How many goals were scored in Match 7?
Solution
Match 7 had only Bimla's goal: .
Answer: (C) 1
CAT 2022 · Slot 1 · DILR
CAT 2022 Slot 1 DILR paper with solutions
4 sets, 20 questions, in exam order.
4
sets
20
questions
Sets in this slot
Goals scored by four hockey players
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.
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.
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.
Set solution: how to crack this set
Four distinct totals (each at least 1) adding to 12: or .
Bimla scored in Match 1 and in three other consecutive matches, so she has at least 4 goals. Harita has more than Bimla, so the totals are 1, 2, 4, 5: Bimla 4, Harita 5 (the highest scorer). Amla and Sarita have 1 and 2.
Even matches have 1 goal each (4 goals), so odd matches have 8.
Harita scored 5 goals in exactly 3 matches including Matches 4 and 8 (1 goal each), so she scored 3 goals in one odd match.
Bimla's three consecutive matches can't include 4 or 8 (Harita's single goals), so they are 5, 6, 7. Bimla: Matches 1, 5, 6, 7.
Harita's 3-goal match: M3 or M7 would force M3 M7 and break the unique-highest rule or the total. M5 can't be the highest match. So it is Match 1: M1 .
Then with M5 M3: M3 M7 , M5 .
| Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Goals | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 |
| Scorers | B, H(3) | A/S | A/S | H | B, A/S | B | B | H |
Amla and Sarita share the three single goals in Matches 2, 3 and 5 (one has 1, the other 2).
Which of the following is the correct sequence of goals scored in matches 1, 3, 5 and 7?
Solution
Matches 1, 3, 5, 7: .
Answer: (D) 4, 1, 2, 1
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.
Solution
Matches 2, 3 and 5 each have exactly one goal from Amla or Sarita, so they never score in the same match. Harita scores only in 1, 4, 8 and Sarita only in 2, 3, 5. Both statements are true.
Answer: (C) Both the statements
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.
Solution
Every match has at least one goal. Matches scored in: Harita 3, Bimla 4, Amla and Sarita 1 and 2, all different. Both statements are true, so none is false.
Answer: (A) None of the statements
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.
Solution
Harita scored in 3 matches, so Sarita scored in 2 (2 goals) and Amla in 1. Sarita's two matches can be , or , so neither statement is necessarily true.
Answer: (B) None of the statements
Get-together: singers, dancers and event days
There are 15 girls and some boys among the graduating students in a class. They are planning a get-together, which can be either a 1-day event, or a 2-day event, or a 3-day event. There are 6 singers in the class, 4 of them are boys. There are 10 dancers in the class, 4 of them are girls. No dancer in the class is a singer.
Some students are not interested in attending the get-together. Those students who are interested in attending a 3-day event are also interested in attending a 2-day event; those who are interested in attending a 2-day event are also interested in attending a 1-day event.
The following facts are also known:
1. All the girls and 80% of the boys are interested in attending a 1-day event. 60% of the boys are interested in attending a 2-day event.
2. Some of the girls are interested in attending a 1-day event, but not a 2-day event; some of the other girls are interested in attending both.
3. 70% of the boys who are interested in attending a 2-day event are neither singers nor dancers. 60% of the girls who are interested in attending a 2-day event are neither singers nor dancers.
4. No girl is interested in attending a 3-day event. All male singers and 2 of the dancers are interested in attending a 3-day event.
5. The number of singers interested in attending a 2-day event is one more than the number of dancers interested in attending a 2-day event.
Set solution: how to crack this set
Singers: 4 boys, 2 girls. Dancers: 6 boys, 4 girls.
Let there be boys. 2-day boys , of whom 30% (i.e. ) are singers or dancers. These include all 4 male singers (3-day implies 2-day) and the 2 dancers on the 3-day list (boys, since no girl is on it), so .
must be a whole number, so is a multiple of 50. gives 18, more than the 10 male singers and dancers. So : 9 boys in the 2-day group are singers or dancers (4 singers, 5 dancers).
Girls in the 2-day group: , of whom are singers or dancers.
Singers (2-day) dancers (2-day) : , . So and .
2-day group: 30 boys 5 girls . Class .
How many boys are there in the class?
Solution
Only satisfies all conditions.
Answer: 50
Which of the following can be determined from the given information?
I. The number of boys who are interested in attending a 1-day event and are neither dancers nor singers.
II. The number of female dancers who are interested in attending a 1-day event.
Solution
I: 1-day boys , but the male dancers among them are 5 or 6, so I can't be fixed. II: all girls attend the 1-day event, so all 4 female dancers do. Only II.
Answer: (C) Only II
What fraction of the class are interested in attending a 2-day event?
Solution
.
Answer: (B)
What BEST can be concluded about the number of male dancers who are interested in attending a 1-day event?
Solution
5 male dancers are in the 2-day group (hence the 1-day group). The sixth may or may not be among the 80% of boys in the 1-day group: or .
Answer: (A) 5 or 6
How many female dancers are interested in attending a 2-day event?
Solution
.
Answer: (C) 0
Interview tokens and funding amounts
Adhara, Bithi, Chhaya, Dhanavi, Esther, and Fathima are the interviewers in a process that awards funding for new initiatives. Every interviewer individually interviews each of the candidates individually and awards a token only if she recommends funding. A token has a face value of 2, 3, 5, 7, 11, or 13. Each interviewer awards tokens of a single face value only.
Once all six interviews are over for a candidate, the candidate receives a funding that is Rs.1000 times the product of the face values of all the tokens. For example, if a candidate has tokens with face values 2, 5, and 7, then they get a funding of Rs.1000 × (2 × 5 × 7) = Rs.70,000.
