbers hain? 1 se 19 tak odd numbers: (19 + 1) ÷ 2 = 10. (Kyun? 1 se 20 tak 20 numbers hain, unme aadhe odd hain: 20 ÷ 2 = 10. Isliye aakhri odd 19 mein 1 jod ke 2 se bhaag.)
Step 2: pehle n odd numbers ka jod = n² → 10² = 100.
Q5. 72 ko kis sabse chhote number se guna karein ki perfect square ban jaaye?
Hal (step by step) dekho
Sahi jawab: (b) 2
Step 1: 72 = 2 × 2 × 2 × 3 × 3.
Step 2: jodiyaan banao: (2 × 2) × (3 × 3) × 2. Ek 2 akela bacha.
Step 3: akele 2 ko jodi dene ke liye 2 se guna karo: 72 × 2 = 144 = 12 × 12 ✅ Jawab 2.
Q6. ∛2744 = ?
Hal (step by step) dekho
Sahi jawab: (c) 14
Step 1: 2744 = 2 × 2 × 2 × 7 × 7 × 7 (2744 ÷ 2 = 1372, ÷ 2 = 686, ÷ 2 = 343; teen baar 2 se yaani 8 se bhaag: 2744 ÷ 8 = 343. Phir 343 ÷ 7 = 49, 49 ÷ 7 = 7, to 343 = 7 × 7 × 7).
Step 2: do toliyaan: (2 × 2 × 2) aur (7 × 7 × 7). Har toli se ek: 2 × 7 = 14.
Check: 14 × 14 × 14 = 196 × 14 = 2744 ✅
Q7. 23 ke cube ki aakhri digit kya hogi?
Hal (step by step) dekho
Sahi jawab: (c) 7
Table yaad karo: aakhri digit 3 → cube ki aakhri digit 7 (3 × 3 × 3 = 27). To 23³ ki aakhri digit 7. (Poora: 23 × 23 × 23 = 12167.)
Q8. Kaunsa number perfect cube hai?
Hal (step by step) dekho
Sahi jawab: (d) 216
216 = 2 × 2 × 2 × 3 × 3 × 3 = (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6 ✅ To 216.
Baaki: 256 = aath 2 ka guna (8 factors, 3 ki toli mein 2 bachte hain); 196 = 2 × 2 × 7 × 7; 500 = 2 × 2 × 5 × 5 × 5 (2 ki toli adhoori). Teeno perfect cube nahi.
Q9. Kaunsa Pythagorean triplet hai jiska ek member 16 hai (2m wala tareeka)?
Hal (step by step) dekho
Sahi jawab: (a) 16, 63, 65
Step 1: 2m = 16 → m = 8.
Step 2: m² − 1 = 64 − 1 = 63, m² + 1 = 64 + 1 = 65.
Check: 16² + 63² = 256 + 3969 = 4225, aur 65² = 4225 ✅ Jawab 16, 63, 65. (16, 30, 34 bhi asli triplet hai, par woh 8, 15, 17 ko 2 se guna karke bana hai, 2m wale tareeke se nahi; 16, 20, 25 aur 16, 65, 67 triplet hi nahi.)
Q10. 1323 ko kis sabse chhote number se bhaag dein ki perfect cube mile?
Hal (step by step) dekho
Sahi jawab: (c) 49
Step 1: 1323 = 3 × 3 × 3 × 7 × 7 (1323 ÷ 3 = 441, ÷ 3 = 147, ÷ 3 = 49; teen baar 3 yaani 27 se bhaag: 1323 ÷ 27 = 49, aur 49 = 7 × 7).
Step 2: 3 ki toli poori hai, par 7 sirf do baar aaya (toli adhoori).
Step 3: dono 7 hataane ke liye 7 × 7 = 49 se bhaag do: 1323 ÷ 49 = 27 = 3 × 3 × 3 ✅ Jawab 49.
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✍️ Section B: Fill in the Blanks
Khaali jagah (_____) mein sahi shabd ya number bharo.
Q1. Koi perfect square kabhi digit _____ pe khatam nahi hota.
Jawab aur kaaran
Jawab: 2, 3, 7 ya 8
Squares ki aakhri digit sirf 0, 1, 4, 5, 6, 9 hoti hai (0 × 0, 1 × 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25 …). Isliye 2, 3, 7, 8 kabhi nahi aate.
