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CAT 2023 Slot 1 DILR · Set 3 of 4Logical ReasoningHard DifficultyQ11–Q15 · 3 MCQ + 2 TITA

CAT 2023 Slot 1 DILR Set 3: Optimization (5 Questions) with Solutions

The Optimization set from the official CAT 2023 Slot 1 DILR paper: the caselet, all 5 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 2023 Slot 1 DILR · Questions 11–15
A visa processing office (VPO) accepts visa applications in four categories – US, UK, Schengen, and Others. The applications are scheduled for processing in twenty 15-minute slots starting at 9:00 am and ending at 2:00 pm. Ten applications are scheduled in each slot.
There are ten counters in the office, four dedicated to US applications, and two each for UK applications, Schengen applications and Others applications. Applicants are called in for processing sequentially on a first-come-first-served basis whenever a counter gets freed for their category. The processing time for an application is the same within each category. But it may vary across the categories. Each US and UK application requires 10 minutes of processing time. Depending on the number of applications in a category and time required to process an application for that category, it is possible that an applicant for a slot may be processed later.
On a particular day, Ira, Vijay and Nandini were scheduled for Schengen visa processing in that order. They had a 9:15 am slot but entered the VPO at 9:20 am. When they entered the office, exactly six out of the ten counters were either processing applications, or had finished processing one and ready to start processing the next.
Mahira and Osman were scheduled in the 9:30 am slot on that day for visa processing in the Others category.
The following additional information is known about that day.
1
All slots were full.
2
The number of US applications was the same in all the slots. The same was true for the other three categories.
3
50% of the applications were US applications.
4
All applicants except Ira, Vijay and Nandini arrived on time.
5
Vijay was called to a counter at 9:25 am.

In the exam this set competed with 3 others for 40 minutes. Budget about 10 minutes if you attempt it.

Q11. Optimization

TITA · +3 / 0
How many UK applications were scheduled on that day? (in numerial value)
Correct Answer
0
Step-by-step solution
📌 Core Concept
Per-slot counts are equal for each category, so the slot totals and the named applicants fix how many applications each category has per slot.
🔢 Step-by-Step Solution
1
There are 20 slots with 10 applications each, so 200 applications in all. Clue 3 makes 50% of them US, which is 100. Clue 2 makes the count per slot the same, so each slot has 5 US applications.
2
The other three categories (UK, Schengen, Others) therefore share the remaining 5 applications in every slot.
3
Ira, Vijay and Nandini were all scheduled in the 9:15 slot for Schengen. So every slot has at least 3 Schengen applications.
4
Mahira and Osman were both scheduled in the 9:30 slot for Others. So every slot has at least 2 Others applications.
5
Schengen (at least 3) plus Others (at least 2) already use all 5 spare places in each slot. That leaves 0 for UK, with exactly 3 Schengen and 2 Others per slot.
Quick Check
5 US + 3 Schengen + 2 Others + 0 UK = 10 per slot, so the slot is full as clue 1 requires.
Correct Answer: 0

Q12. Optimization

TITA · +3 / 0
What is the maximum possible value of the total time (in minutes, nearest to its integer value) required to process all applications in the Others category on that day? (in numerial value)
Correct Answer
200
Step-by-step solution
📌 Core Concept
The condition that exactly six counters are active at 9:20 limits the Others processing time. The total Others time is then the number of Others applications times that time.
🔢 Step-by-Step Solution
1
From the previous question, each slot has 5 US, 3 Schengen, 2 Others and 0 UK applications. The 20 slots give 40 Others applications.
2
At 9:20, the 4 US counters are busy because US demand is 5 per slot, which creates a queue. The 2 Schengen counters are busy or ready: Ira enters at 9:20 and the earlier Schengen applications are still being worked through. That makes six counters.
3
The two Others counters must therefore be idle at 9:20. The 9:15 slot's two Others applicants arrive on time (clue 4) and start at 9:15. They must be finished by 9:20, so the Others processing time tt satisfies t≤5t \le 5 minutes.
4
If tt were above 5, both Others counters would still be processing at 9:20. That would make eight active counters, contradicting the given six.
5
With t≤5t \le 5, the two applicants in each 15-minute slot never queue. The maximum total time is 40×5=20040 \times 5 = 200 minutes.
Quick Check
t=5t = 5 gives the 9:15 pair finishing exactly at 9:20, so those counters are idle. The total is 40×5=20040 \times 5 = 200.
Correct Answer: 200

