Number Systems — Previous Year Questions (Class 9 Mathematics)
Number Systems form the foundation of mathematics, covering natural, whole, integer, rational, and irrational numbers. Master operations on different numbe
TL;DR: Number Systems form the foundation of mathematics, covering natural, whole, integer, rational, and irrational numbers. Master operations on different…
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Number Systems form the foundation of mathematics, covering natural, whole, integer, rational, and irrational numbers. Master operations on different numbe
Number Systems — Previous Year Questions with Solutions
Q (2023, 2 marks): Find the value of (5 + sqrt(3))(5 - sqrt(3)).
Answer: Using the identity (a + b)(a - b) = a^2 - b^2:
(5 + sqrt(3))(5 - sqrt(3)) = 5^2 - (sqrt(3))^2
= 25 - 3
= 22
Q (2022, 3 marks): Rationalize the denominator: 1 / (sqrt(5) - sqrt(3))
Answer: Multiply numerator and denominator by (sqrt(5) + sqrt(3)):
1 / (sqrt(5) - sqrt(3)) × (sqrt(5) + sqrt(3)) / (sqrt(5) + sqrt(3))
= (sqrt(5) + sqrt(3)) / ((sqrt(5))^2 - (sqrt(3))^2)
= (sqrt(5) + sqrt(3)) / (5 - 3)
= (sqrt(5) + sqrt(3)) / 2
Q (2023, 2 marks): Express 0.3̅ (where 3 is repeating) as a fraction.
Answer: Let x = 0.3̅ = 0.333...
Multiply both sides by 10:
10x = 3.333...
Subtract: 10x - x = 3.333... - 0.333...
9x = 3
x = 3/9 = 1/3
Q (2022, 3 marks): Simplify: (sqrt(45) - sqrt(20)) / sqrt(5)
Answer: sqrt(45) = sqrt(9 × 5) = 3sqrt(5)
sqrt(20) = sqrt(4 × 5) = 2sqrt(5)
(3sqrt(5) - 2sqrt(5)) / sqrt(5)
= sqrt(5) / sqrt(5)
= 1
Q (2021, 5 marks): Find 'a' and 'b' if (7 + sqrt(5)) / (7 - sqrt(5)) = a + b*sqrt(5)
Answer: Rationalize by multiplying numerator and denominator by (7 + sqrt(5)):
= (7 + sqrt(5))^2 / ((7)^2 - (sqrt(5))^2)
= (49 + 14sqrt(5) + 5) / (49 - 5)
= (54 + 14sqrt(5)) / 44
= 27/22 + 7sqrt(5)/22
Comparing with a + b*sqrt(5):
a = 27/22, b = 7/22
Q (2023, 3 marks): Locate sqrt(2), sqrt(3), and sqrt(5) on the number line using the method of successive magnification.
Answer: sqrt(1) = 1 and sqrt(4) = 2, so sqrt(2) lies between 1 and 2 (approximately 1.414).
For sqrt(3): lies between sqrt(1)=1 and sqrt(4)=2, closer to 2 (approximately 1.732).
For sqrt(5): lies between sqrt(4)=2 and sqrt(9)=3, closer to 2 (approximately 2.236).
On number line: 1 < sqrt(2) ≈ 1.414 < sqrt(3) ≈ 1.732 < 2 < sqrt(5) ≈ 2.236 < 3
Frequently Asked Questions
What is the difference between rational and irrational numbers?
Rational numbers can be expressed as p/q where p and q are integers and q ≠ 0 (e.g., 1/2, 0.5). Irrational numbers cannot be expressed as fractions; they have non-terminating, non-repeating decimals (e.g., π, sqrt(2)).
How do you identify if a decimal is terminating or non-terminating?
If the denominator (in lowest form) has only factors of 2 and 5, the decimal is terminating. If it has any other prime factors, it's non-terminating. For example, 1/4 = 0.25 (terminating), 1/3 = 0.3̅ (non-terminating).
More Class 9 Mathematics PYQs
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