UPSC CSE Prelims 2020

Correct Option (3)

Statement S1 indicates that each of the 51 pages has 0, 1, or 2 figures. Statement S2 indicates that each of the 51 pages has at least 1 figure. When both statements are considered together, it implies that each of the 51 pages must contain either 1 or 2 figures.

Let 'x' represent the number of pages that have 2 figures. Consequently, (51 - x) pages will have 1 figure.

The total number of figures in the book can be calculated as: (x × 2) + ((51 - x) × 1) = 2x + 51 - x = 51 + x.

Given that 'x' is the number of pages with 2 figures, its value can range from 0 (if all pages have 1 figure) to 51 (if all pages have 2 figures). Thus, 0 ≤ x ≤ 51.

The question asks: "Are there more than 100 figures in that book?" This translates to the inequality: 51 + x > 100.

Solving for x, we get: x > 49.

Since 'x' can be any integer between 0 and 51, we cannot definitively answer whether x > 49. For instance, if x = 40, the total figures would be 51 + 40 = 91, which is not greater than 100. If x = 50, the total figures would be 51 + 50 = 101, which is greater than 100.

Therefore, even with both statements S1 and S2 combined, it is not possible to determine conclusively if there are more than 100 figures in the book. Hence, S1 and S2 together are not sufficient to answer the question.

Incorrect Options:

Option (1) Both S1 and S2 are sufficient to answer the question, but neither S1 alone nor S2 alone is sufficient to answer the question.

• This option is incorrect because, as demonstrated above, even when S1 and S2 are considered together, they are not sufficient to provide a definitive answer to the question.

Option (2) S1 alone is sufficient to answer the question.

• Statement S1 specifies that there are not more than two figures on any page (i.e., 0, 1, or 2 figures per page). For a 51-page book, the total number of figures could range from 0 (if all pages have 0 figures) to 102 (if all pages have 2 figures).
• Since the total number of figures can be less than or equal to 100 (e.g., 51 figures if each page has 1 figure) or greater than 100 (e.g., 102 figures if each page has 2 figures), S1 alone is insufficient to determine if the total is more than 100.

Option (4) S2 alone is sufficient to answer the question.

• Statement S2 specifies that there is at least one figure on every page (i.e., ≥ 1 figure per page). For a 51-page book, the minimum total number of figures would be 51 × 1 = 51.
• However, S2 alone does not provide an upper limit on the number of figures per page. A page could theoretically have 1, 2, 3, or more figures. If each page has 1 figure, the total is 51 (not > 100). If each page has 3 figures, the total is 153 (is > 100).
• Since the total number of figures cannot be uniquely determined to be greater than 100, S2 alone is insufficient.