Directions for the following 5 (five) items: Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly.
Question: X receives three coins of different denominations: 1, 2, 5, 10 and 20. If the total amount received by X is m, does X receive a coin of denomination 5?
- Statement I: m is not a prime number.
- Statement II: The sum of the digits of m is greater than 5.
Answer & explanation
Answer: (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
There are only ten ways to pick three different coins, so list their totals. Every total whose digits add up to more than 5 includes the 5-coin, so Statement II settles it; Statement I leaves 32 (2 + 10 + 20) as an exception.
- Three different coins from {1, 2, 5, 10, 20} can be chosen in 10 ways.
- With the 5-coin: 1+2+5 = 8, 1+5+10 = 16, 1+5+20 = 26, 2+5+10 = 17, 2+5+20 = 27, 5+10+20 = 35.
- Without the 5-coin: 1+2+10 = 13, 1+2+20 = 23, 1+10+20 = 31, 2+10+20 = 32.
- Statement I (m not prime): m can be 8, 16, 26, 27, 35 (5-coin present) or 32 (5-coin absent). Not sufficient.
- Statement II (digit sum > 5): 8, 16, 26, 17, 27, 35 have digit sums 8, 7, 8, 8, 9, 8; the totals without the 5-coin have digit sums 4, 5, 4, 5. So m always includes the 5-coin. Sufficient.
- One statement alone works and the other does not.
- ✗ Statement I Not sufficient alone: non-prime totals include 32 = 2 + 10 + 20, which has no 5-coin, as well as 8, 16, 26, 27 and 35, which do.
- ✓ Statement II Sufficient alone: every total with digit sum above 5 (8, 16, 17, 26, 27, 35) uses the 5-coin; totals without it (13, 23, 31, 32) have digit sums of only 4 or 5.
Remember · In data sufficiency with a small finite set, list every case first, then test each statement against the list.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·