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Class 9 Mathematics: Number Systems

Irrational numbers proof, representation on number line, real numbers decimal expansions (terminating, non-terminating repeating), rationalizing denominators, and laws of exponents.

Class 9 Mathematics 5 Generated Practice Exercises NCERT Exercise Solutions

⚡ Key Formulas & Core Relations for Number Systems

a^p × a^q = a^(p+q), (a^p)^q = a^(pq)
Conjugate rationalization: 1/(√a + √b) = (√a - √b)/(a - b)
p/q terminates if prime factors of q are ONLY 2^m × 5^n

NCERT Textbook Exercise Solutions

1 Solutions
Exercise 1.5 - Q5

Textbook Exercise Problem

Rationalize the denominator of: 1 / (√7 - √6).

Step-by-Step Verified Solution:

  1. Step 1: Identify conjugate of denominator (√7 - √6), which is (√7 + √6).
  2. Step 2: Multiply numerator and denominator by (√7 + √6):
  3. Step 3: [1 × (√7 + √6)] / [(√7 - √6)(√7 + √6)].
  4. Step 4: Apply (a - b)(a + b) = a² - b²: Denominator = (√7)² - (√6)² = 7 - 6 = 1.
  5. Step 5: Result = (√7 + √6) / 1 = √7 + √6.
Final Answer: √7 + √6
Key Concept / Exam Strategy: Conjugate multiplication eliminates radical square roots from the denominator.

5 AI-Generated Chapter Exercises (Foundation to HOTS)

5 Problems with Full Steps
Generated Exercise #1 Foundation
Question: Express 0.2353535... (0.235 bar over 35) in the form p/q where p and q are integers and q ≠ 0.
Core Concept Tested: Non-terminating repeating decimal conversion to rational fraction
Mentor Hint: Multiply by 10 then by 1000, then subtract.

Step-by-Step Solution:

  1. Step 1: Let x = 0.2353535...
  2. Step 2: 10x = 2.353535... (Equation 1)
  3. Step 3: 1000x = 235.3535... (Equation 2)
  4. Step 4: Subtract Eq 1 from Eq 2: 1000x - 10x = 235.3535... - 2.3535...
  5. Step 5: 990x = 233 => x = 233 / 990.
Verified Answer: 233 / 990
Generated Exercise #2 Intermediate
Question: Simplify: (16)^(3/4) × (81)^(3/4).
Core Concept Tested: Fractional rational exponents and prime factor decomposition
Mentor Hint: Write 16 as 2⁴ and 81 as 3⁴.

Step-by-Step Solution:

  1. Step 1: 16^(3/4) = (2⁴)^(3/4) = 2^(4 × 3/4) = 2³ = 8.
  2. Step 2: 81^(3/4) = (3⁴)^(3/4) = 3^(4 × 3/4) = 3³ = 27.
  3. Step 3: Product = 8 × 27 = 216.
Verified Answer: 216
Generated Exercise #3 Intermediate
Question: Find values of a and b if: (3 + √7) / (3 - √7) = a + b√7.
Core Concept Tested: Comparing coefficients of rationalized radical expressions
Mentor Hint: Rationalize the LHS by multiplying by (3 + √7).

Step-by-Step Solution:

  1. Step 1: LHS = (3 + √7)² / (3² - 7) = (9 + 7 + 6√7) / (9 - 7).
  2. Step 2: (16 + 6√7) / 2 = 8 + 3√7.
  3. Step 3: Compare with a + b√7: a = 8, b = 3.
Verified Answer: a = 8, b = 3
Generated Exercise #4 JEE-NEET Foundation
Question: If x = 3 + 2√2, find the value of (x + 1/x) and (x² + 1/x²).
Core Concept Tested: Conjugate reciprocal pairs in surds
Mentor Hint: 1/x = 3 - 2√2 since 3² - (2√2)² = 9 - 8 = 1.

Step-by-Step Solution:

  1. Step 1: 1/x = 1 / (3 + 2√2) = (3 - 2√2) / (9 - 8) = 3 - 2√2.
  2. Step 2: x + 1/x = (3 + 2√2) + (3 - 2√2) = 6.
  3. Step 3: x² + 1/x² = (x + 1/x)² - 2 = 6² - 2 = 36 - 2 = 34.
Verified Answer: x + 1/x = 6, x² + 1/x² = 34
Generated Exercise #5 Advanced HOTS
Question: Prove whether the square root of 5 is irrational using contradiction.
Core Concept Tested: Fundamental proof of irrationality by contradiction
Mentor Hint: Assume √5 = a/b where a, b are co-prime integers.

Step-by-Step Solution:

  1. Step 1: Assume √5 = a/b where a, b are co-prime integers (b ≠ 0).
  2. Step 2: Squaring gives 5 = a²/b² => a² = 5b². Thus 5 divides a², so 5 divides a.
  3. Step 3: Let a = 5c. Then (5c)² = 5b² => 25c² = 5b² => b² = 5c². Thus 5 divides b.
  4. Step 4: This implies 5 is a common factor of both a and b, contradicting that a and b are co-prime.
  5. Step 5: Hence, √5 must be an irrational number.
Verified Answer: Proved: √5 is irrational.