In this aptitude (simplification: quadratic comparison) question, two quadratic equations in x and y are given. Solve both equations, list all possible values of x and y, and then compare them to decide the correct overall relationship between x and y. I. 3x^2 - 14x + 16 = 0 II. 5y^2 - 16y + 12 = 0

Introduction / Context:
Here we again compare the roots of two quadratic equations, but in this case a consistent inequality does exist. These types of questions appear in quantitative aptitude tests to assess algebraic manipulation, root comparison, and careful reasoning about all possible values of variables, not just one solution pair.

Given Data / Assumptions:
Equation I: 3x^2 - 14x + 16 = 0.
Equation II: 5y^2 - 16y + 12 = 0.
All real roots of each equation are considered valid values of x and y respectively.

Concept / Approach:
We factorise each quadratic to find explicit roots. Then we compare every x root with every y root. If the smallest x value is still greater than or equal to the largest y value, then x >= y. If instead there are mixed comparisons, the relationship would not be determined. Thus the key is to compute roots accurately and inspect their order on the number line.

Step-by-Step Solution:
Consider 3x^2 - 14x + 16 = 0. Factorise: 3x^2 - 14x + 16 = (3x - 8)(x - 2) = 0. Therefore x = 8/3 or x = 2. Now consider 5y^2 - 16y + 12 = 0. Factorise: 5y^2 - 16y + 12 = (5y - 6)(y - 2) = 0. Therefore y = 6/5 or y = 2. Now order the values: 6/5 is 1.2, 2 is 2, and 8/3 is approximately 2.67. Thus y takes values 1.2 and 2, while x takes values 2 and about 2.67. So each x value is greater than or equal to each y value.

Verification / Alternative check:
We can verify by forming all pairs. For x = 2 and y = 2, we get x = y. For x = 2 and y = 6/5, we have 2 > 1.2, so x > y. For x = 8/3 and y = 2, x > y again. For x = 8/3 and y = 6/5, x is still greater. In no valid pair is x less than y. Hence x is always greater than or equal to y, which confirms the relationship x >= y.

Why Other Options Are Wrong:
Option x > y ignores the equality case where both variables take value 2. Option x < y is never true because x is never smaller than y. Option x = y fails because there exist pairs with strict inequality. Relationship cannot be determined is not correct because there is a clear consistent inequality once all root pairs are examined carefully.

Common Pitfalls:
Learners sometimes forget to convert fractions to decimals for easy comparison or they misfactor the quadratic expressions, leading to wrong roots. Another common mistake is to compare only one root pair rather than all possible combinations. To avoid errors, always factor step by step and place all roots in increasing order on the number line before deciding the inequality relation.

Final Answer:
All valid x values are greater than or equal to all valid y values. Therefore the correct choice is x >= y.