Q13. Optimization

MCQ · +3 / −1
Which of the following is the closest to the time when Nandini's application process got over?
  1. A) 9:45 am
  2. B) 9:37 am
  3. C) 9:50 am
  4. D) 9:35 am
Correct Answer
A) 9:45 am
Step-by-step solution
📌 Core Concept
Schengen has 2 counters and 3 applications per slot. Vijay being called at 9:25 fixes the Schengen processing time, and Nandini's finish time follows from it.
🔢 Step-by-Step Solution
1
Let the Schengen time be ss minutes. The 9:00 slot has 3 Schengen applicants: two start at 9:00, and the third starts when a counter frees at 9:00+s9{:}00 + s.
2
Ira, Vijay and Nandini arrive at 9:20. Ira is called first. Vijay, the second of the three, is called at 9:25 (clue 5). That is when a counter frees up.
3
The third applicant of the 9:00 slot uses the counter that frees at time ss and finishes at 2s2s (after 9:00). Vijay takes that counter at 9:25, so 2s=252s = 25 and s=12.5s = 12.5 minutes.
4
The other counter has been free since 9:12:30, so it is ready when Ira enters at 9:20. Ira runs from 9:20 to 9:32:30.
5
Nandini takes that counter at 9:32:30 and finishes at 9:32:30+12.5=9:459{:}32{:}30 + 12.5 = 9{:}45.
Why Not the Others
9:37 am and 9:35 am are too early. Nandini cannot start before 9:32:30, when Ira finishes.
9:50 am would need a 17.5-minute processing time, which does not fit s=12.5s = 12.5.
Correct Answer: A

Q14. Optimization

MCQ · +3 / −1
Which of the following statements is false?
  1. A) The application process of Mahira was completed before Nandini's.
  2. B) The application process of Mahira started after Nandini's.
  3. C) The application process of Osman was completed before 9:45 am.
  4. D) The application process of Osman was completed before Vijay's.
Correct Answer
B) The application process of Mahira started after Nandini's.
Step-by-step solution
📌 Core Concept
Work out the per-slot category counts, then the Schengen processing time from Vijay's call at 9:25. That gives the start and finish times of Nandini, Mahira and Osman, which can be compared.
🔢 Step-by-Step Solution
1
All 20 slots are full, so there are 200 applications. Half are US, which is 5 per slot. The other 5 per slot are split equally across slots within each category.
2
Ira, Vijay and Nandini share one 9:15 Schengen slot, so Schengen has at least 3 per slot. Mahira and Osman share one 9:30 Others slot, so Others has at least 2 per slot. Together these use all 5 non-US places: Schengen 3, Others 2.
3
In the 9:00 slot, Schengen applicants A, B and C arrive on time. A and B start at 9:00 on the two Schengen counters. C starts when one of them frees, at ss minutes, where ss is the Schengen processing time.
4
The 9:20 clue counts six busy counters: four US and the two Schengen counters. One Schengen counter is processing C. The other is ready for its next applicant. Both Others counters are therefore idle at 9:20, so the 9:15 Others applications were already finished.
5
Ira enters at 9:20 and is called at once on the free counter. Vijay is called at 9:25, when C's counter frees. C started at ss and finished at 2s2s, so 2s=252s = 25 and s=12.5s = 12.5 minutes.
6
Ira runs 9:20 to 9:32:30 and Vijay runs 9:25 to 9:37:30. Nandini is called when Ira's counter frees. She starts at 9:32:30 and finishes at 9:45.
7
Mahira and Osman arrive on time at 9:30, and both Others counters are idle. Both start at 9:30. The Others time is at most 5 minutes, because those counters were clear by 9:20. Both therefore finish by 9:35.
Why Not the Others
A: Mahira finishes by 9:35, before Nandini's 9:45, so A is true.
B: Mahira starts at 9:30, before Nandini's 9:32:30, so B is false.
COsman finishes by 9before 9
C3545, so C is true.
D35, before Vijay's 9:37:30, so D is true.-
Quick Check
Only B contradicts the timeline, since 9:30 comes before 9:32:30.
Correct Answer: B

Q15. Optimization

MCQ · +3 / −1
When did the application processing for all US applicants get over on that day?
  1. A) 3:40 pm
  2. B) 2:05 pm
  3. C) 2:00 pm
  4. D) 2:25 pm
Correct Answer
B) 2:05 pm
Step-by-step solution
📌 Core Concept
US applications arrive 5 per 15-minute slot and are served by 4 counters at 10 minutes each. Tracking when the counters free up shows a repeating pattern, and the last slot's backlog sets the end time.
🔢 Step-by-Step Solution
1
There are 100 US applications, 5 per slot. Each counter does 1.5 applications per 15 minutes, so the 4 counters can handle 6 per slot. Demand is below capacity, so the queue does not keep growing.
2
Slot 1 (9:00): four applicants start at 9:00. The fifth waits until 9:10.
3
Slot 2 (9:15): the counters free up at 9:20, 9:25, 9:25 and 9:30, since three start at 9:15 and one waits for a counter.
4
Carrying this forward, the state at each slot start repeats every 15 minutes. Counters become free at 5 minutes after the slot start, then 0 minutes (at the slot start), and twice 5 minutes before it.
5
The last slot is 1:45 pm (285 minutes after 9:00). Counters are free at 1:40, 1:40, 1:45 and 1:50. Applicants start at 1:45, 1:45, 1:45 and 1:50 and finish at 1:55, 1:55, 1:55 and 2:00.
6
The fifth applicant then starts at 1:55 on the first counter to free and finishes at 2:05 pm.
Quick Check
The last applicant starts 20 minutes after the slot start of 1:45, giving 285+20=305285 + 20 = 305 minutes after 9:00, which is 2:05 pm.
Correct Answer: B
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CAT 2023 Slot 1 DILR at a Glance

Sets in this section4
Questions in this section20
Section time40 minutes
Convenor IIMIIM Lucknow

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