JEE Advanced Maths: Logarithms and Equations questions with solutions

Q1. Let x1 and x2 be the values of x satisfying the equation [log₂(x - 2)]² + log₂(x - 2) * log₂(3/x) - 2 * [log₂(3/x)]² = 0. Which of the following is/are correct?

1. x1 + x2 = 9
2. x1 * x2 = 18
3. |x1 - x2| = 3
4. x1² + x2² = 45

Answer: x1 + x2 = 9

Setting a = log₂(x-2) and b = log₂(3/x), the equation factors as (a + 2b)(a - b) = 0. Solving each case with domain restrictions x > 2 and x < 3 gives two valid solutions whose sum is 9.