Data interpretation & sufficiency

Answer: (d) The Question can be answered even without using any of the Statements.

Among one-digit numbers only 6 (factors 1, 2, 3, 6) and 8 (factors 1, 2, 4, 8) have exactly four factors, so the smallest is 6. The question answers itself; the statements add nothing.

1. List factor counts for 1–9: only 6 (1, 2, 3, 6) and 8 (1, 2, 4, 8) have exactly 4 factors.
2. The smallest such number is 6 — found from the question alone.
3. The statements (2 and 3 are factors) are true of 6 but are not needed.

• ✗ Statement I Not needed: the question already fixes the answer as 6.
• ✗ Statement II Not needed: 6 is found without it.

Remember · In data sufficiency, first try the question alone; if it is already settled, choose 'even without using any of the Statements'.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) The Question can be answered even without using any of the Statements.

Dividing both sides by 11 gives P × PQ = 100R + S, which must be at least 100. With P ≤ 3 and Q ≤ 4, only 33 × 34 = 1122 works, so Q = 4 without using either statement.

1. PP = 11 × P and RRSS = 1100R + 11S = 11 × (100R + S). So P × PQ = 100R + S.
2. R is non-zero, so P × PQ must be at least 100. P = 1 gives at most 1 × 14 = 14; P = 2 gives at most 2 × 24 = 48. Too small.
3. P = 3: 3 × (30 + Q) = 90 + 3Q reaches 100 only for Q = 4, giving 102, so R = 1 and S = 2.
4. Check: 33 × 34 = 1122, and the digits 3, 4, 1, 2 are distinct.
5. Q = 4 follows from the question alone; neither statement is needed.

• ✗ Statement I Not needed: R = 1 already follows from the question itself.
• ✗ Statement II Not needed: S = 2 also follows from the question itself.

Remember · PP and RRSS are multiples of 11 — divide it out first. Digit limits often settle the question on their own.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) The Question cannot be answered even using any of the Statements.

Together the statements show that Q is a brother or sister of P's father, but Q's gender is never given. Q could be P's uncle or aunt, so the relation cannot be fixed even with both statements.

1. Statement I alone tells us about P's sisters but nothing about Q.
2. Statement II alone links Q to R's father, but not to P.
3. Together: R is P's sister, so R's father is P's father, and Q is a sibling of P's father.
4. Q's gender is not given, so Q could be P's paternal uncle or aunt.
5. The question cannot be answered even using both statements.

• ✗ Statement I Says nothing about Q.
• ✗ Statement II Links Q only to R's father; even combined with I, Q may be P's uncle or aunt.

Remember · In relation-based sufficiency items, check gender: a relation that depends on an unknown gender is not determined.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) The Question cannot be answered even using any of the Statements.

Even combined, the statements allow two outcomes. If Q had 3 goals with 10 minutes left, the match ends 4–4; if Q already had 4, P wins 5–4. So whether P won cannot be decided.

1. Let Q have g goals with 10 minutes left; P then had g − 3.
2. Statement I alone: P ends with g + 1 goals, but Q may also have scored in those 10 minutes. Not sufficient.
3. Statement II alone: Q ends with 4 goals, but P's final score is unknown. Not sufficient.
4. Together: P ends with g + 1 and Q with 4, where g is 3 or 4 (P had g − 3 ≥ 0, and Q's final 4 ≥ g).
5. g = 3 gives a 4–4 draw; g = 4 gives a 5–4 win for P. Both fit, so the question cannot be answered.

• ✗ Statement I P's 4 late goals could be matched by Q's own late goals.
• ✗ Statement II Q's total is known but P's is not; even with I, the result can be 4–4 or 5–4.

Remember · Before choosing 'both together', build two cases that fit both statements; if they give different answers, the data are insufficient.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) The Question can be answered by using either Statement alone.

The expression simplifies to (p − q)², which is positive unless p = q. Each statement on its own rules out p = q, so the question can be answered using either statement alone.

1. (p + q)² − 4pq = p² + 2pq + q² − 4pq = (p − q)².
2. (p − q)² is positive exactly when p ≠ q; it is 0 when p = q.
3. Without the statements, p = q is possible, so the question cannot be answered alone.
4. Statement I (p < q) gives p ≠ q, so the expression is positive. Statement II (p > q) also gives p ≠ q.
5. Either statement alone is sufficient.

• ✓ Statement I p < q means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
• ✓ Statement II p > q also means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.

Remember · Simplify before judging sufficiency: (p + q)² − 4pq = (p − q)², so only whether p ≠ q matters.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error