Comprehension
The following is the table of points drawn at the end of all the matches in a six-nation Hockey tournament, in which each country played with every other country exactly once. The table gives the positions of the countries in terms of their respective total points scored (i.e., in the decreasing order of their total points). Each win was worth three points, each draw one point, and there were no points for a loss. Some information in the table has been intentionally left out. The results of none of the individual matches are known, except that Pakistan beat India and no two teams finished with the same number of points.
\[ \begin{array}{|c|l|c|c|c|c|c|c|} \hline \textbf{Position} & \textbf{Country} & \textbf{Won} & \textbf{Drawn} & \textbf{Lost} & \textbf{Goals For} & \textbf{Goals Against} & \textbf{Total Points} \\ \hline 1 & \text{Australia} & & & & 17 & 5 & 15 \\ 2 & \text{Netherlands} & & & & 9 & 6 & 10 \\ 3 & \text{Pakistan} & & & & & 2 & 8 \\ 4 & \text{India} & & & & 2 & 5 & \\ 5 & \text{South Korea} & & & & 7 & 11 & 2 \\ 6 & \text{Spain} & & & & 8 & 16 & \\ \hline \end{array} \]
\[ \begin{array}{|c|l|c|c|c|c|c|c|} \hline \textbf{Position} & \textbf{Country} & \textbf{Won} & \textbf{Drawn} & \textbf{Lost} & \textbf{Goals For} & \textbf{Goals Against} & \textbf{Total Points} \\ \hline 1 & \text{Australia} & & & & 17 & 5 & 15 \\ 2 & \text{Netherlands} & & & & 9 & 6 & 10 \\ 3 & \text{Pakistan} & & & & & 2 & 8 \\ 4 & \text{India} & & & & 2 & 5 & \\ 5 & \text{South Korea} & & & & 7 & 11 & 2 \\ 6 & \text{Spain} & & & & 8 & 16 & \\ \hline \end{array} \]
Question: 1
Which of the following matches was a draw?
Which of the following matches was a draw?
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Use the total point logic with draw and win values to back-calculate the possible outcomes.
Updated On: Jul 28, 2025
- India vs South Korea
- Spain vs Netherlands
- Netherlands vs South Korea
- Spain vs South Korea
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The Correct Option is D
Solution and Explanation
Each team plays every other team once in a 6-nation tournament →
Total matches per team = 5
Total matches in tournament = \( \binom{6}{2} = 15 \)
Scoring system: Win = 3 pts, Draw = 1 pt, Loss = 0 pt
From the table:
- Australia = 15 pts (max = 5 matches × 3 = 15) → Won all
- Netherlands = 10 pts
- Pakistan = 8 pts
- South Korea = 2 pts
- Draws = only 2 pts for South Korea → Must be from 2 draws or 1 draw + 1 win
But if South Korea had 1 win → 3 pts → Contradiction
So, South Korea must have had 2 draws → \(2 \times 1 = 2\) pts
From “Goals For” and “Against”, Spain had 8 GF and 16 GA but no points → All losses → 0 pts
So, only draw possible for South Korea = against Spain (since Spain has 0 pts, can't be win) \[ \Rightarrow \boxed{\text{Spain vs South Korea was a draw}} \]
Total matches per team = 5
Total matches in tournament = \( \binom{6}{2} = 15 \)
Scoring system: Win = 3 pts, Draw = 1 pt, Loss = 0 pt
From the table:
- Australia = 15 pts (max = 5 matches × 3 = 15) → Won all
- Netherlands = 10 pts
- Pakistan = 8 pts
- South Korea = 2 pts
- Draws = only 2 pts for South Korea → Must be from 2 draws or 1 draw + 1 win
But if South Korea had 1 win → 3 pts → Contradiction
So, South Korea must have had 2 draws → \(2 \times 1 = 2\) pts
From “Goals For” and “Against”, Spain had 8 GF and 16 GA but no points → All losses → 0 pts
So, only draw possible for South Korea = against Spain (since Spain has 0 pts, can't be win) \[ \Rightarrow \boxed{\text{Spain vs South Korea was a draw}} \]
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Question: 2
The total number of points won by India is:
The total number of points won by India is:
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Use total point pool and subtract known scores to infer missing ones.
Updated On: Jul 28, 2025
- 5
- 6
- 7
- Cannot be determined
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The Correct Option is B
Solution and Explanation
Total points available = \(15 \text{ matches} \times 3 = 45 \text{ points}\)
From table:
\[
\text{Australia} = 15
\text{Netherlands} = 10
\text{Pakistan} = 8
\text{South Korea} = 2
\text{Sum so far} = 35 \Rightarrow \text{Remaining for India + Spain} = 10 \] Spain has 0 points (all losses). So India = \(45 - 35 - 0 = \boxed{6 \text{ points}}\)
\text{Netherlands} = 10
\text{Pakistan} = 8
\text{South Korea} = 2
\text{Sum so far} = 35 \Rightarrow \text{Remaining for India + Spain} = 10 \] Spain has 0 points (all losses). So India = \(45 - 35 - 0 = \boxed{6 \text{ points}}\)
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Question: 3
Total number of goals scored in Netherlands vs Pakistan match is:
Total number of goals scored in Netherlands vs Pakistan match is:
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Cross-reference “Goals For” and “Goals Against” to infer likely scorelines.
Updated On: Jul 28, 2025
- 0
- 1
- 2
- Cannot be determined
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The Correct Option is C
Solution and Explanation
From table:
- Netherlands GF = 9, GA = 6
- Pakistan GA = 2 (i.e., goals conceded across all matches)
Pakistan played 5 matches. Only Netherlands could have scored 2 goals against Pakistan and still have 6 goals conceded overall. Also, if Netherlands scored 2 goals against Pakistan and conceded none → Valid Thus, likely: \[ \text{Netherlands 2 – 0 Pakistan} \Rightarrow \boxed{2 \text{ goals}} \]
- Netherlands GF = 9, GA = 6
- Pakistan GA = 2 (i.e., goals conceded across all matches)
Pakistan played 5 matches. Only Netherlands could have scored 2 goals against Pakistan and still have 6 goals conceded overall. Also, if Netherlands scored 2 goals against Pakistan and conceded none → Valid Thus, likely: \[ \text{Netherlands 2 – 0 Pakistan} \Rightarrow \boxed{2 \text{ goals}} \]
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Question: 4
The number of goals scored by Australia against India is at most:
The number of goals scored by Australia against India is at most:
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Use the constraints on each team's total GA or GF to find upper bounds.
Updated On: Jul 28, 2025
- 5
- 4
- 3
- 2
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The Correct Option is A
Solution and Explanation
Australia’s total “Goals For” = 17
India’s “Goals Against” = 5
\[
\text{So, vs India, Australia could score at most } 5 \text{ goals}
\Rightarrow \boxed{5 \text{ is the maximum possible}}
\]
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