Q2. Odd number ka square hamesha _____ hota hai.
Jawab aur kaaran
Jawab: odd
Odd × odd = odd. Jaise 7 × 7 = 49, 9 × 9 = 81.
Q3. √1600 = _____
Jawab aur kaaran
Jawab: 40
1600 = 16 × 100. √16 = 4 aur √100 = 10, to √1600 = 4 × 10 = 40. Check: 40 × 40 = 1600 ✅
Q4. Pehle 12 odd numbers ka jod = _____
Jawab aur kaaran
Jawab: 144
Pehle n odd numbers ka jod n² hota hai: 12 × 12 = 144.
Q5. ∛9261 = _____
Jawab aur kaaran
Jawab: 21
9261 = 3 × 3 × 3 × 7 × 7 × 7 (9261 ÷ 3 = 3087, ÷ 3 = 1029, ÷ 3 = 343; yaani 9261 ÷ 27 = 343 = 7 × 7 × 7). Toli se: 3 × 7 = 21. Check: 21 × 21 × 21 = 441 × 21 = 9261 ✅
Q6. (−4)³ = _____
Jawab aur kaaran
Jawab: −64
(−4) × (−4) = 16 (minus × minus = plus; pattern dekho: 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Har baar jawab 2 badhta hai, to agla (−1) × (−2) = +2. Isliye minus × minus = plus), phir 16 × (−4) = −64. Negative number ka cube negative.
Q7. Agar kisi number ke end mein 3 zero hon, to uske cube ke end mein _____ zero honge.
Jawab aur kaaran
Jawab: 9
Cube mein number teen baar guna hota hai, to zero bhi 3 + 3 + 3 = 9. Jaise 1000 × 1000 × 1000 = 1000000000 (9 zero).
Q8. 12² aur 13² ke beech _____ non-square numbers hain.
Jawab aur kaaran
Jawab: 24
2n niyam: 2 × 12 = 24. Check: 144 aur 169 ke beech 145 se 168 tak: 168 − 145 + 1 = 24 ✅
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✅❌ Section C: True / False
Batao statement sahi (True) hai ya galat (False). Galat ho to use sahi karke likho.
Q1. 5 pe khatam hone wale har number ka square 25 pe khatam hota hai.
Jawab aur kaaran
✅ True
5 × 5 = 25, aur bache hisse ka guna sirf aage ke digits badalta hai: 15 × 15 = 225, 35 × 35 = 1225, 105 × 105 = 11025. Hamesha …25.
Q2. 1 pe khatam hone wala har number perfect square hai.
Jawab aur kaaran
❌ False
Galat. 11, 21, 31 sab 1 pe khatam hain par square nahi. Aakhri digit ka niyam sirf yeh batata hai ki kaun square nahi ho sakta; yeh pakka nahi karta ki kaun square hai. Sahi statement: “Perfect square 1 pe khatam ho sakta hai.”
Q3. 8000 ek perfect cube hai.
Jawab aur kaaran
✅ True
8000 = 8 × 1000 = (2 × 2 × 2) × (10 × 10 × 10) = 20 × 20 × 20 ✅
Q4. Even number ka cube odd hota hai.
Jawab aur kaaran
❌ False
Galat. Even × even × even = even. Jaise 2 × 2 × 2 = 8, 4 × 4 × 4 = 64. Sahi: “Even number ka cube even hota hai.”
Q5. √(16 + 9) = √16 + √9
Jawab aur kaaran
❌ False
Galat. Baayan: √(16 + 9) = √25 = 5. Daayan: √16 + √9 = 4 + 3 = 7. 5 aur 7 barabar nahi. Root jod pe aise nahi baant-ta (par guna pe baant-ta hai: √(4 × 9) = √36 = 6, aur √4 × √9 = 2 × 3 = 6, dono barabar).
Q6. Jis number ki aakhri digit 6 hai, uske cube ki aakhri digit bhi 6 hogi.
Jawab aur kaaran
✅ True
6 × 6 × 6 = 216, aakhri digit 6. Isliye 16³ = 4096, 26³ = 17576, sab 6 pe khatam ✅
Q7. 9000 ek perfect square hai.
