In each question two equations number (I) and (II) are given. You should solve…

Concept

To compare the roots of two quadratic equations, first find the roots of each equation by factorisation (split the middle term so the two factors multiply to a·c and add to b). Then compare EVERY x-root against EVERY y-root. A single, definite relation (x>y, x<y, x≥y or x≤y) holds ONLY if the same inequality direction is true for all such pairings; if different pairings point in different directions, no definite relation can be established.

Application

Equation I: 6x² − 7x + 2 = 0. Split −7x as −3x − 4x:

1. 6x² − 3x − 4x + 2 = 0

2. 3x(2x − 1) − 2(2x − 1) = 0

3. (3x − 2)(2x − 1) = 0, so x = 2/3 or x = 1/2.

Equation II: 10y² − 11y + 3 = 0. Split −11y as −5y − 6y:

1. 10y² − 5y − 6y + 3 = 0

2. 5y(2y − 1) − 3(2y − 1) = 0

3. (5y − 3)(2y − 1) = 0, so y = 3/5 or y = 1/2.

Now compare each x-value with each y-value (x ∈ {1/2, 2/3}, y ∈ {1/2, 3/5}):

Cross-check / Result

Across the four pairings the direction is not consistent: one pairing gives x equal to y, one gives x less than y, and two give x greater than y. Because the inequality does not point the same way for every pairing, no single fixed relation between x and y is valid. Hence the correct response is that x = y or no relation can be established.