CAT 2026 Slot 2 DILR: Set 2: Set 2 – Six Friends and Mobile Networks with Solutions

Answer: D — 150

1. Priya makes 150 minutes of calls to Vertex friends in total. Therefore Priya→Rahul cannot exceed 150. Rahul also receives exactly 150 minutes from Vertex friends, so the same upper bound follows independently.

2. The upper bound must be shown attainable; merely quoting a row total is not enough. Set the Vertex-only calls to Rahul→Vikram=25, Rahul→Ananya=150, Priya→Rahul=150, Vikram→Priya=100 and Ananya→Vikram=350; all other Vertex-only directed calls are zero.

3. Their outgoing totals are Rahul 175, Priya 150, Vikram 100, Ananya 350. Incoming totals are Rahul 150, Priya 100, Vikram 375, Ananya 150, matching the table.

4. Complete the cross-network calls with Kabir→Rahul 250, Kabir→Priya 275, Zoya→Vikram 100, Zoya→Ananya 100; Rahul→Kabir 50, Priya→Kabir 100, Vikram→Kabir 75 and Vikram→Zoya 125. Also Kabir→Zoya 100 and Zoya→Kabir 50. All other calls are zero.

5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.

G Strategy shortcut

Method: Bound + construction: min(Priya’s outgoing-to-Vertex 150, Rahul’s incoming-from-Vertex 150)=150. Exhibit a valid allocation attaining it.
Verification: 5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.

Answer: 50

1. Kabir and Zoya are the only AeroNet users. The group is closed and self-calls are forbidden.

2. Kabir receives 50 minutes from AeroNet. Those calls can only originate from Zoya.

3. Therefore Zoya→Kabir=50 minutes. No individual outgoing=incoming assumption is needed.

Detailed setup and verification:
Follow the direction carefully: the question asks for calls made by Zoya and received by Kabir. In a two-person network with self-calls excluded, Kabir’s incoming AeroNet cell identifies that directed pair uniquely. Kabir’s outgoing 100 instead describes the opposite direction, Kabir→Zoya.

Verification: set Zoya→Kabir=50 and Kabir→Zoya=100. Kabir’s listed incoming 50 and outgoing 100 are both met. There is no reason for the two directions to have the same duration. The remaining cross-network and Vertex-only calls do not alter either number.

G Strategy shortcut

Method: Two-person network: each incoming internal call must come from the other member.
Verification: 3. Therefore Zoya→Kabir=50 minutes. No individual outgoing=incoming assumption is needed.
Trap to avoid: Do not reverse the caller and receiver.

Answer: 525

1. Calls received from AeroNet by Vertex users total 250+275+100+100=725 minutes.

2. They are made by Kabir or Zoya, the only AeroNet users. Zoya contributes 200 minutes.

3. Kabir therefore contributes 725−200=525 minutes. This is network-flow conservation, not equality of an individual’s outgoing and incoming calls.

Detailed setup and verification:
Write the relevant four destination cells before subtracting anything: Rahul 250, Priya275, Vikram100 and Ananya100 incoming minutes from AeroNet. These count the same calls as the two AeroNet users’ outgoing-to-Vertex cells, just from the receivers’ side. Each minute is counted once on each side.

Thus Kabir + Zoya =725. Zoya’s 200 is given, so Kabir = 525. Verification:525+200=725. Do not use Kabir’s 225 incoming from Vertex: that is the reverse direction and is a separate collection of calls.

G Strategy shortcut

Method: Cross-network total 725 minus Zoya 200 gives Kabir 525.
Verification: 3. Kabir therefore contributes 725−200=525 minutes. This is network-flow conservation, not equality of an individual’s outgoing and incoming calls.
Trap to avoid: Balance only AeroNet→Vertex calls.

Answer: 350

1. Incoming-from-Vertex totals for the four Vertex friends sum to 150+100+375+150=775.

2. Every Vertex-to-Vertex minute must also occur once in their combined outgoing-to-Vertex total.

3. Known outgoing totals are 175+150+100=425, so Ananya contributes 775−425=350 minutes.

Detailed setup and verification:
Restrict both totals to calls whose caller and receiver are Vertex users. Incoming 150+100+375+150=775 counts every such minute once. Outgoing 175+150+100+Ananya must therefore also total 775. Calls to AeroNet are outside this balance and must not be mixed in.

Subtract the known 425 outgoing minutes to get 350. Check 175+150+100+350=775. This determines Ananya’s aggregate even though her individual pairwise calls need not be uniquely determined by the table.

G Strategy shortcut

Method: Conserve within-network calls: total incoming 775 minus other outgoing 425=350.
Verification: 3. Known outgoing totals are 175+150+100=425, so Ananya contributes 775−425=350 minutes.
Trap to avoid: Balance only Vertex→Vertex calls.