Jawab aur kaaran
❌ False
Galat. 9000 ke end mein 3 zero hain (odd ginti). Perfect square ke end mein zero hamesha jodiyon mein (even) aate hain. (900 = 30 × 30 square hai, 9000 nahi.)
Q8. Har perfect cube ka cube root uski aakhri digit se poori tarah pata chal jaata hai.
Jawab aur kaaran
❌ False
Galat. Aakhri digit se sirf root ki aakhri digit pata chalti hai. Jaise 2744 aur 85184 dono ki aakhri digit 4 hai (root 4 pe khatam), par roots alag hain: 14 aur 44 (14 × 14 × 14 = 2744, 44 × 44 × 44 = 85184). Baaki digit ke liye aage ke group (number ko right se 3-3 digit ke tukdon mein todna: 85184 → 85 | 184) se andaaza lagana padta hai.
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🟢 Section D: Short Answer Questions
2 se 3 marks wale sawaal. Har step likhne ki aadat daalo.
Q1. Prime factorisation se √5184 nikaalo.
Step-by-step hal
Step 1: 5184 ko baar-baar 2 se bhaag do: 5184 → 2592 → 1296 → 648 → 324 → 162 → 81. Chhe baar 2 se bhaag gaya.
Step 2: 81 = 3 × 3 × 3 × 3.
Step 3: 5184 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3). Paanch jodiyaan.
Step 4: har jodi se ek: 2 × 2 × 2 × 3 × 3 = 72.
Check: 72 × 72 = 5184 ✅ To √5184 = 72.
Q2. 1008 ko kis sabse chhote number se guna karein ki perfect square mile? Woh square aur uska square root bhi likho.
Step-by-step hal
Step 1: 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7.
Step 2: jodiyaan: (2 × 2) × (2 × 2) × (3 × 3) × 7. 7 akela hai.
Step 3: 7 se guna: 1008 × 7 = 7056.
Step 4: √7056 = 2 × 2 × 3 × 7 = 84. Check: 84 × 84 = 7056 ✅
Jawab: multiplier 7, square 7056, root 84.
Q3. Odd numbers ko baar-baar ghata kar dikhao ki 36 perfect square hai aur √36 kitna hai.
Step-by-step hal
36 − 1 = 35 → 35 − 3 = 32 → 32 − 5 = 27 → 27 − 7 = 20 → 20 − 9 = 11 → 11 − 11 = 0.
6 step mein 0 aaya, to 36 perfect square hai aur √36 = 6. (Kyunki 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6².)
Q4. Prime factorisation se ∛13824 nikaalo.
Step-by-step hal
Step 1: 13824 ko 2 se bhaag dete raho: 13824 → 6912 → 3456 → 1728 → 864 → 432 → 216 → 108 → 54 → 27. Nau baar 2.
Step 2: 27 = 3 × 3 × 3.
Step 3: toliyaan: (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3).
Step 4: har toli se ek: 2 × 2 × 2 × 3 = 24.
Check: 24 × 24 = 576, 576 × 24 = 13824 ✅ To ∛13824 = 24.
Q5. √300 kin do lagaataar (consecutive) poore numbers ke beech hai? Kiske zyada paas hai?
Step-by-step hal
Step 1: paas ke squares: 17 × 17 = 289 aur 18 × 18 = 324.
Step 2: 289 < 300 < 324, to 17 < √300 < 18.
Step 3: 300 − 289 = 11 aur 324 − 300 = 24. 11 chhota hai, to √300 17 ke zyada paas hai (asal mein lagbhag 17.3).
Q6. Kya 243 perfect cube hai? Agar nahi, to sabse chhota number batao jisse guna karne pe perfect cube bane.
Step-by-step hal
Step 1: 243 = 3 × 3 × 3 × 3 × 3 (paanch 3).
Step 2: ek toli (3 × 3 × 3) poori, phir 3 × 3 bache: toli ke liye ek aur 3 chahiye. To 243 perfect cube nahi.
Step 3: 3 se guna: 243 × 3 = 729 = 9 × 9 × 9 ✅
Jawab: nahi; multiplier 3 (cube 729, root 9).
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📘 Section E: Long Answer Questions
4 se 5 marks wale sawaal. Hal mein har step ka kaaran bhi likha hai.
