CAT 2022 Slot 1 · DILR · Set 1: Goals Scored by Four Hockey Players
The full set and all 5 questions (Q1–Q5) from CAT 2022 Slot 1, with answers and solutions.
The management of a university hockey team was evaluating performance of four women players - Amla, Bimla, Harita and Sarita for their possible selection in the university team for next year. For this purpose, the management was looking at the number of goals scored by them in the past 8 matches, numbered 1 through 8. The four players together had scored a total of 12 goals in these matches. In the 8 matches, each of them had scored at least one goal. No two players had scored the same total number of goals.
The following facts are known about the goals scored by these four players only. All the questions refer only to the goals scored by these four players.
1. Only one goal was scored in every even numbered match.
2. Harita scored more goals than Bimla.
3. The highest goal scorer scored goals in exactly 3 matches including Match 4 and Match 8.
4. Bimla scored a goal in Match 1 and one each in three other consecutive matches.
5. An equal number of goals were scored in Match 3 and Match 7, which was different from the number of goals scored in either Match 1 or Match 5.
6. The match in which the highest number of goals was scored was unique and it was not Match 5.
Question 1 · Caselet DI
- A3
- B2
- C1
- DCannot be determined
Answer & solution
Answer: Option C
Solving the set
Step 1 – Goals per player. Four different positive totals adding to 12: {1, 2, 3, 6} or {1, 2, 4, 5}.
Step 2 – Even matches (2, 4, 6, 8) have 1 goal each (fact 1), so 4 goals; the odd matches (1, 3, 5, 7) have 8 goals.
Step 3 – Bimla. She scores in Match 1 and exactly one goal in each of three other consecutive matches (fact 4), so she has at least 4 goals; with Harita above her (fact 2), Bimla = 4 and Harita = 5. Amla and Sarita have 1 and 2.
Step 4 – Harita is the top scorer: she scores in exactly 3 matches including 4 and 8 (fact 3). Those two even matches give her 1 goal each, so her third match gives 3 goals – an odd match. Matches 4 and 8 are 'used up' by Harita, so Bimla's run of three consecutive matches avoids 4 and 8 and excludes Match 1: it must be Matches 5, 6, 7.
Step 5 – Match totals. Matches 1, 5 and 7 each have at least Bimla's goal; Match 3 = Match 7 and both differ from Matches 1 and 5 (fact 5); the largest match total is unique and is not Match 5 (fact 6). Where is Harita's 3-goal match?
• Match 3 or Match 7: then Match 3 = Match 7 ≥ 3 and the odd matches already need 1 + 3 + 1 + 3 = 8 goals, so the top total (3) is shared by Matches 3 and 7 – not unique.
• Match 5: then Match 5 ≥ 4, and the other odd matches share at most 4 goals with Match 3 = Match 7, so Match 5 would be the unique maximum – not allowed.
• So it is Match 1 (at least 4 goals). With Match 1 + 2 × Match 3 + Match 5 = 8 and Match 5 ≠ Match 3, the only option is Match 1 = 4, Match 3 = Match 7 = 1, Match 5 = 2. Amla and Sarita's 3 goals therefore fall in Matches 2, 3 and 5 (Match 6's goal is Bimla's, Matches 4 and 8 are Harita's).
Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
Bimla | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 |
Harita | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 5 |
Amla + Sarita | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 |
Goals in match | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 12 |
Step 6 – Amla and Sarita's 3 goals (one player 1, the other 2) are spread over Matches 2, 3 and 5 in six possible ways:
Case | Amla scored in | Sarita scored in | Totals (A, S) |
1 | Match 5 | Matches 2, 3 | 1, 2 |
2 | Match 3 | Matches 2, 5 | 1, 2 |
3 | Matches 3, 5 | Match 2 | 2, 1 |
4 | Match 2 | Matches 3, 5 | 1, 2 |
5 | Matches 2, 5 | Match 3 | 2, 1 |
6 | Matches 2, 3 | Match 5 | 2, 1 |
This question
From Step 5, Match 7 had 1 goal.
Hence, the answer is option 3.
Question 2 · Caselet DI
- A5, 1, 0, 1
- B3, 1, 2, 1
- C3, 2, 1, 2
- D4, 1, 2, 1
Answer & solution
Answer: Option D
Solving the set
Step 1 – Goals per player. Four different positive totals adding to 12: {1, 2, 3, 6} or {1, 2, 4, 5}.
