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?

1. A. I only
2. B. II only
3. C. Both I and II
4. D. Neither I nor II

Answer: C

Explanation

Let’s break down the information:
* **Teams:** P, Q, R. Each plays every other team exactly once (P vs Q, P vs R, Q vs R – 3 matches total).
* **Points:** Win = 2, Draw = 1, Loss = 0.
* **Goals:** Each team scored exactly one goal in the entire tournament.
* **Scores:** P = 3 points, Q = 2 points, R = 1 point.

**Deductions based on points:**
* **P (3 points):** Must have 1 Win (2 pts) and 1 Draw (1 pt).
* **Q (2 points):** Must have 2 Draws (1+1 pts).
* **R (1 point):** Must have 1 Draw (1 pt) and 1 Loss (0 pts).

**Deducing Match Outcomes:**
1. **P vs R:** Since P won one match and R lost one match, P must have defeated R. P scored exactly one goal in the tournament. If P won against R, P must have scored its only goal in this match. So, the score for P vs R is 1-0 (P wins).

2. **P vs Q:** P has one win (vs R) and one draw. Q has two draws. Since P already scored its only goal against R, P cannot score any goals in the P vs Q match. For P vs Q to be a draw, and P scoring 0 goals, Q must also have scored 0 goals in this match. So, P vs Q = 0-0 (Draw). **Statement I is correct.**

3. **Q vs R:** Q has two draws (one with P, one with R). R has one draw (with Q) and one loss (with P). Since Q scored 0 goals against P, Q’s only goal must have been scored against R. Since R lost 0-1 to P, R’s only goal must have been scored against Q. For Q vs R to be a draw, and both Q and R having scored their only goal in this match, the score must be 1-1. So, Q vs R = 1-1 (Draw). **Statement II is correct** (R scored 1 goal against Q).

Both statements I and II are correct. This is a complex logical reasoning problem requiring careful deduction and consistency checking, typical of higher difficulty CSAT questions.