Q1. Ek maali ke paas 1000 paudhe hain. Woh unhe aise lagaana chahta hai ki jitni lines (rows) hon, har line mein utne hi paudhe hon. (a) Sabse kam kitne paudhe aur laaye ki saare paudhe lag jaayein? (b) Agar woh naye paudhe na laaye, to kitne paudhe bach jaayenge?
Step-by-step hal
Samjho: rows = har row ke paudhe, to kul paudhe ek perfect square hone chahiye.
Step 1: 1000 ke paas ke squares: 31 × 31 = 961 aur 32 × 32 = 1024.
Step 2 (a): saare 1000 lagaane hain to agla square 1024 chahiye: 1024 − 1000 = 24 paudhe aur laane honge. Tab 32 rows, har row mein 32 paudhe.
Step 3 (b): naye na laaye to 961 tak hi square bana sakta hai: 1000 − 961 = 39 paudhe bachenge. Tab 31 rows, har row mein 31.
Q2. Woh sabse chhota perfect square number nikaalo jo 6, 9 aur 15 teeno se poora bhaag ho jaaye.
Step-by-step hal
Step 1: teeno se bhaag hone wala sabse chhota number = LCM. 6 = 2 × 3, 9 = 3 × 3, 15 = 3 × 5 → LCM = 2 × 3 × 3 × 5 = 90.
Step 2: 90 = 2 × (3 × 3) × 5. 3 ki jodi hai, par 2 aur 5 akele hain.
Step 3: unhe jodi dene ke liye 2 × 5 = 10 se guna: 90 × 10 = 900.
Step 4: 900 = 30 × 30 ✅ aur 900 ÷ 6 = 150, 900 ÷ 9 = 100, 900 ÷ 15 = 60, sab poore.
Jawab: 900.
Q3. Aakhri digit aur andaaze ke tareeke se cube root nikaalo (yeh dono perfect cubes hain): (a) 175616 (b) 42875
Step-by-step hal
Tareeka: number ko right se 3-3 digit ke group mein todo. (Ikai = number ki aakhri digit, jaise 56 mein 6; dahai = usse pehle wali digit, jaise 56 mein 5.) Right group ki aakhri digit se root ki ikai, aur left group se root ki dahai (andaaze se).
(a) 175616: groups 175 | 616. Aakhri digit 6 → root ki ikai 6 (6 ka cube 6 pe khatam). Left group 175: 5 × 5 × 5 = 125 aur 6 × 6 × 6 = 216, to 125 < 175 < 216 → chhota wala 5 lo. Root = 56. Check: 56 × 56 = 3136, 3136 × 56 = 175616 ✅
(b) 42875: groups 42 | 875. Aakhri digit 5 → ikai 5. Left group 42: 3 × 3 × 3 = 27 aur 4 × 4 × 4 = 64, 27 < 42 < 64 → 3. Root = 35. Check: 35 × 35 = 1225, 1225 × 35 = 42875 ✅
Dhyaan: yeh tareeka sirf tab, jab pata ho ki number perfect cube hai.
Q4. Ek square (varg) aakaar ke khet ka area 2025 m² hai. (a) Khet ki ek bhuja (side) kitni hai? (b) Chaaron taraf baad (fence) lagaane ka kharcha ₹25 prati metre ke hisaab se kitna hoga?
Step-by-step hal
Step 1: square ka area = side × side, to side = √2025.
Step 2: 2025 = 3 × 3 × 3 × 3 × 5 × 5 (2025 ÷ 25 = 81). Jodiyaan: (3 × 3) × (3 × 3) × (5 × 5) → √2025 = 3 × 3 × 5 = 45. Side = 45 m. Check: 45 × 45 = 2025 ✅
Step 3: baad ki lambai = perimeter = 4 × 45 = 180 m.
Step 4: kharcha = 180 × 25 = ₹4500.
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🧩 Section F: HOTS (Soch Wale Sawaal)
HOTS matlab Higher Order Thinking Skills: aise sawaal jinme formula se aage thoda dimaag lagana padta hai.
Q1. 1 ke baad woh sabse chhota number batao jo perfect square bhi ho aur perfect cube bhi.