Pragnyaa, Qahira, Rasheeda, Smera, and Tantra were five candidates who received funding. The funds they received, in descending order, were Rs.390,000, Rs.210,000, Rs.165,000, Rs.77,000, and Rs.66,000.
The following additional facts are known:
1. Fathima awarded tokens to everyone except Qahira, while Adhara awarded tokens to no one except Pragnyaa.
2. Rashida received the highest number of tokens that anyone received, but she did not receive one from Esther.
3. Bithi awarded a token to Smera but not to Qahira, while Dhanavi awarded a token to Qahira but not to Smera.
Set solution: how to crack this set
Fundings in thousands: , , , , .
Each prime appears at most once per candidate, so each interviewer has a different value.
Fathima gives to four candidates (all but Qahira): only 3 is in four amounts, so Fathima and Qahira .
Adhara gives to Pragnyaa only: only 13 appears once, so Adhara and Pragnyaa .
Rasheeda has the most tokens (4): Rasheeda . Smera and Tantra are 165 and 66.
Esther gave nothing to Rasheeda (2, 3, 5, 7), so Esther . Dhanavi gave to Qahira (7, 11), so Dhanavi . Bithi and Chhaya are 2 and 5 in some order.
| Candidate | Funding | Tokens |
|---|---|---|
| Pragnyaa | 390 | 4 |
| Rasheeda | 210 | 4 |
| Smera / Tantra | 165 or 66 | 3 |
| Qahira | 77 | 2 |
How many tokens did Qahira receive?
Solution
: tokens.
Answer: 2
Who among the following definitely received a token from Bithi but not from Dhanavi?
Solution
Pragnyaa holds both 2 and 5, so she got Bithi's token whatever Bithi's value, and no 7 (Dhanavi). Rasheeda and Qahira got 7 from Dhanavi.
Answer: (A) Pragnyaa
How many tokens did Chhaya award?
Solution
If Chhaya is 2: Pragnyaa, Rasheeda and the 66 candidate. If 5: Pragnyaa, Rasheeda and the 165 candidate. Either way .
Answer: 3
How many tokens did Smera receive?
Solution
Smera has 165 or 66, both three tokens: .
Answer: 3
Which of the following could be the amount of funding that Tantra received?
(a) Rs. 66,000
(b) Rs. 165,000
Solution
Bithi (2 or 5) gives to Smera, and either value fits. So Tantra could have 66 or 165: both.
Answer: (D) Both (a) and (b)
Metro lines and train timings
Given above is the schematic map of the metro lines in a city with rectangles denoting terminal stations (e.g. A), diamonds denoting junction stations (e.g. R) and small filled-up circles denoting other stations. Each train runs either in east-west or north-south direction, but not both. All trains stop for 2 minutes at each of the junction stations on the way and for 1 minute at each of the other stations. It takes 2 minutes to reach the next station for trains going in east-west direction and 3 minutes to reach the next station for trains going in northsouth direction. From each terminal station, the first train starts at 6 am; the last trains leave the terminal stations at midnight. Otherwise, during the service hours, there are metro service every 15 minutes in the north-south lines and every 10 minutes in the east-west lines. A train must rest for at least 15 minutes after completing a trip at the terminal station, before it can undertake the next trip in the reverse direction. (All questions are related to this metro service only. Assume that if someone reaches a station exactly at the time a train is supposed to leave, (s)he can catch that train.)
Set solution: how to crack this set
From the map, intermediate stations: on A–B and C–D, 2 (terminal to junction), 3 (junction to junction), 2 (junction to terminal). On M–N and P–Q, 3, 9, 5.
North-south: 3 min per hop. East-west: 2 min per hop. Stops: 1 min (ordinary), 2 min (junction).
A→R: min; R→T: ; T→B: 11. Full trip with junction stops: min.
M→R: ; R→S: ; S→N: . Full trip: min.
So a train arrives at the first junction 11 minutes after leaving its terminal and departs 2 minutes later.
If Hari is ready to board a train at 8:05 am from station M, then when is the earliest that he can reach station N?
Solution
Next train from M after 8:05 is 8:10. Arrives N at am.
Answer: (A) 9:11 am
If Priya is ready to board a train at 10:25 am from station T, then when is the earliest that she can reach station S?
Solution
Route T→R→S: northbound trains leave B at :00, :15, …, and depart T 13 min later. The 10:15 train leaves T at 10:28 and reaches R at 10:43.
Eastbound trains from M depart R at M-time : the 10:30 train leaves R at 10:43. It reaches S at .
Route T→V→S reaches S only at 11:28. Earliest: 11:12 am.
Answer: (A) 11:12 am
Haripriya is expected to reach station S late. What is the latest time by which she must be ready to board at station S if she must reach station B before 1 am via station R?
Solution
The last southbound train leaves A at midnight, departs R at 00:13 and reaches B at 00:41.
Westbound trains depart S 19 min after leaving N and reach R 29 min later (48 min after N). To reach R by 00:13, the N departure must be 23:20 or earlier. That train leaves S at 23:39, so she must be ready by 11:39 pm.
Answer: (A) 11:39 pm
What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?
Solution
A train takes 41 min and rests 15, so the earliest it can leave again is 56 min later; the next 15-minute slot is 60 min after its departure. Each train therefore leaves each end once every 120 min.
Departures every 15 min: trains.
Answer: 8
What is the minimum number of trains that are required to provide the service in this city?
Solution
North-south lines: 8 each (above).
East-west lines: min, so the next 10-minute slot is 80 min after departure. Each train leaves each end once every 160 min: trains per line.
Total .
Answer: 48
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