Square Root Ka Andaaza | Class 8 Maths Book 1 Ch 1

📘 Class 8 · Ganita Prakash Book 1Ch 1 · A Square and a CubePost 5 / 8⏱ 25–30 min

Square Root Ka Andaaza 🎯

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap Like & Share

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

40² = ?
√2025 = ?
Kaunsa bada: 15² ya 230?

Answers dekho

1600
45
15² = 225, to 230 bada

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 4 — Square Root Kya Hai? aur Post 2 — Square Numbers Ki Properties ka aakhri digit table dekh lijiye.

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🎬 Kahani: Aribam Aur Bijou Ka Root Game

Aribam aur Bijou ek game khel rahe the: ek number bolta, doosra uska square root. “25!” “5!” “81!” “9!” Phir Aribam bola: “250!” 😈

Bijou atak gaya: 250 perfect square nahi! Aribam: “Theek hai, kam se kam paas ka number to batao.” Aur doosri taraf Akhil ke paas 125 sq cm ka square kapda hai: kya 15 cm ka rumaal katega? Dono ko andaaza chahiye.

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📖 Concept: Do Squares Ke Beech Ghero, Phir Pass Karo

💡 Is post ki ek badi baat: Root ka andaaza = number ko do nazdeeki perfect squares ke beech gherna: a² < y < b² to a < √y < b. Phir aakhri digit se options kam karo aur beech ka square (jaise 45²) nikaal ke interval aadha karo.

Step 1: Pehla ghera: das ke squares

√1936? 40² = 1600 < 1936 < 2500 = 50² → 40 < √1936 < 50.

WHY? Bada number = bada root. Square badhta to root bhi badhta, isliye ghera sahi.

Step 2: Aakhri digit se do options

1936 ka aakhri digit 6 → root 4 ya 6 pe khatam (4² = 16, 6² = 36). Options: 44 ya 46.

Table: square 1 → root 1/9; 4 → 2/8; 9 → 3/7; 6 → 4/6; 5 → 5; 0 → 0.

Step 3: Beech ka square: interval aadha

45² = (40 + 5)² = 1600 + 400 + 25 = 2025 > 1936 → root 40 aur 45 ke beech → 44.

Check: 44² = 1936 ✅ (Shortcut: 5 pe khatam ka square a(a + 1)|25, Post 2.)

Step 4: Jo perfect square nahi: √250

10² = 100, 20² = 400 → 10 < √250 < 20. Paas aao: 15² = 225, 16² = 256 → 15 < √250 < 16.

250 se 256 sirf 6 dur, 225 se 25 dur → √250 ≈ 16 (thoda kam). Asli ≈ 15.8.

Step 5: Rumaal ka sawaal (integer side)

Kapda 125 sq cm. 15 cm ka rumaal = 225 sq cm > 125 → nahi katega.

Sabse bada integer side: 11² = 121 ≤ 125 < 144 = 12² → 11 cm. Yahan “paas wala” nahi, neeche wala chahiye, kyunki kapda kam nahi pad sakta.

🪜 Atak Gaye? Aise Socho

Pehle 10, 20, 30… ke squares se ghero.
Phir ek-ek se paas: 15², 16²…
Perfect square hai? Aakhri digit se 2 options.
Beech ka 5 wala square nikaal ke ek option hatao.
Sawaal “sabse bada fit” poochhe to neeche wala integer lo.

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🔐 Deep Concept: √2 Ka Andaaza Aur Decimal Tak Pass Jaana

1² = 1 < 2 < 4 = 2² → 1 < √2 < 2. Ab decimal: 1.4² = 1.96, 1.5² = 2.25 → 1.4 < √2 < 1.5. Phir 1.41² = 1.9881, 1.42² = 2.0164 → √2 ≈ 1.41.

Gehri baat: har kadam pe ghera 10 guna chhota. Yeh “ghero aur pass karo” soch computers bhi use karte hain. √2 kabhi khatam nahi hota (na integer, na fraction), par andaaza jitna chahe utna pass.

Kab kaunsa andaaza? “Lagbhag kitna” poochha → nazdeeki integer. “Kitna fit hoga” → neeche wala. “Kam se kam kitna chahiye” → upar wala.

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✏️ Solved Examples

Example 1 (Easy): √250 sabse paas ka integer.

15² = 225, 16² = 256 → 16 (6 dur vs 25 dur).

Check: 16² = 256 ≈ 250 ✅

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Example 2 (Medium): √5329 (perfect square hai).

70² = 4900, 80² = 6400 → 70–80. Aakhri 9 → 73 ya 77. 75² = 5625 > 5329 → 73.

Check: 73² = 4900 + 420 + 9 = 5329 ✅

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Example 3 (NCERT style): NCERT style: 500 sq m zameen, sabse bada integer side ka square bagicha?

22² = 484, 23² = 529 → 22 m.

