Q1. If logₓ(81) = 4/3, find the value of x.
Solution:
We know that logₓ(81) = 4/3 means:
x^(4/3) = 81
Now, 81 = 3⁴. So:
x^(4/3) = 3⁴
Taking both sides to the power of 3/4:
x = (3⁴)^(3/4) = 3³ = 27
Final Answer: x = 27 ✅
Q2. Solve for x: 2^(x+1) - 3·2^x + 4 = 0
Solution:
Let y = 2^x. Then the equation becomes:
2y - 3y + 4 = 0 => -y + 4 = 0 => y = 4
Since y = 2^x:
2^x = 4 = 2² => x = 2
Final Answer: x = 2 ✅
Q3. If a + b + c = 0, prove that a³ + b³ + c³ = 3abc.
Solution:
We use the identity:
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
Given a + b + c = 0, the RHS becomes:
0 × (a² + b² + c² - ab - bc - ca) = 0
Thus:
a³ + b³ + c³ - 3abc = 0 => a³ + b³ + c³ = 3abc
Proved ✅
Q4. The sum of the first n terms of a GP is Sₙ = 3(2ⁿ - 1). Find its common ratio and first term.
Solution:
The formula for sum of GP: Sₙ = a(rⁿ - 1)/(r - 1)
Comparing with Sₙ = 3(2ⁿ - 1), we see:
a = 3, r = 2
Final Answer: First term a = 3, Common ratio r = 2 ✅
Q5. If α and β are the roots of 3x² - 5x + 2 = 0, form a quadratic equation whose roots are α² and β².
Solution:
From the given equation:
α + β = 5/3 αβ = 2/3
For α² and β²:
Sum = α² + β² = (α + β)² - 2αβ = (25/9) - (4/3) = 25/9 - 12/9 = 13/9 Product = α²β² = (αβ)² = (4/9)
Equation: x² - (13/9)x + (4/9) = 0
Multiply through by 9:
9x² - 13x + 4 = 0
Final Answer: 9x² - 13x + 4 = 0 ✅