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CAT 2019 Slot 2 DILR · Set 5 of 8Logical ReasoningModerate DifficultyQ17–Q20 · 4 MCQ

CAT 2019 Slot 2 DILR Set 5: Optimization (4 Questions) with Solutions

The Optimization set from the official CAT 2019 Slot 2 DILR paper: the caselet, all 4 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2019 Slot 2 DILR · Questions 17–20
Three doctors, Dr. Ben, Dr. Kane and Dr. Wayne visit a particular clinic Monday to Saturday to see patients. Dr. Ben sees each patient for 10 minutes and charges Rs. 100/-. Dr. Kane sees each patient for 15 minutes and charges Rs. 200/-, while Dr. Wayne sees each patient for 25 minutes and charges Rs. 300/-.
The clinic has three rooms numbered 1, 2 and 3 which are assigned to the three doctors as per the following table.
Room No.Monday & TuesdayWednesday & ThursdayFriday & Saturday
1BenWayneKane
2KaneBenWayne
3WayneKaneBen
The clinic is open from 9 a.m. to 11.30 a.m. every Monday to Saturday.On arrival each patient is handed a numbered token indicating their position in the queue, starting with token number 1 every day. As soon as any doctor becomes free, the next patient in the queue enters that emptied room for consultation. If at any time, more than one room is free then the waiting patient enters the room with the smallest number. For example, if the next two patients in the queue have token numbers 7 and 8 and if rooms numbered 1 and 3 are free, then patient with token number 7 enters room number 1 and patient with token number 8 enters room number 3.

In the exam this set competed with 7 others for 60 minutes. Budget about 8 minutes if you attempt it.

Q17. Optimization

MCQ · +3 / −1
What is the maximum number of patients that the clinic can cater to on any single day?
  1. A) 30
  2. B) 12
  3. C) 15
  4. D) 31
Correct Answer
D) 31
Step-by-step solution
📌 Core Concept
Each doctor works on his own, so the day's total is the sum of what each doctor can start. The clinic runs from 9:00 to 11:30, which is 150 minutes. A patient who enters a room before 11:30 is seen.
🔢 Step-by-Step Solution
1
Dr. Ben takes 10 minutes per patient. He can start patients at minutes 0, 10, 20, ..., 140, which gives 15 patients.
2
Dr. Kane takes 15 minutes per patient. He can start patients at minutes 0, 15, ..., 135, which gives 10 patients.
3
Dr. Wayne takes 25 minutes per patient. He can start patients at minutes 0, 25, 50, 75, 100, 125, which gives 6 patients.
4
A start at minute 150 (11:30) is too late. The next start after each doctor's last one above falls at 150 or later (Ben 150, Kane 150, Wayne 150).
5
The maximum happens when the queue is never empty, so every doctor is busy from 9:00 onward. The total is 15+10+6=3115 + 10 + 6 = 31.
Why Not the Others
30: this drops one patient who can still be started before 11:30.
12: this is far too low.
15: this counts only Dr. Ben's patients.
Quick Check
The room assignment only changes which doctor is in which room. It does not change each doctor's capacity, so the answer is the same on every day.
Correct Answer: D

Q18. Optimization

MCQ · +3 / −1
The queue is never empty on one particular Saturday. Which of the three doctors would earn the maximum amount in consultation charges on that day?
  1. A) Dr. Ben
  2. B) Dr. Wayne
  3. C) Both Dr. Wayne and Dr. Kane
  4. D) Dr. Kane
Correct Answer
D) Dr. Kane
Step-by-step solution
📌 Core Concept
When the queue never empties, each doctor is busy from 9:00 to 11:30. Earnings are then (patients seen) × (fee) for each doctor.
🔢 Step-by-Step Solution
1
The clinic is open for 150 minutes. Counting patients started before 11:30, Ben sees 15, Kane sees 10 and Wayne sees 6.
2
Ben earns 15×100=150015 \times 100 = 1500.
3
Kane earns 10×200=200010 \times 200 = 2000.
4
Wayne earns 6×300=18006 \times 300 = 1800.
5
Kane's total of 2000 is the highest. It is 200 more than Wayne's and 500 more than Ben's.
6
On Saturday the room allocation is Room 1 Kane, Room 2 Wayne, Room 3 Ben. The allocation does not change the counts, because every doctor is busy throughout.
Why Not the Others
Dr. Ben: 1500 is the lowest of the three.
Dr. Wayne: 1800 is below Kane's 2000.
Both Wayne and Kane: their totals differ (1800 vs 2000), so they do not tie.
Quick Check
2000>1800>15002000 > 1800 > 1500.
Correct Answer: D