Check: 484 ≤ 500 < 529 ✅

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Bijou ka jawab: “√250 lagbhag 16, par 16 se thoda kam!” Aur Akhil ka rumaal: 11 cm. 🎯

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🏠 Andaaza Kahan Kaam Aata

🧵 Darzi: kapde se sabse bada square tukda.
🏡 Plot: 300 sq yd ka square plot ≈ 17.3 yd side.
📺 Screen: area se side ka andaaza.
🧮 Exam check: √1000 ≈ 31.6; jawab 100 aaya to galat.
🌳 Bagicha: 200 paudhe square mein: 14 × 14 = 196, 4 bachenge.

Ghero, pass karo, aur dekho sawaal kaunsa integer maang raha.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Perfect square (4 digit)	✅ Ghera + aakhri digit + 5 wala square	Seedha jawab
Perfect square nahi, “lagbhag”	✅ Nazdeeki integer	Kam doori wala
“Sabse bada fit”	✅ Neeche wala integer	Upar wala fit nahi hoga
Decimal tak chahiye	✅ 0.1 ke kadam	Ghera chhota karte jao
Aakhri digit 2, 3, 7, 8	❌ Integer root nahi	Sirf andaaza

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❌→✅ Worked Mistake

Sawaal: 150 sq cm kapde se sabse bada integer side ka square?

❌ Galat: “√150 ≈ 12.2, paas ka 12… nahi 12² = 144 chalega. Par 13 bhi paas hai, 13 lo.”

✅ Sahi: 13² = 169 > 150, kapda kam padega → 12 cm. Check: 144 ≤ 150 ✅

🧠 Rule of thumb: Fit karna ho to hamesha neeche wala square; lagbhag ho to paas wala.

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📝 10 MCQs

Q1. √90 kin integers ke beech?

(a) 9 aur 10

(b) 8 aur 9

(c) 10 aur 11

(d) 45 aur 46

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 81 < 90 < 100 ✅
(b) galat: 64 aur 81 dono 90 se chhote.
(c) galat: 100 aur 121 dono 90 se bade.
(d) galat: Aadha karna root nahi hota.

Q2. √1936 = ?

(a) 44

(b) 46

(c) 34

(d) 54

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 44² = 1936 ✅
(b) galat: 46² = 2116, zyada hai.
(c) galat: 34² = 1156, kam hai, isliye yeh galat.
(d) galat: 54² = 2916, bahut zyada.

Q3. Square 9 pe khatam, root ka aakhri digit?

(a) 3 ya 7

(b) 9

(c) 1 ya 9

(d) Sirf 3

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 3² = 9, 7² = 49 ✅
(b) galat: 9² = 81, 1 pe khatam, isliye yeh galat.
(c) galat: 1 ya 9 to 1 wale square ke.
(d) galat: 7 bhi option hai, sirf 3 nahi.

Q4. √250 ka paas wala integer?

(a) 16

(b) 15

(c) 25

(d) 125

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 256 sirf 6 dur ✅
(b) galat: 225 to 25 dur hai, isliye yeh galat.
(c) galat: 25² = 625, bahut zyada.
(d) galat: Aadha karna root nahi hota.

Q5. 125 sq cm se sabse bada integer square rumaal?

(a) 11 cm

(b) 12 cm

(c) 15 cm

(d) 10 cm

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 121 ≤ 125 ✅
(b) galat: 144 > 125, fit nahi hoga.
(c) galat: 225 to bahut bada hai.
(d) galat: 10 fit hai, par 11 bada fit.

Q6. 45² = ?

(a) 2025

(b) 2005

(c) 1625

(d) 2125

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 4 × 5 | 25 = 2025 ✅
(b) galat: 5 wale square 25 pe khatam.
(c) galat: 16 to 4 × 4 se aata, galat.
(d) galat: 21 ka koi aadhar nahi.

Q7. √3000 lagbhag?

(a) 55

(b) 30

(c) 300

(d) 17

Solution dekho

Sahi jawab: (a)

(a) sahi: 55² = 3025, sabse paas ✅
(b) galat: 30² = 900, bahut kam, isliye yeh galat.
(c) galat: 300² = 90000, bahut zyada.
(d) galat: 17² = 289, bahut kam, isliye yeh galat.

Q8. √2 kin decimals ke beech?

(a) 1.4 aur 1.5

(b) 1.5 aur 1.6

(c) 1 aur 1.2

(d) 2 aur 3

Solution dekho

Sahi jawab: (a)

(a) sahi: 1.96 < 2 < 2.25 ✅
(b) galat: 1.5² = 2.25 pehle se zyada.
(c) galat: 1.2² = 1.44, abhi door.
(d) galat: 2² = 4, bahut zyada, isliye yeh galat.

Q9. √7056 (perfect square)?

(a) 84

(b) 86

(c) 76

(d) 94

Solution dekho

Sahi jawab: (a)

(a) sahi: 80–90, 6 → 84/86, 85² = 7225 → 84 ✅
(b) galat: 86 > 85, aur 85² pehle se zyada.
(c) galat: 76² = 5776, kam hai, isliye yeh galat.
(d) galat: 94² = 8836, zyada hai.

Q10. 200 paudhe square mein, kitne bachenge?

