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?

The correct option is Both I and II.

Step 1: Understand the Tournament Format and Point System

There are 3 teams: P, Q, and R.

Each team plays every other team exactly once. Therefore, there are 3 total matches in the tournament:

1. P vs Q
2. P vs R
3. Q vs R

Point Rules:

• Win = 2 points
• Draw = 1 point
• Loss = 0 points

Points obtained by each team:

• Team P = 3 points
• Team Q = 2 points
• Team R = 1 point

Step 2: Determine the Outcomes of the Matches

Let us analyze the possible results for each team based on their points:

• For Team P to get 3 points from 2 matches, P must have 1 win and 1 draw (2 + 1 = 3).

• For Team R to get 1 point from 2 matches, R must have 1 draw and 1 loss (1 + 0 = 1).

• For Team Q to get 2 points from 2 matches, Q could either have 1 win and 1 loss (2 + 0 = 2) OR 2 draws (1 + 1 = 2).

Let's deduce the match results:

1. Team P has 1 win and 1 draw. Team R has 1 draw and 1 loss.
Since Team P won one match, P must have beaten Team R (because if P beat Q, Q would have a loss, but Team P is the only team with a win, which means P defeated R).

2. Therefore, the match P vs R was a win for P (P won, R lost).

3. Since P's other match was a draw, the match P vs Q must be a draw.

4. Since R's other match was a draw, the match Q vs R must be a draw.

Summary of Match Results:

• P vs Q : Draw
• Q vs R : Draw
• P vs R : P won (R lost)

This gives P: 1 win, 1 draw = 3 points.
Q: 2 draws = 2 points.
R: 1 draw, 1 loss = 1 point.
All point tallies match the given data perfectly!

Step 3: Analyze Goals Scored and Match Scores

We are given that each team scored exactly 1 goal in the entire tournament.

• Total goals scored in the tournament by P, Q, and R combined = 1 + 1 + 1 = 3 goals.

Let's look at the matches individually:

1. Match P vs R (P won):
For P to win against R, P must have scored at least 1 goal against R, and R must have scored fewer goals than P.
Since P only scored 1 goal in the entire tournament, P scored 1 goal against R and 0 goals against Q.
Since P scored 1 goal against R and won, R could not have scored against P (if R scored 1, it would be 1-1, a draw). Thus, R scored 0 goals against P.
So, the score for P vs R is 1-0.

2. Match P vs Q (Draw):
We know P scored 0 goals in this match.
Since the match ended in a draw, Q must also have scored 0 goals in this match.
So, the score for P vs Q is 0-0.

3. Match Q vs R (Draw):
Team Q scored 1 goal in total in the tournament. Since Q scored 0 goals against P, Q must have scored 1 goal against R.
Team R also scored 1 goal in total in the tournament. Since R scored 0 goals against P, R must have scored 1 goal against Q.
Since both Q and R scored 1 goal against each other, the match ended in a draw.
So, the score for Q vs R is 1-1.

Step 4: Evaluate the Given Statements

• Statement I: "The result of the match between P and Q is a draw with the score 0-0."
As derived in Step 3, P vs Q was indeed a 0-0 draw. Therefore, Statement I is correct.

• Statement II: "The number of goals scored by R against Q is 1."
As derived in Step 3, R scored 1 goal against Q in the 1-1 draw. Therefore, Statement II is correct.

Hence, both statements I and II are correct.