Step 2 – Even matches (2, 4, 6, 8) have 1 goal each (fact 1), so 4 goals; the odd matches (1, 3, 5, 7) have 8 goals.
Step 3 – Bimla. She scores in Match 1 and exactly one goal in each of three other consecutive matches (fact 4), so she has at least 4 goals; with Harita above her (fact 2), Bimla = 4 and Harita = 5. Amla and Sarita have 1 and 2.
Step 4 – Harita is the top scorer: she scores in exactly 3 matches including 4 and 8 (fact 3). Those two even matches give her 1 goal each, so her third match gives 3 goals – an odd match. Matches 4 and 8 are 'used up' by Harita, so Bimla's run of three consecutive matches avoids 4 and 8 and excludes Match 1: it must be Matches 5, 6, 7.
Step 5 – Match totals. Matches 1, 5 and 7 each have at least Bimla's goal; Match 3 = Match 7 and both differ from Matches 1 and 5 (fact 5); the largest match total is unique and is not Match 5 (fact 6). Where is Harita's 3-goal match?
• Match 3 or Match 7: then Match 3 = Match 7 ≥ 3 and the odd matches already need 1 + 3 + 1 + 3 = 8 goals, so the top total (3) is shared by Matches 3 and 7 – not unique.
• Match 5: then Match 5 ≥ 4, and the other odd matches share at most 4 goals with Match 3 = Match 7, so Match 5 would be the unique maximum – not allowed.
• So it is Match 1 (at least 4 goals). With Match 1 + 2 × Match 3 + Match 5 = 8 and Match 5 ≠ Match 3, the only option is Match 1 = 4, Match 3 = Match 7 = 1, Match 5 = 2. Amla and Sarita's 3 goals therefore fall in Matches 2, 3 and 5 (Match 6's goal is Bimla's, Matches 4 and 8 are Harita's).
Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
Bimla | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 |
Harita | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 5 |
Amla + Sarita | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 |
Goals in match | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 12 |
Step 6 – Amla and Sarita's 3 goals (one player 1, the other 2) are spread over Matches 2, 3 and 5 in six possible ways:
Case | Amla scored in | Sarita scored in | Totals (A, S) |
1 | Match 5 | Matches 2, 3 | 1, 2 |
2 | Match 3 | Matches 2, 5 | 1, 2 |
3 | Matches 3, 5 | Match 2 | 2, 1 |
4 | Match 2 | Matches 3, 5 | 1, 2 |
5 | Matches 2, 5 | Match 3 | 2, 1 |
6 | Matches 2, 3 | Match 5 | 2, 1 |
This question
From the match table, Matches 1, 3, 5 and 7 had 4, 1, 2 and 1 goals.
Hence, the answer is option 4.
Question 3 · Caselet DI
Which of the following statement(s) is/are true?
Statement-1: Amla and Sarita never scored goals in the same match.
Statement-2: Harita and Sarita never scored goals in the same match.
- AStatement-1 only
- BStatement-2 only
- CBoth the statements
- DNone of the statements
Answer & solution
Answer: Option C
Solving the set
Step 1 – Goals per player. Four different positive totals adding to 12: {1, 2, 3, 6} or {1, 2, 4, 5}.
Step 2 – Even matches (2, 4, 6, 8) have 1 goal each (fact 1), so 4 goals; the odd matches (1, 3, 5, 7) have 8 goals.
Step 3 – Bimla. She scores in Match 1 and exactly one goal in each of three other consecutive matches (fact 4), so she has at least 4 goals; with Harita above her (fact 2), Bimla = 4 and Harita = 5. Amla and Sarita have 1 and 2.
Step 4 – Harita is the top scorer: she scores in exactly 3 matches including 4 and 8 (fact 3). Those two even matches give her 1 goal each, so her third match gives 3 goals – an odd match. Matches 4 and 8 are 'used up' by Harita, so Bimla's run of three consecutive matches avoids 4 and 8 and excludes Match 1: it must be Matches 5, 6, 7.
Step 5 – Match totals. Matches 1, 5 and 7 each have at least Bimla's goal; Match 3 = Match 7 and both differ from Matches 1 and 5 (fact 5); the largest match total is unique and is not Match 5 (fact 6). Where is Harita's 3-goal match?