(a) 4

(b) 0

(c) 25

(d) 14

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 14² = 196, 4 bache ✅
(b) galat: 200 perfect square nahi.
(c) galat: 225 ke liye 25 kam padte.
(d) galat: 14 to qatar ki ginti hai.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Andaaza aur asli root mein kitna farak?

Integer andaaza 1 ke andar; decimal kadam se aur pass.

2. Aakhri digit 5 ho to?

Root 5 pe hi khatam.

3. Paas wala kaise tay karein?

Dono squares se doori ginti: kam doori wala.

4. √ fraction ka andaaza?

√(1/2) ≈ 0.7, kyunki 0.7² = 0.49.

5. Calculator kyun nahi?

Andaaza calculator ka jawab check karne ke liye bhi zaroori.

6. Aage kya?

Ab cube: Post 6 — Cube Numbers Aur Taxicab 1729.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy √50 kin integers ke beech?

Solution dekho

7 aur 8. Check: 49, 64 ✅

Q2. Easy √30 ka paas wala integer?

Solution dekho

5 (25 se 5, 36 se 6). Check ✅

Q3. Easy Square 4 pe khatam: root ke options?

Solution dekho

2 ya 8. Check ✅

Q4. Easy 65² = ?

Solution dekho

4225. Check: 6 × 7 | 25 ✅

Q5. Medium √2116 = ?

Solution dekho

40–50, 6 → 44/46, 45² = 2025 < 2116 → 46. Check ✅

Q6. Medium √3249 = ?

Solution dekho

50–60, 9 → 53/57, 55² = 3025 → 57. Check: 57² = 3249 ✅

Q7. Medium √600 lagbhag?

Solution dekho

24² = 576, 25² = 625 → 24 (24 dur vs 25 dur). Check ✅

Q8. Medium 300 sq m mein sabse bada integer square?

Solution dekho

17 m (289). Check: 18² = 324 ✅

Q9. NCERT style Kya 2032 square? Andaaza bhi.

Solution dekho

45² = 2025, 46² = 2116 → nahi, √2032 ≈ 45. Check ✅

Q10. NCERT style √8281 = ?

Solution dekho

90–100, 1 → 91/99, 95² = 9025 → 91. Check: 91² = 8281 ✅

Q11. NCERT style √10 ek decimal tak.

Solution dekho

3.1² = 9.61, 3.2² = 10.24 → 3.2 paas (0.24 vs 0.39). Check ✅

Q12. NCERT style 1000 bachche square mein: kam se kam kitne aur?

Solution dekho

32² = 1024 → 24. Check ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Kaunsa bada: √50 + √50 ya √100?

Solution dekho

2√50 ≈ 14.1 > 10. Check ✅

Q2. √99 aur √101: kitna farak lagbhag?

Solution dekho

9.95 aur 10.05 → ≈ 0.1. Check ✅

Q3. 5-digit perfect square jo 1 pe khatam aur 100 ke paas root?

Solution dekho

10201 = 101². Check ✅

Q4. Square farsh 150 sq ft, side ek decimal tak?

Solution dekho

12.2² = 148.84, 12.3² = 151.29 → ≈ 12.2. Check ✅

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🏆 Olympiad Corner

Q1. √(10⁶ + 1) lagbhag?

Solution dekho

1000 (thoda zyada).

Q2. Kitne integers ka √ 5 aur 6 ke beech?

Solution dekho

26 se 35: 10.

Q3. √12321?

Solution dekho

111.

Q4. √(0.5) ek decimal?

Solution dekho

0.7.

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💗 CharuMam Motivation

Bijou ne “nahi pata” ki jagah “lagbhag 16” kaha. Andaaza lagana bhi maths hai, aur zindagi mein sabse zyada kaam aata! 🌟

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🟢 Master Recap

💡 Chaabi: Root ka andaaza = number ko do nazdeeki perfect squares ke beech gherna: a² < y < b² to a < √y < b. Phir aakhri digit se options kam karo aur beech ka square (jaise 45²) nikaal ke interval aadha karo.
a² < y < b² → a < √y < b.
Aakhri digit se 2 options.
Beech ka 5 wala square se interval aadha.
Lagbhag → paas wala integer.
Fit → neeche wala.
Decimal tak: 0.1 ke kadam.

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👉 Aage Kya?

Square se cube ki taraf: Post 6 — Cube Numbers Aur Taxicab 1729.

← Pichla: Post 4 — Square Root Kya Hai?📚 Chapter 1 ke saare posts Agla: Post 6 — Cube Numbers Aur Taxicab 1729 →

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✅ Pehle Yeh Aata Hai?

🎬 Kahani: Aribam Aur Bijou Ka Root Game

📖 Concept: Do Squares Ke Beech Ghero, Phir Pass Karo

Step 1: Pehla ghera: das ke squares

Step 2: Aakhri digit se do options

Step 3: Beech ka square: interval aadha

Step 4: Jo perfect square nahi: √250

Step 5: Rumaal ka sawaal (integer side)

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: √2 Ka Andaaza Aur Decimal Tak Pass Jaana

✏️ Solved Examples

🏠 Andaaza Kahan Kaam Aata

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

💛 Post Achha Laga? Like, Share, Subscribe, Follow!

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