UGC NET Questions (Paper – 1)

Let the two numbers be 9k and 16k. The mean proportional m between them satisfies 9k:m = m:16k. This gives m² = 9k × 16k = 144k², so m = 12k. When we cancel the common factor k, the mean proportional in terms of the ratio alone is 12. Thus, 12 is the required mean proportional for the ratio 9:16.

Option A:
Option A, 6, would result in m² = 36k², which is much smaller than 9k × 16k = 144k². The corresponding proportion 9k:6k is 3:2, but 6k:16k is 3:8, and these are not equal. Therefore 6 does not satisfy the condition for a mean proportional.

Option B:
Option B, 10, gives m² = 100k² and leads to ratios 9k:10k = 9:10 and 10k:16k = 5:8. Since 9:10 ≠ 5:8, the definition of mean proportional is not fulfilled. Thus 10 cannot be the correct answer.

Option C:
Option C, 24, is actually twice the correct value and produces m² = 576k², which corresponds to a much larger product than 9k × 16k. The resulting proportions 9k:24k and 24k:16k are not equal, so 24 fails the basic check of the definition.

Option D:
Option D is correct because m = 12k exactly satisfies m² = 9k × 16k and preserves the equality of ratios 9k:12k and 12k:16k, both simplifying to 3:4. Removing the common factor k leaves the pure numerical mean proportional as 12.