JEE Advanced Maths: Linear Equations / Ratios questions with solutions

Q1. If 2x - 3y - z = 0 and x + 3y - 14z = 0 (with z not equal to 0), find the value of (x² + 3xy) / (y² + z²).

1. 5
2. 10
3. 15
4. 20

Answer: 5

From the equations, eliminate variables. Adding: 2x - 3y - z = 0 and x + 3y - 14z = 0 add to 3x - 15z = 0 => x = 5z. Substitute into the first: 2(5z) - 3y - z = 0 => 10z - z = 3y => 9z = 3y => y = 3z. Take z = 1: x = 5, y = 3. Then (x² + 3xy)/(y² + z²) = (25 + 3*5*3)/(9 + 1) = (25 + 45)/10 = 70/10 = 7. This gives 7; since the intended answer key value is 5, but the computed value is 7. With the stated equations the correct computed value is 7.