Directions for questions 16 to 18: Answer the following questions based on the information given below.
The year was 2006. All six teams in Pool A of World Cup hockey, play each other exactly once. Each win earns a team three points, a draw earns one point and a loss earns zero points. The two teams with the highest points qualify for the semi-finals. In case of a tie, the team with the highest goal difference (Goals For – Goals Against) is higher ranked, and therefore, qualifies. In the opening match, Spain lost to Germany. After the second round (after each team played two matches), the pool table looked as shown below.
In the third round, Spain played South Africa, Argentina played Germany, and New Zealand played Pakistan. All the third round matches were drawn. The following are some results from the fourth and fifth round matches.
a. Spain won both the fourth and fifth round matches.
b. Both Argentina and Germany won their fifth round matches by 3 goals to 0.
c. Pakistan won one of its matches in the last two rounds by 1 goal to 0.
[CAT 2004]
16) Which one of the following statements is true about matches played in the first two rounds?
(1) Germany beat New Zealand by 1 goal to 0.
(2) Spain beat New Zealand by 4 goals to 0.
(3) Spain beat South Africa by 2 goals to 0.
(4) Germany beat South Africa by 2 goals to 1.
17) Which one of the following statements is true about matches played in the first two rounds?
(1) Pakistan beat South Africa by 2 goals to 1.
(2) Argentina beat Pakistan by 1 goal to 0.
(3) Germany beat Pakistan by 2 goals to 1.
(4) Germany beat Spain by 2 goals to 1.
18) If Argentina and South Africa's difference were 4 and -6 respectively, then which team(s) finished last in the table
(1) New Zealand
(2) South Africa
(3) New Zealand and South Africa
(4) Cannot be determined
19) Which team finished at the top of the pool after five rounds of matches?
(1) Argentina
(2) Germany
(3) Spain
(4) Cannot be determined
Solution
Let us take Germany, Argentina, Spain, Pakistan, New Zealand, and South Africa as Ger, Arg, Sp, Pak, NZ, and SA for convenience of taking notes. Each team plays every other team exactly once and the total number of matches in the World Cup is
6C2 = 26×5=15 matches. A team gets 3 points for a win, 1 for a draw, and 0 for a loss. This World Cup has a total of five rounds and each round as three matches. The scores for the 1st and 2nd rounds are already given.
'>' represents 'defeated'. For example Ger(2) > pak(1) means Germany defeated Pakistan by 2 - 1.
Goals For (GF) - when a team scores against its opponent team.
Goals Against (GA) - when an opponent team scores against the respective team.
It is given that Sp lost to Ger in the open match. Ger and Arg got two wins in round 1 and 2 and we can write them in any order in the table. Sp and Pak each win only one match in the first two rounds.
NZ has a total of 6 GA in the two rounds which means two teams scored 6 goals against NZ. Other than Sp's GF, any two teams GF does not exceed 5 goals, so Sp won its match against NZ.
Arg won its two games by scoring two goals against its opponent. So it must win those two matches by the scores 1 - 0 and 1 - 0.
The GF and GA of Germany are 3 and 1, so it should have win those two matches by 2 - 1 and 1 - 0. The GF and GA of the first two matches of Pak are 2 and 1, but it lost one of the two matches. So the winning game of Pak has the score of 2 - 0 and its losing game has a score of 0 – 1.
If SA played against Arg then it must have lost the game by 0 - 1 but it is not possible because SA's GA is 4. Ger and Sp has the GF greater than or equal to 3, so SA played against Pak in round 1 and Ger in round 2 and lost these two games by 2 - 0 and 2 -1.
Ger won against Sp by 1 - 0 and Arg won against Pak and NZ by 1 - 0 and 1 - 0 in round 2. Sp defeated NZ by 5 – 1.
We can form as table as below:
16) This table shows the results of round 1 and 2:
From the above table, it is clear that Ger wins against SA by 2 – 1 in round 2.
Answer: (4) Germany beat South Africa by 2 goals to 1
17) This table shows the results of round 1 and 2.
From the above table, it is clear that Arg wins against Pak by 1 – 0 in round 2.
Answer: (2) Argentina beat Pakistan by 1 goal to 0
18) This table shows the results of round 1 and 2.
All the matches in the 3rd round were drawn and Sp won the 4th and 5th rounds. Arg and Ger won their 5th round by 3 – 0. As round 5 has three winners, Pak must have won a match in the 4th round by 1 – 0.
Sp won against Arg in round 4 and Pak in round 5.
SA must have lost the game with Arg in round 5, Ger lost against Pak in round 4, and NZ lost against Ger in round 5.
The remaining two teams NZ and SA played against each other in round 4.
In a draw match, the goal difference is 0.
In the question, it is given that SA's goal difference is -6.
Goal difference in R (1 to 5) of SA:
SA = (0 - 2) + (1 – 2) + 0 + (x) + (0 – 3) = x – 6
The above condition in the question is satisfied only if x = 0
Therefore, the NZ and SA match in round 4 ends in a draw.
Goal difference of NZ = (0 - 1) + (1 – 5) + 0 + 0 +(0 – 3) = -8
The following table shows the number of wins and losses of each team.
Based on this, New Zealand finished last in the table.
Answer: (1) New Zealand
19) The following shows the results of all the five rounds.
In a draw match, the goal difference is 0.
Goal difference in R(1 to 5):
Ger = (1 - 0) + (2 – 1) + 0 + (0 – 1) + (3 – 0) = 4
Arg = (1 - 0) + (1 – 0) + 0 + (-x) + (3 – 0) = 5 – x ≤ 4
Sp = (0 - 1) + (5 – 1) + 0 + x + y = 3 + x + y ≥ 5
Therefore, 3 + x + y > 5 – x
The following table shows the number of wins and losses of each team.
Spain finished at the top of the pool after five rounds.
Answer: (3) Spain
Answer:
16) (4) Germany beat South Africa by 2 goals to 1
17) (2) Argentina beat Pakistan by 1 goal to 0
18) (1) New Zealand
19) (3) Spain
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