Number Systems — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)
Free step-by-step Karnataka (SSLC) Class 9 Mathematics solutions for "Number Systems" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Karnataka (SSLC) Class 9 Mathematics solutions for "Number Systems" — important questions with detailed answers, download PDF for bo…
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Q1: Classify √2 as rational or irrational. Justify your answer.
√2 is irrational.
Justification (Proof by contradiction):
Suppose √2 is rational. Then √2 = p/q where p and q are coprime integers (gcd(p,q) = 1).
Squaring both sides: 2 = p²/q²
Therefore: p² = 2q²
This means p² is even, so p must be even. Let p = 2m.
Substituting: (2m)² = 2q²
4m² = 2q²
2m² = q²
This means q² is even, so q is even.
But if both p and q are even, they have a common factor of 2, contradicting our assumption that gcd(p,q) = 1.
Therefore, √2 cannot be rational.
Conclusion: √2 is irr…
Q2: Express 2.3̄ (2.333...) as a fraction in lowest terms.
Let x = 2.3̄ = 2.333...
Step 1: Write x = 2 + 0.333...
Let y = 0.333...
Step 2: Multiply y by 10
10y = 3.333...
Step 3: Subtract to eliminate the repeating part
10y - y = 3.333... - 0.333...
9y = 3
y = 3/9 = 1/3
Step 4: Find x
x = 2 + 1/3 = 6/3 + 1/3 = 7/3
Answer: 2.3̄ = 7/3
Q3: Rationalize the denominator: 1/(√5 - √3)
Expression: 1/(√5 - √3)
Step 1: Identify the conjugate
Conjugate of (√5 - √3) is (√5 + √3)
Step 2: Multiply numerator and denominator by the conjugate
= 1/(√5 - √3) × (√5 + √3)/(√5 + √3)
= (√5 + √3)/[(√5 - √3)(√5 + √3)]
Step 3: Apply difference of squares formula (a - b)(a + b) = a² - b²
= (√5 + √3)/[(√5)² - (√3)²]
= (√5 + √3)/(5 - 3)
= (√5 + √3)/2
Answer: (√5 + √3)/2
Q4: If a = 1/(2 - √3), find the value of a² - 2a.
Step 1: Rationalize a
a = 1/(2 - √3) × (2 + √3)/(2 + √3)
= (2 + √3)/[(2 - √3)(2 + √3)]
= (2 + √3)/(4 - 3)
= 2 + √3
Step 2: Calculate a²
a² = (2 + √3)²
= 4 + 4√3 + 3
= 7 + 4√3
Step 3: Calculate 2a
2a = 2(2 + √3) = 4 + 2√3
Step 4: Find a² - 2a
a² - 2a = (7 + 4√3) - (4 + 2√3)
= 7 + 4√3 - 4 - 2√3
= 3 + 2√3
Answer: a² - 2a = 3 + 2√3
Q5: Locate √5 on the number line using the Pythagorean theorem.
Step 1: Express √5 using Pythagorean theorem
√5 = √(1² + 2²)
This is the hypotenuse of a right triangle with legs 1 and 2.
Step 2: Construction on number line
- Mark point 0 (origin) and point 2 on the number line.
- At point 2, draw a perpendicular line of length 1 unit.
- Using compass, measure the hypotenuse from 0 to the endpoint of the perpendicular.
- This distance equals √5.
Step 3: Locate √5
- With center at 0 and radius equal to the hypotenuse length, draw an arc on the number line.
-…
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