Three teams P, Q, R participated in a tournament in which the teams play with one another exactly once. A win fetches a team 2 points and a draw 1 point. A team gets no point for a loss. Each team scored exactly one goal in the tournament. The team P got 3 points, Q got 2 points and R got 1 point. Which of the following statements is/are correct ? I. The result of the match between P and Q is a draw with the score 0 – 0. II. The number of goals scored by R against Q is 1. Which of the statements given above is/are correct ?

• There are 3 matches in total (P vs Q, Q vs R, R vs P), and each team scored exactly one goal, meaning the total goals in the tournament is 3.

• Since P has 3 points, P must have won one match (2 points) and drawn one match (1 point), as a loss would yield 0 points and two wins would yield 4 points.

• Q has 2 points, which implies Q won one match and lost one match (2+0), or drew two matches (1+1). However, if Q drew two matches, P and R would also have draws, conflicting with P's 3 points (win+draw) and R's 1 point (draw+loss or 3 losses). Let's analyze the points distribution: Total points = 3+2+1 = 6. Each match distributes 2 points (win/loss) or 2 points (draw/draw). Thus, all 3 matches must be decisive (no draws) OR some draws exist. Wait, a draw gives 1+1=2 points. A win/loss gives 2+0=2 points. So total points is always 6. This doesn't restrict draws yet.

• Let's look at goals. Each team scored 1 goal. Total goals = 3. This means there are no 0-0 draws because a 0-0 draw contributes 0 goals. If a match is a draw, it must be 1-1 (contributing 2 goals) or 0-0 (0 goals). Since total goals is 3, we cannot have a 1-1 draw (would use 2 goals, leaving 1 for two other matches, impossible as goals are integers and min 0). We cannot have a 0-0 draw if it's the only draw? Let's check. If P vs Q is 0-0, that's 0 goals. Remaining goals = 3. Q vs R and R vs P must account for 3 goals. Points: P=3, Q=2, R=1.

• If P vs Q is 0-0 (Draw), P gets 1, Q gets 1. P needs 2 more points from R. So P beats R (2 pts). P total = 3. Q has 1 pt, needs 1 more from R. So Q draws with R? If Q draws with R, Q total = 2. R gets 1 from P loss (0) and 1 from Q draw (1). R total = 1. This fits the points: P(3), Q(2), R(1).

• Now check goals for this scenario: P vs Q is 0-0. P beats R. Q draws R. Goals: P scored 1 total. In P vs Q (0-0), P scored 0. So P must have scored 1 in P vs R. Q scored 1 total. In P vs Q (0-0), Q scored 0. So Q must have scored 1 in Q vs R. R scored 1 total. In P vs R, R scored 0 (since P won and P scored 1, if R scored 1 it would be 1-1 draw, but P won). So R must have scored 1 in Q vs R. But Q vs R is a draw. If Q scored 1 and R scored 1, the score is 1-1. This is a valid draw. Total goals: P(1) + Q(1) + R(1) = 3. Matches: P-Q (0-0), P-R (1-0), Q-R (1-1). This scenario is consistent.

• Statement I says P vs Q is a draw with score 0-0. In our derived scenario, this is true.

• Statement II says R scored 1 goal against Q. In our derived scenario, Q vs R is 1-1, so R scored 1 goal against Q. This is true.

• Are there other scenarios? If P vs Q is not 0-0. Suppose P vs Q is a win/loss. P wins (2 pts). P needs 1 more from R, so P draws R. Q has 0 from P, needs 2 from R, so Q beats R. Points: P(2+1=3), Q(0+2=2), R(0+0=0). But R has 1 point. Contradiction. Suppose Q wins P. Q(2), P(0). P needs 3 from R, impossible (max 2). So P must win or draw Q. If P draws Q (1-1), P(1), Q(1). P needs 2 from R (Win). Q needs 1 from R (Draw). R gets 0 from P, 1 from Q. R total 1. Points fit. Goals: P-Q is 1-1. P scored 1, Q scored 1. P has 0 goals left for P-R. Q has 0 goals left for Q-R. R must score 1 goal total. In P-R (P wins), if P scored 0, R must score 0 for P to win? No, P wins means P > R. If P scored 0, R must score negative, impossible. So P must score > R. If P scored 0 in P-Q, P must score 1 in P-R. But P already scored 1 in P-Q (1-1). Total P goals = 2. Contradiction (P scored exactly 1). Thus, P-Q cannot be 1-1. The only valid draw for P-Q is 0-0.