Class X Math all chapters Question Solved NCERT

Chapter 1: REAL NUMBERS Class X Math(বাস্তৱ সংখ্যা)

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🔹 Important Topics Covered

• Euclid’s Division Lemma
• HCF by Euclid’s Algorithm
• Fundamental Theorem of Arithmetic
• Decimal Expansion of Rational Numbers
• Proof-based Questions (Very Important)

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🟢 Q1. State Euclid’s Division Lemma.

✅ Answer:

Euclid’s Division Lemma:
For any two positive integers a and b, there exist unique integers q and r such that$$a = bq + r,\quad 0 \le r < b$$a=bq+r,0≤r<b

ইউক্লিডৰ ভাগ উপপাদ্য:
যিকোনো দুটা ধনাত্মক পূৰ্ণসংখ্যা a আৰু b ৰ বাবে এককভাবে নিৰ্দিষ্ট q আৰু r আছে যেনে$$a = bq + r,\; 0 \le r < b$$a=bq+r, 0≤r<b

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🟢 Q2. Use Euclid’s Division Lemma to find the HCF of 135 and 225.

✅ Solution:

225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0

👉 HCF = 45

উত্তৰ (অসমীয়া):
শেষ ভাগফল 0 হোৱাৰ আগৰ ভাগক = 45
অতএব, HCF = 45

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🟢 Q3. Find the HCF of 306 and 657 by Euclid’s Algorithm.

✅ Solution:

657 = 306 × 2 + 45
306 = 45 × 6 + 36
45 = 36 × 1 + 9
36 = 9 × 4 + 0

👉 HCF = 9

অসমীয়া ব্যাখ্যা:
ইউক্লিড পদ্ধতি ব্যৱহাৰ কৰি শেষ ভাগক = 9

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🟢 Q4. State the Fundamental Theorem of Arithmetic.

✅ Answer:

Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a product of primes, and this factorisation is unique.

মৌলিক গণিত উপপাদ্য:
প্ৰত্যেক সংযোজিত সংখ্যা এককভাবে মৌলিক সংখ্যাৰ গুণফল হিচাপে প্ৰকাশ কৰিব পাৰি।

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🟢 Q5. Find the prime factorisation of 360.

✅ Solution:

360 = 2 × 2 × 2 × 3 × 3 × 5$$= 2^3 \times 3^2 \times 5$$=23×32×5

অসমীয়া:
360 ৰ মৌলিক গুণনীয়ক =$$2^3 × 3^2 × 5$$23×32×5

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🟢 Q6. Check whether 6ⁿ can end with digit 0 for any natural number n.

✅ Solution:

6ⁿ = (2 × 3)ⁿ
To end with 0, number must have factor 10 = 2 × 5
But no factor 5 is means present.

👉 Therefore, 6ⁿ cannot end with digit 0.

অসমীয়া:
0 ত শেষ হ’বলৈ 5 গুণনীয়ক থাকিব লাগিব।
6ⁿ ত 5 নাই।
অতএব, অসম্ভৱ।

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🟢 Q7. Find whether 13/125 has a terminating decimal expansion.

✅ Solution:

125 = 5³
Denominator has only prime factor 5.

👉 Decimal expansion is terminating.

অসমীয়া:
হৰফত কেৱল 5 থাকিলে দশমিক সমাপ্ত হয়।
অতএব, সমাপ্ত দশমিক সংখ্যা।

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🟢 Q8. Write the decimal expansion of 17/8.

✅ Solution:

$$\frac{17}{8} = 2.125$$817​=2.125

👉 Terminating decimal.

অসমীয়া:$$17 ÷ 8 = 2.125$$17÷8=2.125

এটা সমাপ্ত দশমিক।

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🟢 VERY IMPORTANT QUESTIONS (HSLC)

✔ Euclid’s Division Algorithm (Numerical)
✔ Proof: 6ⁿ cannot end with 0
✔ Decimal expansion questions
✔ HCF problems (3–5 marks)

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📌 Chapter 1 – Quick Revision Formula

• Euclid’s Lemma: a = bq + r
• Terminating Decimal: Denominator = 2ᵐ × 5ⁿ
• Non-terminating: Other prime factors present