JEE Advanced Maths: Vectors / Inequalities questions with solutions

Q1. Given that a² + b² + c² = 9*(x² + y² + z²), find the maximum value of (a*x + b*y + c*z) / (x² + y² + z²).

1. 3
2. 4
3. 5
4. 6

Answer: 3

By Cauchy-Schwarz, (ax+by+cz)² <= (a²+b²+c²)(x²+y²+z²) = 9(x²+y²+z²)². Dividing both sides by (x²+y²+z²)² gives [(ax+by+cz)/(x²+y²+z²)]² <= 9, so the maximum is 3.