Direct variation with square of variable – UGC NET Question

The relation “x varies directly as the square of y” can be written as x = k·y² for some constant k. Using x = 50 when y = 5 gives 50 = k·25, so k = 50/25 = 2. For y = 7, we have x = 2·7² = 2·49 = 98. Thus, the new value of x corresponding to y = 7 is 98.

Option A:
Option A, 70, might result from incorrectly assuming x varies directly as y (not y²), which would give x proportional to 5 → 50 and 7 → 70. However, this ignores the square dependence stated in the problem, so 70 cannot satisfy the given variation law.

Option B:
Option B, 84, could arise from partially accounting for the square by mixing linear and quadratic reasoning, but substituting y = 7 into x = 2y² clearly yields 98, not 84. Therefore, 84 fails the direct substitution test in the correct formula.

Option C:
Option C is correct because it uses the exact variation relationship x = 2y² determined from the first pair (x, y) = (50, 5). Substituting y = 7 gives 98, which is consistent with the nature of direct variation with the square and the original data point.

Option D:
Option D, 112, would correspond to x = 2·8² if y were 8, or to another mistaken scaling, but does not arise from y = 7 in the function x = 2y². Hence, 112 is incompatible with the established proportional rule.