In aptitude (simplification: quadratic comparison), two quadratic equations in real variables x and y are given. First solve each equation to find all possible values of x and y, and then compare these values to decide the correct relationship between x and y. I. 2x^2 - 13x - 189 = 0 II. 2y^2 - 3y - 189 = 0

Introduction / Context:
This question tests comparison of roots of two quadratic equations in aptitude level simplification. Instead of directly guessing the relationship between x and y, we must first find all possible roots of each quadratic, then carefully compare every possible pair of x and y values. Only after that analysis can we decide whether x is always greater than y, always less, always equal, or if no single consistent relationship exists.

Given Data / Assumptions:
We have two equations in real variables x and y.
I. 2x^2 - 13x - 189 = 0
II. 2y^2 - 3y - 189 = 0
Assume x and y are real numbers and any root of the equations is a valid value.

Concept / Approach:
For a quadratic ax^2 + bx + c = 0, roots are obtained by factorisation or by the quadratic formula. Once we compute the two roots of the x equation and the two roots of the y equation, we form all root pairs (x, y) and compare them. If we can find even one pair where x > y and another pair where x < y, then no unique relationship can be fixed and the correct conclusion is that the relationship cannot be determined in a strict sense.

Step-by-Step Solution:
For equation I: 2x^2 - 13x - 189 = 0. Factor 2x^2 - 13x - 189 as (2x + 21)(x - 9) = 0. Hence x = -21/2 or x = 9. For equation II: 2y^2 - 3y - 189 = 0. Factor 2y^2 - 3y - 189 as (2y + 27)(y - 7) = 0. Hence y = -27/2 or y = 7. Now compare pairs: for example x = -21/2 and y = -27/2 gives x > y, but x = -21/2 and y = 7 gives x < y. Because different choices of roots lead to different inequalities, no single global relation between x and y can be fixed.

Verification / Alternative check:
We can quickly approximate the roots as decimals to cross check. For x: -21/2 is -10.5 and 9 is 9. For y: -27/2 is -13.5 and 7 is 7. By pairing 9 with -13.5, x is greater. By pairing -10.5 with 7, x is smaller. This confirms that both x > y and x < y situations are possible, so there is no unique comparison that holds for all combinations of x and y that satisfy the equations.

Why Other Options Are Wrong:
Options x > y and x < y each claim a strict direction for all pairs, which is not true because we found counterexamples. The option x = y would require every root pair to be equal, which is clearly false because the sets of roots are different. The option x >= y also fails because we have at least one pair where x < y. Therefore none of these directional statements holds for every valid combination of x and y.

Common Pitfalls:
A common mistake in such comparison questions is to compare only one root pair, for example taking the larger root of x with the smaller root of y and concluding x > y. Another pitfall is to ignore the fact that quadratics generally have two roots, not one, so conclusions must be based on all combinations. Students sometimes also approximate too roughly and miscompare numbers, so it is safer to work with exact fractions or clearly written decimals.

Final Answer:
The only safe conclusion is that no single consistent inequality describes all possible pairs. Hence the correct option is Relationship cannot be determined.