Number Systems — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)

√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 irrational.

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

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

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

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. - The arc intersects the number line at point √5 ≈ 2.236. Conclusion: √5 lies between 2 and 3 on the number line, closer to 2.

Expression: (5 + 2√3)/(7 + 4√3) Step 1: Multiply by conjugate of denominator Conjugate of (7 + 4√3) is (7 - 4√3) = (5 + 2√3)/(7 + 4√3) × (7 - 4√3)/(7 - 4√3) Step 2: Calculate denominator using (a + b)(a - b) = a² - b² = (5 + 2√3)(7 - 4√3)/[(7)² - (4√3)²] = (5 + 2√3)(7 - 4√3)/(49 - 48) = (5 + 2√3)(7 - 4√3)/1 Step 3: Expand numerator = 5(7) + 5(-4√3) + 2√3(7) + 2√3(-4√3) = 35 - 20√3 + 14√3 - 8(3) = 35 - 20√3 + 14√3 - 24 = 11 - 6√3 Answer: (5 + 2√3)/(7 + 4√3) = 11 - 6√3

Let x = 0.142857142857... = 0.1̄4̄2̄8̄5̄7̄ (6 digits repeat) Step 1: The block 142857 repeats every 6 decimal places Multiply x by 10⁶ = 1,000,000 1,000,000x = 142857.142857... Step 2: Subtract the original equation from this 1,000,000x - x = 142857.142857... - 0.142857... 999,999x = 142857 x = 142857/999,999 Step 3: Simplify the fraction Note: 999,999 = 7 × 142,857 Verify: 7 × 142,857 = 999,999 ✓ Therefore: x = 142,857/(7 × 142,857) = 1/7 Conclusion: 0.142857̄ = 1/7

Given: x = 3 + 2√2 Step 1: Find 1/x 1/x = 1/(3 + 2√2) Rationalize by multiplying with (3 - 2√2)/(3 - 2√2) = (3 - 2√2)/[(3 + 2√2)(3 - 2√2)] = (3 - 2√2)/(9 - 8) = 3 - 2√2 Step 2: Calculate x + 1/x x + 1/x = (3 + 2√2) + (3 - 2√2) = 3 + 2√2 + 3 - 2√2 = 6 Answer: x + 1/x = 6