• Match 3 or Match 7: then Match 3 = Match 7 ≥ 3 and the odd matches already need 1 + 3 + 1 + 3 = 8 goals, so the top total (3) is shared by Matches 3 and 7 – not unique.
• Match 5: then Match 5 ≥ 4, and the other odd matches share at most 4 goals with Match 3 = Match 7, so Match 5 would be the unique maximum – not allowed.
• So it is Match 1 (at least 4 goals). With Match 1 + 2 × Match 3 + Match 5 = 8 and Match 5 ≠ Match 3, the only option is Match 1 = 4, Match 3 = Match 7 = 1, Match 5 = 2. Amla and Sarita's 3 goals therefore fall in Matches 2, 3 and 5 (Match 6's goal is Bimla's, Matches 4 and 8 are Harita's).
Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
Bimla | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 |
Harita | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 5 |
Amla + Sarita | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 |
Goals in match | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 12 |
Step 6 – Amla and Sarita's 3 goals (one player 1, the other 2) are spread over Matches 2, 3 and 5 in six possible ways:
Case | Amla scored in | Sarita scored in | Totals (A, S) |
1 | Match 5 | Matches 2, 3 | 1, 2 |
2 | Match 3 | Matches 2, 5 | 1, 2 |
3 | Matches 3, 5 | Match 2 | 2, 1 |
4 | Match 2 | Matches 3, 5 | 1, 2 |
5 | Matches 2, 5 | Match 3 | 2, 1 |
6 | Matches 2, 3 | Match 5 | 2, 1 |
This question
Statement-1: Amla and Sarita score only in Matches 2, 3 and 5. Matches 2 and 3 have just one goal each, and Match 5 has 2 goals of which one is Bimla's. So they can never both score in one match – true in all six cases.
Statement-2: Harita scores only in Matches 1, 4, 8; Sarita never scores in these – true.
Hence, the answer is option 3.
Question 4 · Caselet DI
Which of the following statement(s) is/are false?
Statement-1: In every match at least one player scored a goal.
Statement-2: No two players scored goals in the same number of matches.
- ANone of the statements
- BStatement-1 only
- CBoth the statements
- DStatement-2 only
Answer & solution
Answer: Option A
Solving the set
Step 1 – Goals per player. Four different positive totals adding to 12: {1, 2, 3, 6} or {1, 2, 4, 5}.
Step 2 – Even matches (2, 4, 6, 8) have 1 goal each (fact 1), so 4 goals; the odd matches (1, 3, 5, 7) have 8 goals.
Step 3 – Bimla. She scores in Match 1 and exactly one goal in each of three other consecutive matches (fact 4), so she has at least 4 goals; with Harita above her (fact 2), Bimla = 4 and Harita = 5. Amla and Sarita have 1 and 2.
Step 4 – Harita is the top scorer: she scores in exactly 3 matches including 4 and 8 (fact 3). Those two even matches give her 1 goal each, so her third match gives 3 goals – an odd match. Matches 4 and 8 are 'used up' by Harita, so Bimla's run of three consecutive matches avoids 4 and 8 and excludes Match 1: it must be Matches 5, 6, 7.
Step 5 – Match totals. Matches 1, 5 and 7 each have at least Bimla's goal; Match 3 = Match 7 and both differ from Matches 1 and 5 (fact 5); the largest match total is unique and is not Match 5 (fact 6). Where is Harita's 3-goal match?
• Match 3 or Match 7: then Match 3 = Match 7 ≥ 3 and the odd matches already need 1 + 3 + 1 + 3 = 8 goals, so the top total (3) is shared by Matches 3 and 7 – not unique.
• Match 5: then Match 5 ≥ 4, and the other odd matches share at most 4 goals with Match 3 = Match 7, so Match 5 would be the unique maximum – not allowed.
• So it is Match 1 (at least 4 goals). With Match 1 + 2 × Match 3 + Match 5 = 8 and Match 5 ≠ Match 3, the only option is Match 1 = 4, Match 3 = Match 7 = 1, Match 5 = 2. Amla and Sarita's 3 goals therefore fall in Matches 2, 3 and 5 (Match 6's goal is Bimla's, Matches 4 and 8 are Harita's).
Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
Bimla | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 |
Harita | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 5 |
Amla + Sarita | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 |
Goals in match | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 12 |
Step 6 – Amla and Sarita's 3 goals (one player 1, the other 2) are spread over Matches 2, 3 and 5 in six possible ways:
Case | Amla scored in | Sarita scored in | Totals (A, S) |
1 | Match 5 | Matches 2, 3 | 1, 2 |
2 | Match 3 | Matches 2, 5 | 1, 2 |
3 | Matches 3, 5 | Match 2 | 2, 1 |
4 | Match 2 | Matches 3, 5 | 1, 2 |
5 | Matches 2, 5 | Match 3 | 2, 1 |
6 | Matches 2, 3 | Match 5 | 2, 1 |
This question
Statement-1: every match total is at least 1 – true.
Statement-2: numbers of matches with goals: Bimla 4, Harita 3, and Amla/Sarita 1 and 2 (from Step 6) – all different, so true.
Neither statement is false.
Hence, the answer is option 1.
Question 5 · Caselet DI
If Harita scored goals in one more match as compared to Sarita, which of the following statement(s) is/are necessarily true?
Statement-1: Amla scored goals in consecutive matches.
Statement-2: Sarita scored goals in consecutive matches.
- AStatement-2 only
- BNone of the statements
- CStatement-1 only
- DBoth the statements
Answer & solution
Answer: Option B
Solving the set
Step 1 – Goals per player. Four different positive totals adding to 12: {1, 2, 3, 6} or {1, 2, 4, 5}.
Step 2 – Even matches (2, 4, 6, 8) have 1 goal each (fact 1), so 4 goals; the odd matches (1, 3, 5, 7) have 8 goals.
Step 3 – Bimla. She scores in Match 1 and exactly one goal in each of three other consecutive matches (fact 4), so she has at least 4 goals; with Harita above her (fact 2), Bimla = 4 and Harita = 5. Amla and Sarita have 1 and 2.
Step 4 – Harita is the top scorer: she scores in exactly 3 matches including 4 and 8 (fact 3). Those two even matches give her 1 goal each, so her third match gives 3 goals – an odd match. Matches 4 and 8 are 'used up' by Harita, so Bimla's run of three consecutive matches avoids 4 and 8 and excludes Match 1: it must be Matches 5, 6, 7.
Step 5 – Match totals. Matches 1, 5 and 7 each have at least Bimla's goal; Match 3 = Match 7 and both differ from Matches 1 and 5 (fact 5); the largest match total is unique and is not Match 5 (fact 6). Where is Harita's 3-goal match?
• Match 3 or Match 7: then Match 3 = Match 7 ≥ 3 and the odd matches already need 1 + 3 + 1 + 3 = 8 goals, so the top total (3) is shared by Matches 3 and 7 – not unique.
• Match 5: then Match 5 ≥ 4, and the other odd matches share at most 4 goals with Match 3 = Match 7, so Match 5 would be the unique maximum – not allowed.
• So it is Match 1 (at least 4 goals). With Match 1 + 2 × Match 3 + Match 5 = 8 and Match 5 ≠ Match 3, the only option is Match 1 = 4, Match 3 = Match 7 = 1, Match 5 = 2. Amla and Sarita's 3 goals therefore fall in Matches 2, 3 and 5 (Match 6's goal is Bimla's, Matches 4 and 8 are Harita's).
Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
Bimla | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 |
Harita | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 5 |
Amla + Sarita | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 |
Goals in match | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 12 |
Step 6 – Amla and Sarita's 3 goals (one player 1, the other 2) are spread over Matches 2, 3 and 5 in six possible ways:
Case | Amla scored in | Sarita scored in | Totals (A, S) |
1 | Match 5 | Matches 2, 3 | 1, 2 |
2 | Match 3 | Matches 2, 5 | 1, 2 |
3 | Matches 3, 5 | Match 2 | 2, 1 |
4 | Match 2 | Matches 3, 5 | 1, 2 |
5 | Matches 2, 5 | Match 3 | 2, 1 |
6 | Matches 2, 3 | Match 5 | 2, 1 |
This question
Harita scored in 3 matches, so Sarita scored in 2 (Cases 1, 2 and 4 of Step 6). Amla then scored in only one match, so Statement-1 is false in every case. Sarita's matches are 2, 3 (consecutive) in Case 1 but 2, 5 or 3, 5 in Cases 2 and 4, so Statement-2 is not necessarily true.
Hence, the answer is option 2.
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