Q19. Optimization

MCQ · +3 / −1
Mr. Singh visited the clinic on Monday, Wednesday, and Friday of a particular week, arriving at 8:50 a.m. on each of the three days. His token number was 13 on all three days. On which day was he at the clinic for the maximum duration?
  1. A) Monday
  2. B) Wednesday
  3. C) Same duration on all three days
  4. D) Friday
Correct Answer
A) Monday
Step-by-step solution
📌 Core Concept
Simulate the queue. All three doctors start at 9:00, so token 13's start time is the same on every day. Which doctor sees token 13 depends on the room-number tie-break, and that decides the finish time.
🔢 Step-by-Step Solution
1
Patients 1, 2 and 3 enter at 9:00, one per doctor. After that each doctor takes a new patient as soon as the previous one ends.
2
Start times in minutes after 9:00:
Ben: 0, 10, 20, 30, 40, 50, ...
Kane: 0, 15, 30, 45, 60, ...
Wayne: 0, 25, 50, 75, ...
3
In order, tokens 1-3 start at 0, token 4 at 10, token 5 at 15, token 6 at 20, token 7 at 25, tokens 8-9 at 30 (Ben and Kane), token 10 at 40 and token 11 at 45.
4
Only Ben and Wayne are free at minute 50, so tokens 12 and 13 start at 9:50. The smaller-numbered room takes token 12, and the larger-numbered room takes token 13.
5
Monday has Ben in room 1 and Wayne in room 3. Ben takes token 12 and Wayne takes token 13. Wayne needs 25 minutes, so Mr. Singh leaves at 10:15. He arrived at 8:50, so he stays 85 minutes.
6
Wednesday has Wayne in room 1 and Ben in room 2. Wayne takes token 12 and Ben takes token 13. Ben needs 10 minutes, so Mr. Singh leaves at 10:00, a stay of 70 minutes.
7
Friday has Wayne in room 2 and Ben in room 3. Wayne takes token 12 and Ben takes token 13. Mr. Singh again leaves at 10:00, a stay of 70 minutes.
Why Not the Others
B: Wednesday gives only 70 minutes, since Ben sees him.
C: The durations are 85, 70 and 70 minutes, so they are not equal.
D: Friday gives only 70 minutes, the same as Wednesday and less than Monday.
Quick Check
Monday is the only day on which Wayne, the slowest doctor, sees token 13, so Monday is the longest stay.
Correct Answer: A

Q20. Optimization

MCQ · +3 / −1
On a slow Thursday, only two patients are waiting at 9a.m. After that two patients keep arriving at exact 15 minute intervals starting at 9:159:15 a.m. -- i.e. at 9:159:15 a.m., 9:309:30 a.m., 9:459:45 a.m. etc. Then the total duration in minutes when all three doctors are simultaneously free is
  1. A) 0
  2. B) 15
  3. C) 10
  4. D) 30
Correct Answer
A) 0
Step-by-step solution
📌 Core Concept
Thursday's rooms are Room 1 Wayne (25 min), Room 2 Ben (10 min) and Room 3 Kane (15 min). Track whether any patient is waiting each time a doctor frees up, using the rule that the smallest free room number is filled first.
🔢 Step-by-Step Solution
1
At 9:00 two patients are waiting. They enter Room 1 (Wayne, until 9:25) and Room 2 (Ben, until 9:10). Wayne is busy, so the three doctors are not all free.
2
At 9:15 two new patients arrive. Ben (free since 9:10) and Kane (free from the start) are free, so they take Rooms 2 and 3: Ben is busy until 9:25 and Kane until 9:30. Wayne is still busy until 9:25.
3
At 9:25 Wayne and Ben become free, but Kane is busy until 9:30.
4
At 9:30 Kane becomes free and two more patients arrive at the same instant. They enter Room 1 (Wayne, until 9:55) and Room 2 (Ben, until 9:40). Wayne is busy again, so there is no gap with all three free.
5
At 9:45 two patients arrive. Ben and Kane take them, Ben until 9:55 and Kane until 10:00, while Wayne is busy until 9:55.
6
At 10:00 two patients arrive as Kane becomes free, and the situation matches 9:30. The pattern repeats every 30 minutes.
7
In every cycle one doctor is always busy whenever the others are free. The three are never free together for any length of time, so the total is 0 minutes.
Why Not the Others
10, 15 and 30: each would need a stretch with no doctor occupied, but in every 30-minute cycle the arrivals are served just as the last busy doctor finishes.
Correct Answer: A
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CAT 2019 Slot 2 DILR at a Glance

Sets in this section8
Questions in this section32
Section time60 minutes
Convenor IIMIIM Kozhikode

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