Maps, paths, distances and timings between connected points.
Appeared in 5 of 18 slots. Most recent: CAT 2024 Slot 2.
| Year | Slot 1 | Slot 2 | Slot 3 | Sets |
|---|---|---|---|---|
| CAT 2025 | 0 | 0 | 0 | 0 |
| CAT 2024 | 0 | 1 | 1 | 2 |
| CAT 2023 | 0 | 0 | 1 | 1 |
| CAT 2022 | 1 | 1 | 0 | 2 |
| CAT 2021 | 0 | 0 | 0 | 0 |
| CAT 2020 | 0 | 0 | 0 | 0 |
| Total | 1 | 2 | 2 | 5 |
The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes (3 of them, each in the shape of rectangles - shaded in the diagram) of a gated area. Different points on the walkway are indicated by letters (A through P) with distances being OP = 150 m, ON = MN = 300 m, ML = 400 m, EL = 200 m, DE = 400 m.
The following additional information about the facilities in the area is known.
1. The only entry/exit point is at C.
2. There are many residences within the gated area; all of them are located on the path AH and ML with four of them being at A, H, M, and L.
3. The post office is located at P and the bank is located at B.
Use coordinates: columns at (A, H, I, P), 150 (B, G, J, O), 450 (C, F, K, N), 750 (D, E, L, M); rows at (A–D), 600 (H–E), 400 (I–L), 0 (P–M).
Horizontal pieces: 150, 300, 300. Vertical pieces: 400 (top), 200 (middle), 400 (bottom). Diagonals: IG , OK .
Lakes: CDEF, GFKJ and KLMN.
One resident whose house is located at L, needs to visit the post office as well as the bank. What is the minimum distance (in ) he has to walk starting from his residence and returning to his residence after visiting both the post office and the bank?
Shortest legs: L→P via K, O: . P→B via I, G: . B→L (e.g. B–C–F–K–L): .
Total m.
One person enters the gated area and decides to walk as much as possible before leaving the area without walking along any path more than once and always walking next to one of the lakes. Note that he may cross a point multiple times. How much distance (in ) will he walk within the gated area?
Paths bordering a lake: the three rectangles' sides, total .
In this network every point has an even number of such paths (F and K have 4), so all of them can be walked once in a loop from C: m.
One resident takes a walk within the gated area starting from A and returning to A without going through any point (other than A) more than once. What is the maximum distance (in ) she can walk in this way?
The longest loop through A without repeating a point:
A–B–G–F–C–D–E–L–M–N–K–J–O–P–I–H–A m.
Visitors coming for morning walks are allowed to enter as long as they do not pass by any of the residences and do not cross any point (except C) more than once. What is the maximum distance (in ) that such a visitor can walk within the gated area?
Avoiding A, H, M, L and repeating no point except C, the longest loop is
C–B–G–J–I–P–O–N–K–F–E–D–C m.
The figure below shows a network with three parallel roads represented by horizontal lines R-A, R-B, and R-C and another three parallel roads represented by vertical lines V1, V2, and V3. The figure also shows the distance (in km) between two adjacent intersections. Six ATMs are placed at six of the nine road intersections. Each ATM has a distinct integer cash requirement (in Rs. Lakhs), and the numbers at the end of each line in the figure indicate the total cash requirements of all ATMs placed on the corresponding road. For example, the total cash requirement of the ATM(s) placed on road R-A is Rs. 22 Lakhs.
The following additional information is known.
1. The ATMs with the minimum and maximum cash requirements of Rs. 7 Lakhs and Rs. 15 Lakhs are placed on the same road.
2. The road distance between the ATM with the second highest cash requirement and the ATM located at the intersection of R-C and V3 is 12 km.
Six distinct values between 7 and 15 add to 62 (row totals ).
A row can't hold three ATMs () or one (max 15), so every row has exactly two.
V1 totals 15: it holds either just 15, or 7 and 8.
Case V1 : 7 is on 15's road, so RA has 15 (V1) and 7; that forces 7 at (RA, V3). Then , and each row sums to 20. Two fillings exist, and the 12-km condition (second highest = 12) keeps: RB: 12 (V2), 8 (V3); RC: 9 (V2), 11 (V3).
Case V1 : RA has 7 (V1) and 15 (V3). The 12-km condition keeps: RB: 8 (V1), 12 (V2); RC: 9 (V2), 11 (V3).
| Road | Arrangement 1 | Arrangement 2 |
|---|---|---|
| R-A | V1 15, V3 7 | V1 7, V3 15 |
| R-B | V2 12, V3 8 | V1 8, V2 12 |
| R-C | V2 9, V3 11 | V2 9, V3 11 |
Which of the following statements is correct?
(R-C, V2) holds 9 in both arrangements.
How many ATMs have cash requirements of Rs. 10 Lakhs or more?
15, 12 and 11 are at least 10: ATMs.
Which of the following two statements is/are DEFINITELY true?
Statement A: Each of R-A, R-B, and R-C has two ATMs.
Statement B: Each of V1, V2, and V3 has two ATMs.
Every row always has two ATMs; in Arrangement 1, V1 has only one. Only Statement A.
What best can be said about the road distance (in km) between the ATMs having the second highest and the second lowest cash requirements?
Second highest 12 is at (R-B, V2); second lowest 8 is at (R-B, V3) or (R-B, V1): 7 km or 4 km.
What is the number of ATMs whose locations and cash requirements can both be uniquely determined?
Same in both arrangements: 12 at (R-B, V2), 9 at (R-C, V2), 11 at (R-C, V3), i.e. ATMs.
