CAT 2024 Slot 2 — DILR Question 2
Answer the following questions based on the information given below.
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.
| Segment | Length (m) | Segment | Length (m) |
|---|---|---|---|
| AB, HG, IJ, PO | 150 | HI, GJ, FK, EL | 200 |
| BC, CD | 300 | GF, FE | 300 |
| JK, KL | 300 | ON, NM | 300 |
| AH, IP, BG, JO | 400 | CF, KN, DE, LM | 400 |
| GI (diagonal) | 250 | OK (diagonal) | 500 |
Coordinates (m), origin at P, taking the grid columns at x = 0, 150, 450, 750 and rows at y = 0, 400, 600, 1000: A(0,1000) B(150,1000) C(450,1000) D(750,1000); H(0,600) G(150,600) F(450,600) E(750,600); I(0,400) J(150,400) K(450,400) L(750,400); P(0,0) O(150,0) N(450,0) M(750,0). Lakes: C-D-E-F, G-F-K-J, K-L-M-N.
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 m) will he walk within the gated area?
Answer & solution
- A
3200
3800
- C
2800
- D
3000
Easy
"Always walking next to a lake" restricts you to the walkway segments that form the perimeters of the three rectangular lakes. "No path more than once" means an Euler-trail over those edges. Check the degrees: every junction has even degree, so a closed Euler circuit from C exists and can cover all lake-edges.
The three lakes and their bordering walkway segments (m):
| Lake | Bordering segments | Perimeter (m) |
|---|---|---|
| C-D-E-F | CD 300, DE 400, EF 300, CF 400 | 1400 |
| G-F-K-J | GF 300, FK 200, KJ 300, GJ 200 | 1000 |
| K-L-M-N | KL 300, LM 400, MN 300, KN 400 | 1400 |
All vertices have even degree. In the lake-edge subgraph, the corners F and K each have degree 4 and every other vertex degree 2, so a single closed trail (Euler circuit) covers every lake-edge exactly once, starting and ending at C.
Total lake-edge length. No edge is shared, so add the three perimeters.