JEE Advanced Maths: Algebra / Equations and Inequalities questions with solutions

Q1. Consider the equation 1/(x+p) + 1/(x+q) + 1/(x+r) = 3/x, where p, q, r are distinct positive real numbers. Find the number of positive real roots of this equation.

1. 1
2. 2
3. 3
4. 4

Answer: 2

After cross-multiplying and simplifying, the equation reduces to a cubic in x with one negative root guaranteed for each vertical asymptote (x = -p, -q, -r). For p,q,r > 0, by analyzing the function on (0, +inf) and applying the intermediate value theorem, there are exactly 2 positive real roots.