A shopping mall has a large basement parking lot with parking slots painted in it along a single row. These slots are quite narrow; a compact car can fit in a single slot but an SUV requires two slots. When a car arrives, the parking attendant guides the car to the first available slot from the beginning of the row into which the car can fit.

For our purpose, cars are numbered according to the order in which they arrive at the lot. For example, the first car to arrive is given a number 1, the second a number 2, and so on. This numbering does not indicate whether a car is a compact or an SUV. The configuration of a parking lot is a sequence of the car numbers in each slot. Each single vacant slot is represented by letter V.

For instance, suppose cars numbered 1 through 5 arrive and park, where cars 1, 3 and 5 are compact cars and 2 and 4 are SUVs. At this point, the parking lot would be described by the sequence 1, 2, 3, 4, 5. If cars 2 and 5 now vacate their slots, the parking lot would now be described as 1, V, V, 3, 4. If a compact car (numbered 6) arrives subsequently followed by an SUV (numbered 7), the parking lot would be described by the sequence 1, 6, V, 3, 4, 7.

Answer the following questions INDEPENDENTLY of each other.

**Question 1: **

Initially cars numbered 1, 2, 3, and 4 arrive among which 1 and 4 are SUVs while 2 and 3 are compact cars. Car 1 then leaves, followed by the arrivals of car 5 (a compact car) and car 6 (an SUV). Car 4 then leaves. Then car 7 (an SUV) and car 8 (a compact car) arrive. At this moment, which among the following numbered car is parked next to car 3?

- 8
- 6
- 5
- 7

**Show Answer**

**4**

**Question 2: **

Suppose eight cars have arrived, of which two have left. Also suppose that car 4 is a compact and car 7 is an SUV. Which of the following is a POSSIBLE current configuration of the parking lot?

- V, 2, 3, 7, 5, 6, 8
- 8, 2, 3, V, 5, 6, 7
- 8, 2, 3, V, 6, 5, 7
- 8, 2, 3, V, 5, 7, 6

**Show Answer**

**2**

**Question 3: **

Suppose the sequence at some point of time is 4, 5, 6, V, 3. Which of the following is NOT necessarily true?

- Car 5 is a compact.
- Car 4 is a compact.
- Car 3 is an SUV.
- Car 1 is an SUV.

**Show Answer**

**3**

**Question 4: **

Suppose that car 4 is not the first car to leave and that the sequence at a time between the arrival of the car 7 and car 8 is V, 7, 3, 6, 5. Then which of the following statements MUST be false?

- Car 7 is a compact.
- Car 4 is an SUV.
- Car 6 is a compact.
- Car 2 is a compact.

**Show Answer**

**3**

**Question 1:**

The parking slots will be numbered as follows.

The cars that are parked next to car 3 are car 2 and car 7.

Option: 4**Question 2:**

When cars 5 and 6 are present, car 7 cannot take a place between them. So , 5,7,6 is not possible.

\(8,2,3, V, 5,6,7\) is possible in the following sequence of arrival of cars.

I. Cars 1,2,3,4,5,6 and 7 arrive one after the other.

II. Car 1 leaves

III. Car 8 arrives and is parked in the slot vacated by car 1

IV. Car 4 leaves.

V. \(2,3,7,5,6,8\) is not possible as car 7 (SUV) cannot be parked in place of car 4 (compact) while cars 3 and 5 are still parked.

\(8,2,3, V, 6,5,7\) is not possible as car 6 cannot be between a vacant slot and to the left of car 5 for the following reasons.

If no slot is vacant when car 6 is arrived, it would have been parked to the right of car 5.

If car 4 (compact) left by the time car 6 arrives it would either be parked in the slot vacated by car 4 or to the right of car 5.

Option: 2**Question 3:**

From the given arrangement it can be understood that cars 1, 2 and 3 arrived first and then cars 1 and 2 left before cars 4, 5 and 6 arrived. Now, there are four slots to the left of car 3. Hence, cars 1 and 2 are SUVs. Since three out of four slots are occupied by cars 4, 5 and 6, all these are compact cars. Whether car 3 is a compact car or an SUV, the given arrangement is possible.

Option: 3**Question 4:**

As cars 3 and 5 are still in their positions and that car 4 is not the first car to leave, at least one of cars 1 and 2 left before car 4. Let us consider the following cases. Both cars 1 and 2 left before car 4: In this case car 6, whether it is a compact car or an SUV, would have been parked in the parking lots vacated by cars 1 and 2. Hence, this case is not possible.

Only car 1 left before car 4: As car 6 is parked in the lot vacated by car 4, car 7 would have been parked in place of car 1. But the s lot vacated by car 1 is still vacant. Hence, this case is not possible.

Only 2 left before car 4: If the car 2 is an SUV, car 6 would have been parked in that lot. Hence, cars 2 and 7 are compact cars. Car 1 left after car 7 arrived.

Option: 3### CAT 2020 LRDI sets with Solutions

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