CAT 2022 Slot 1 · DILR · Set 4: Metro Lines in a City

The full set and all 5 questions (Q16–Q20) from CAT 2022 Slot 1, with answers and solutions.

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 north-south 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.)

Question 16 · Caselet DI

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?

A
9:11 am
B
9:06 am
C
9:01 am
D
9:13 am

Pick an option, then check.

Answer & solution

Answer: Option A

Solving the set

Step 1 – Count the stations on each line from the map. M–N: M, 3 stations, R, 9 stations, S, 5 stations, N (20 links). P–Q: same pattern through T and V (20 links). A–B: A, 2, R, 3, T, 2, B (10 links). C–D: same through S and V (10 links).

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

Next M → N train after 8:05 is the 8:10. Trip time 61 minutes (Step 2) ⇒ arrival at N at 9:11 am.

Hence, the answer is option 1.

Question 17 · Caselet DI

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?

A
11:12 am
B
11:22 am
C
11:07 am
D
11:28 am

Pick an option, then check.

Answer & solution

Answer: Option A

Solving the set

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

Two routes from T to S:

• T → R (B → A train) then R → S (M → N train): B → A trains leave T at B + 13. The 10:15 train from B leaves T at 10:28 and reaches R at 10:43. The 10:30 train from M leaves R at 10:43 – she catches it – and reaches S at 11:12.

• T → V (P → Q train) then V → S (D → C train): the 10:20 train from P leaves T at 10:33 and reaches V at 11:02. D → C trains leave V at D + 13; the first after 11:02 is the 11:00 train from D (leaves V at 11:13), reaching S at 11:28.

Earliest = 11:12 am.

Hence, the answer is option 1.

Question 18 · Caselet DI

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?

A
11:39 pm
B
11:49 am
C
11:35 pm
D
11:43 pm

Pick an option, then check.

Answer & solution

Answer: Option A

Solving the set

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

She must go S → R westbound (N → M train) and then R → B southbound (A → B train).

• The last A → B train leaves A at midnight, leaves R at 0:13 and reaches B at 0:41 (before 1 am).

• She must reach R by 0:13. N → M trains reach R at N + 48: the 23:20 train from N reaches R at 0:08 (the 23:30 one would arrive at 0:18, too late). It leaves S at 23:20 + 19 = 23:39.

Latest time = 11:39 pm.

Hence, the answer is option 1.

Question 19 · Caselet DI

What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?

Answer & solution

Answer: 8

Solving the set

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

From Step 4, each A–B train completes a full cycle (A → B → A with rests) every 120 minutes, and a train must leave each terminal every 15 minutes, so 120/15 = 8 trains are needed. (At 6:00 am, 4 trains start from A and 4 from B; from 7:00 the arriving trains cover the departures.)

Hence, the answer is 8.

Question 20 · Caselet DI

What is the minimum number of trains that are required to provide the service in this city?

Answer & solution

Answer: 48

Solving the set

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

From the table in Step 4: A–B 8, C–D 8, M–N 16, P–Q 16. Total = 48.

Hence, the answer is 48.

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