Soch aur hal dekho
Soch: square ke liye har prime factor jodiyon (2 ke group) mein, cube ke liye toliyon (3 ke group) mein chahiye. Dono ek saath tab, jab har factor 6 ke group mein ho.
Sabse chhota prime 2 lo: 2 × 2 × 2 × 2 × 2 × 2 = 64.
Check: 64 = 8 × 8 ✅ aur 64 = 4 × 4 × 4 ✅ Jawab 64. (Agla aisa number 729 = 27 × 27 = 9 × 9 × 9.)
Q2. Bina lamba guna kiye batao: (a) 51² − 50² (b) 1 + 3 + 5 + … + 99
Soch aur hal dekho
(a) Pehle numbers se dekho: 4² − 3² = 16 − 9 = 7 = 4 + 3. Isi tarah lagaataar squares ka antar: (n + 1)² − n² = 2n + 1, yaani dono numbers ka jod. 51² − 50² = 51 + 50 = 101. Check: 2601 − 2500 = 101 ✅
(b) 1 se 99 tak odd numbers: (99 + 1) ÷ 2 = 50. Pehle 50 odd numbers ka jod = 50 × 50 = 2500.
Dekho: dono jawab ek hi baat se jude hain: 50² mein agla odd number 101 jodo to 51² milta hai!
Q3. Kitne 2-digit numbers ka square 3-digit ka hota hai?
Soch aur hal dekho
Step 1: sabse chhota 2-digit number 10: 10 × 10 = 100 (3-digit ✅).
Step 2: 31 × 31 = 961 (3-digit), par 32 × 32 = 1024 (4-digit). To 10 se 31 tak.
Step 3: ginti: 31 − 10 + 1 = 22 numbers.
Q4. Ek number ko double (2 guna) kar dein, to uska cube kitne guna ho jaata hai? Ek example se check karo.
Soch aur hal dekho
Soch: number n ho to cube n × n × n. Double number 2n ka cube = 2n × 2n × 2n = (2 × 2 × 2) × (n × n × n) = 8 × (n ka cube).
To cube 8 guna ho jaata hai.
Check: 3 × 3 × 3 = 27 aur 6 × 6 × 6 = 216 = 8 × 27 ✅
Bonus: square sirf 4 guna hota (2 × 2), kyunki square mein number do hi baar guna hota hai.
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🔑 Answer Key (Ek Nazar Mein)
Pehle saare sawaal khud karo, phir yahan milaao. Jahan galti ho, upar us sawaal ka step-by-step hal dobara padho.
| Section | Answers |
|---|---|
| A. MCQ | 1: (c) · 2: (d) · 3: (b) · 4: (a) · 5: (b) · 6: (c) · 7: (c) · 8: (d) · 9: (a) · 10: (c) |
| B. Fill in the Blanks | 1: 2, 3, 7 ya 8 · 2: odd · 3: 40 · 4: 144 · 5: 21 · 6: −64 · 7: 9 · 8: 24 |
| C. True / False | 1: T · 2: F · 3: T · 4: F · 5: F · 6: T · 7: F · 8: F |
| D. Short Answer | 1: 72 · 2: 7; 7056; 84 · 3: Haan, √36 = 6 · 4: 24 · 5: 17 aur 18 ke beech, 17 ke paas · 6: Nahi; 3 (729 = 9³) |
| E. Long Answer | 1: (a) 24 (b) 39 · 2: 900 · 3: (a) 56 (b) 35 · 4: (a) 45 m (b) ₹4500 |
| F. HOTS | 1: 64 · 2: (a) 101 (b) 2500 · 3: 22 · 4: 8 guna |
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👩🏫 CharuMam Ka Tip
Exam mein square / cube root ke sawaal mein hamesha pehle prime factorisation likho, aur jodiyon (square) ya toliyon (cube) pe gol ghera (circle) lagao. Akela bacha factor hi “chhota multiplier” hai. Aur jawab ke baad ek line ka check (72 × 72 = 5184) zaroor likho: galti pakdi jaati hai aur teacher ko bhi achha lagta hai! 🟦
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👉 Aage Kya?
Pehle chapter A Square and a Cube ke posts padh lo (link neeche), phir yeh practice dobara karo aur apna score milao. Agla Exemplar unit: Exponents and Powers (Power Play ke saath), jald aa raha hai!
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