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.

Correct Option (3)

The tournament involves three teams (P, Q, R) playing each other exactly once, resulting in three matches: P vs Q, P vs R, and Q vs R. A win yields 2 points, a draw 1 point, and a loss 0 points. Each team scored exactly one goal in the tournament.

The total points accumulated are P=3, Q=2, R=1, summing to 6 points. Since each match distributes 2 points (either 2+0 for a win/loss or 1+1 for a draw), the total points distributed across 3 matches is 3 * 2 = 6 points, which is consistent with the given scores.

Let's deduce the match outcomes:

• Team R scored 1 point from two matches. This implies R had one draw (1 point) and one loss (0 points).
• Team P scored 3 points from two matches. This implies P had one win (2 points) and one draw (1 point).
• Team Q scored 2 points from two matches. This implies Q either had two draws (1+1=2) or one win and one loss (2+0=2).

Combining these deductions:

• If P drew with Q (P gets 1 point, Q gets 1 point), then P must have won against R (P gets 2 points, R gets 0 points) to achieve its total of 3 points.
• For R to achieve its total of 1 point (0 from P), R must have drawn with Q (R gets 1 point, Q gets 1 point).
• This scenario for Q means Q got 1 point from P (draw) and 1 point from R (draw), totaling 2 points, which matches Q's score.

Therefore, the match results are:

• P vs Q: Draw (P=1 point, Q=1 point)
• P vs R: P wins (P=2 points, R=0 points)
• Q vs R: Draw (Q=1 point, R=1 point)

Now, let's incorporate the goal information: Each team scored exactly one goal in the tournament.

Statement I: The result of the match between P and Q is a draw with the score 0 -0.

• As established, P vs Q was a draw.
• P scored its only goal in the match against R (P won 1-0 against R).
• Q scored its only goal in the match against R (Q drew 1-1 with R).
• Since P scored its goal against R and Q scored its goal against R, neither P nor Q scored against each other. Thus, the match between P and Q must have ended 0-0.
• Statement I is correct.

Statement II: The number of goals scored by R against Q is 1.

• As established, the match between Q and R was a 1-1 draw.
• This indicates that R scored 1 goal against Q.
• Statement II is correct.

Since both Statement I and Statement II are correct, option 3 is the correct answer.

Incorrect Options:

Options 1, 2, and 4 are incorrect because the analysis demonstrates that both Statement I and Statement II are factually correct based on the given tournament conditions and point distribution. Therefore, selecting only one statement as correct or neither as correct would be erroneous.