Class 10 Mathematics - Chapter 2: Numbers and Sequence - English Medium

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Class 10 Mathematics - Important Questions

Chapter 2: Numbers and Sequences (English Medium Solutions)

Overview

This document contains the step-by-step solutions for important Class 10 Chapter 2 (Numbers and Sequences) mathematics problems as explained in the YouTube lesson by Kalaimagal Academy.

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Problem 1: Division & Word Problem (Euclid's Division Lemma)

Question: A man has 532 flower pots. He wants to arrange them in rows such that each row contains 21 flower pots. Find the number of completed rows and how many flower pots are left over.

Solution Steps:

• Total number of flower pots = 532
• Number of flower pots in each row = 21
• To find the completed rows and remaining pots, divide 532 by 21 using Euclid's Division Lemma:
  a = bq + r, where 0 ≤ r < b
• Divide 532 by 21:
  • 532 ÷ 21 = 25 with a remainder of 7
  • 532 = 21 × 25 + 7

Result:
• Number of completed rows = 25
• Remaining flower pots left over = 7

Problem 2: Prime Factorization (Example 2.10)

Question: If ab × ba = 800, find the values of a and b.

Solution Steps:

• Express 800 in terms of prime factors:
  800 = 2 × 400
800 = 2 × 2 × 200
800 = 2 × 2 × 2 × 100
800 = 2 × 2 × 2 × 2 × 50
800 = 2 × 2 × 2 × 2 × 2 × 25
800 = 25 × 52
• Equating ab × ba = 25 × 52:
  • Case 1: a = 2 and b = 5
  • Case 2: a = 5 and b = 2

Result: a = 2, b = 5 (or a = 5, b = 2)

Problem 3: Finding Exponents using Prime Factorization (Exercise 2.2, Q4)

Question: If 13824 = 2a × 3b, find the values of a and b.

Solution Steps:

• Perform prime factorization on 13824:
  • Divide continuously by 2: 13824 ÷ 29 = 27
  • Divide remainder by 3: 27 ÷ 33 = 1
• Therefore, 13824 = 29 × 33
• Comparing powers with 2a × 3b:
  • a = 9
  • b = 3

Result: a = 9, b = 3

Problem 4: Modular Arithmetic - Congruence (Exercise 2.3, Q3)

Question: Find x such that 104 ≡ x (mod 19).

Solution Steps:

• Calculate 104 = 10,000
• We need to find the remainder x when 10,000 is divided by 19.
• Divide 10,000 by 19:
  • 10000 = (19 × 526) + 6
• Remainder obtained = 6

Result: x = 6

Problem 5: Clock Arithmetic (Exercise 2.3, Q5)

Question: What is the time 100 hours after 7:00 AM?

Solution Steps:

• 1 day = 24 hours.
• Divide 100 hours by 24 to find full days and remaining hours:
  • 100 ÷ 24 = 4 days with a remainder of 4 hours (since 24 × 4 = 96).
• After 4 full days (96 hours), the time will be 7:00 AM again.
• Add the remaining 4 hours to 7:00 AM:
  • 7:00 AM + 4 hours = 11:00 AM

Result: The time will be 11:00 AM

Problem 6: Calendar / Day Calculation

Question: Today is Tuesday. My uncle will come after 45 days. On which day will my uncle come?

Solution Steps:

• 1 week = 7 days.
• Divide 45 by 7 to find the number of completed weeks and leftover days:
  • 45 ÷ 7 = 6 weeks with a remainder of 3 days (since 7 × 6 = 42).
• After 6 complete weeks (42 days), it will be Tuesday again.
• Count 3 days forward from Tuesday:
  • Day 1: Wednesday
  • Day 2: Thursday
  • Day 3: Friday

Result: My uncle will come on Friday

Problem 7: Piecewise Defined Sequence (Example 2.21)

Question: Find the 11th and 18th terms of the sequence whose n^th term is defined by:
an = n(n + 3) if n is odd
an = n2 + 1 if n is even

Solution Steps:

• To find a₁₁ (since 11 is Odd):
  • Use formula: an = n(n + 3)
  • a11 = 11 × (11 + 3) = 11 × 14 = 154
• To find a₁₈ (since 18 is Even):
  • Use formula: an = n2 + 1
  • a18 = 182 + 1 = 324 + 1 = 325

Result: a₁₁ = 154, a₁₈ = 325

Problem 8: Sequence Evaluation

Question: Find a₈ and a₁₅ for the sequence where:
an = (n2 - 1) / (n + 3) if n is even
an = n2 / (2n + 1) if n is odd

Solution Steps:

• To find a₈ (since 8 is Even):
  • Use formula: an = (n2 - 1) / (n + 3)
  • a8 = (82 - 1) / (8 + 3) = (64 - 1) / 11 = 63 / 11
• To find a₁₅ (since 15 is Odd):
  • Use formula: an = n2 / (2n + 1)
  • a15 = 152 / (2(15) + 1) = 225 / (30 + 1) = 225 / 31

Result: a₈ = 63/11, a₁₅ = 225/31

Extracted & Structured from Video: Class 10 Chapter 2 Important Questions - Kalaimagal Academy

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