A, B, C, D, E and F are the six police stations in an area, which are connected by streets as shown below. Four teams - Team 1, Team 2, Team 3 and Team 4 - patrol these streets continuously between 09:00 hrs. and 12:00 hrs. each day.
The teams need 30 minutes to cross a street connecting one police station to another. All four teams start from Station A at 09:00 hrs. and must return to Station A by 12:00 hrs. They can
also pass via Station A at any point on their journeys.
The following facts are known.
1. None of the streets has more than one team traveling along it in any direction at any point in time.
2. Teams 2 and 3 are the only ones in stations E and D respectively at 10:00 hrs.
3. Teams 1 and 3 are the only ones in station E at 10:30 hrs.
4. Teams 1 and 4 are the only ones in stations B and E respectively at 11:30 hrs.
5. Team 1 and Team 4 are the only teams that patrol the street connecting stations A and E.
6. Team 4 never passes through Stations B, D or F.
Streets: A–B, A–C, A–E, A–F, C–D, D–E, E–F (30 minutes each). Teams move every half hour, so each route has 6 steps from A back to A.
Team 4 avoids B, D and F, so it shuttles between A and C or A and E; it must be at E at 11:30.
Team 3 can't use A–E, so reaching D at 10:00 means going A–C–D; it is at E at 10:30 and must get back to A through D and C.
Team 2 reaches E at 10:00 through F. Team 1 is at E at 10:30 and at B at 11:30, so it goes E–A–B–A.
Street clashes at 9:00 force Team 4 onto A–E and Team 1 onto A–B.
| Time | 9:30 | 10:00 | 10:30 | 11:00 | 11:30 | 12:00 |
|---|---|---|---|---|---|---|
| Team 1 | B | A | E | A | B | A |
| Team 2 | F | E | F | A or E | F | A |
| Team 3 | C | D | E | D | C | A |
| Team 4 | E | A | C | A | E | A |
Which one among the following stations is visited the largest number of times?
E is visited 5 or 6 times, more than C (3), D (2) or F (3).
How many times do the teams pass through Station B in a day?
Only Team 1 passes B, twice: .
Which team patrols the street connecting Stations D and E at 10:15 hrs?
Between 10:00 and 10:30, Team 3 goes from D to E.
How many times does Team 4 pass through Station E in a day?
Team 4 is at E at 9:30 and 11:30: times.
How many teams pass through Station C in a day?
Teams 3 and 4 pass through C: teams.
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.)
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?
Next train from M after 8:05 is 8:10. Arrives N at 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?
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.
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?
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.
What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?
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.
What is the minimum number of trains that are required to provide the service in this city?
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 .
Every day a widget supplier supplies widgets from the warehouse (W) to four locations - Ahmednagar (A), Bikrampore (B), Chitrachak (C), and Deccan Park (D). The daily demand for widgets in each location is uncertain and independent of each other. Demands and corresponding probability values (in parenthesis) are given against each location (A, B, C, and D) in the figure below. For example, there is a 40% chance that the demand in Ahmednagar will be 50 units and a 60% chance that the demand will be 70 units. The lines in the figure connecting the locations and warehouse represent two-way roads connecting those places with the distances (in km) shown beside the line. The distances in both the directions along a road are equal. For example, the road from Ahmednagar to Bikrampore and the road from Bikrampore to Ahmednagar are both 6 km long.
Every day the supplier gets the information about the demand values of the four locations and creates the travel route that starts from the warehouse and ends at a location after visiting all the locations exactly once. While making the route plan, the supplier goes to the locations in decreasing order of demand. If there is a tie for the choice of the next location, the supplier will go to the location closest to the current location. Also, while creating the route, the supplier can either follow the direct path (if available) from one location to another or can take the path via the warehouse. If both paths are available (direct and via warehouse), the supplier will choose the path with minimum distance.
Distances between locations (direct road, or via W, whichever is shorter):
| A | B | C | D | |
|---|---|---|---|---|
| W | 5 | 10 | 12 | 2 |
| A | – | 6 | 17 | 7 |
| B | 6 | – | 4 | 12 |
| C | 17 | 4 | – | 6 |
Demands: A 50/70, B 40/60, C 70/100, D 30/50. The lowest total is 190; raising A, B or D adds 20 and raising C adds 30.
If the last location visited is Ahmednagar, then what is the total distance covered in the route (in km)?
For A to be last it must have the lowest demand: A 50, so B 60 and D 50 (tied with A). The intended route is W→C→B→D→A: km.
Note: by the tie rule, from B the closer of A (6) and D (12) is A, so A cannot actually be last. This question was dropped from evaluation; 35 is the official key.
If the total number of widgets delivered in a day is 250 units, then what is the total distance covered in the route (in km)?
needs A, B and D raised and C at 70: A 70, B 60, C 70, D 50.
A and C tie at 70; A is closer to W. Route W→A→C→B→D: km.
What is the chance that the total number of widgets delivered in a day is 260 units and the route ends at Bikrampore?
: C at 100 and two of A, B, D raised. Ending at B needs B lowest, so B 40 with A 70 and D 50.
Chance .
If the first location visited from the warehouse is Ahmednagar, then what is the chance that the total distance covered in the route is 40 km?
A first means A 70 and C 70 (A wins the tie). W→A→C costs .
From C: if B is visited before D, (total 38). If D before B, (total 40). D comes before B only when D 50 and B 40.
Chance .
If Ahmednagar is not the first location to be visited in a route and the total route distance is 29 km, then which of the following is a possible number of widgets delivered on that day?
A isn't first, so C is first (12 km). The remaining 17 km fits only C→B→A→D: .
So B 60, A 50, D 30 or 50 (A wins a tie from B), C 70 or 100. Totals: 210, 230, 240, 260. Of the options, 210.
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