CAT 2023 Slot 2 · DILR · Set 3: Random Draw Competition over Five Days

The full set and all 5 questions (Q11–Q15) from CAT 2023 Slot 2, with answers and solutions.

Three participants - Akhil, Bimal and Chatur participate in a random draw competition for five days. Every day, each participant randomly picks up a ball numbered between 1 and 9. The number on the ball determines his score on that day. The total score of a participant is the sum of his scores attained in the five days. The total score of a day is the sum of participants’ scores on that day. The 2-day average on a day, except on Day 1, is the average of the total scores of that day and of the previous day. For example, if the total scores of Day 1 and Day 2 are 25 and 20, then the 2-day average on Day 2 is calculated as 22.5. Table 1 gives the 2-day averages for Days 2 through 5.

Table 1: 2-day averages for Days 2 through 5

Day 2

Day 3

Day 4

Day 5

15

15.5

16

17

Participants are ranked each day, with the person having the maximum score being awarded the minimum rank (1) on that day. If there is a tie, all participants with the tied score are awarded the best available rank. For example, if on a day Akhil, Bimal, and Chatur score 8, 7 and 7 respectively, then their ranks will be 1, 2 and 2 respectively on that day. These ranks are given in Table 2.

Table 2: Ranks of participants on each day

Day 1

Day 2

Day 3

Day 4

Day 5

Akhil

1

2

2

3

3

Bimal

2

3

2

1

1

Chatur

3

1

1

2

2

The following information is also known.

1. Chatur always scores in multiples of 3. His score on Day 2 is the unique highest score in the competition. His minimum score is observed only on Day 1, and it matches Akhil’s score on Day 4.

2. The total score on Day 3 is the same as the total score on Day 4.

3. Bimal’s scores are the same on Day 1 and Day 3.

Question 11 · Caselet DI

What is Akhil's score on Day 1?
  1. A
    5
  2. B
    7
  3. C
    6
  4. D
    8
Pick an option, then check.
Answer & solution

Answer: Option B

Solving the set

Step 1 – Day totals from the 2-day averages.

Day

2-day average

Equation

Day total

1

—

T1 + T2 = 30

15

2

15

T2 + T3 = 31

15

3

15.5

T3 + T4 = 32, T3 = T4

16

4

16

16

5

17

T4 + T5 = 34

18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day

Total

Ranks (A, B, C)

Possible scores (Akhil, Bimal, Chatur)

1

15

1, 2, 3

(7, 5, 3) or (8, 4, 3)

2

15

2, 3, 1

(4, 2, 9), (5, 1, 9) or (5, 4, 6)

3

16

2, 2, 1

(5, 5, 6)

4

16

3, 1, 2

(1, 9, 6), (2, 8, 6) or (3, 7, 6)

5

18

3, 1, 2

(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case

Akhil (D1–D5)

Bimal (D1–D5)

Chatur (D1–D5)

Totals A / B / C

1

7, 4, 5, 3, 4

5, 2, 5, 7, 8

3, 9, 6, 6, 6

23 / 27 / 30

2

7, 4, 5, 3, 5

5, 2, 5, 7, 7

3, 9, 6, 6, 6

24 / 26 / 30

3

7, 5, 5, 3, 4

5, 1, 5, 7, 8

3, 9, 6, 6, 6

24 / 26 / 30

4

7, 5, 5, 3, 5

5, 1, 5, 7, 7

3, 9, 6, 6, 6

25 / 25 / 30

This question

From Step 3, Day 1 is (7, 5, 3). Akhil scored 7.

Hence, the answer is option 2.

Question 12 · Caselet DI

Who attains the maximum total score?
  1. A
    Cannot be determined
  2. B
    Akhil
  3. C
    Bimal
  4. D
    Chatur
Pick an option, then check.
Answer & solution

Answer: Option D

Solving the set

Step 1 – Day totals from the 2-day averages.

Day

2-day average

Equation

Day total

1

—

T1 + T2 = 30

15

2

15

T2 + T3 = 31

15

3

15.5

T3 + T4 = 32, T3 = T4

16

4

16

16

5

17

T4 + T5 = 34

18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day

Total

Ranks (A, B, C)

Possible scores (Akhil, Bimal, Chatur)

1

15

1, 2, 3

(7, 5, 3) or (8, 4, 3)

2

15

2, 3, 1

(4, 2, 9), (5, 1, 9) or (5, 4, 6)

3

16

2, 2, 1

(5, 5, 6)

4

16

3, 1, 2

(1, 9, 6), (2, 8, 6) or (3, 7, 6)

5

18

3, 1, 2

(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case

Akhil (D1–D5)

Bimal (D1–D5)

Chatur (D1–D5)

Totals A / B / C

1

7, 4, 5, 3, 4

5, 2, 5, 7, 8

3, 9, 6, 6, 6

23 / 27 / 30

2

7, 4, 5, 3, 5

5, 2, 5, 7, 7

3, 9, 6, 6, 6

24 / 26 / 30

3

7, 5, 5, 3, 4

5, 1, 5, 7, 8

3, 9, 6, 6, 6

24 / 26 / 30

4

7, 5, 5, 3, 5

5, 1, 5, 7, 7

3, 9, 6, 6, 6

25 / 25 / 30

This question

From the four cases, Chatur always totals 30, while Akhil (23–25) and Bimal (25–27) are lower.

Hence, the answer is option 4.

Question 13 · Caselet DI

What is the minimum possible total score of Bimal?
Answer & solution

Answer: 25

Solving the set

Step 1 – Day totals from the 2-day averages.

Day

2-day average

Equation

Day total

1

—

T1 + T2 = 30

15

2

15

T2 + T3 = 31

15

3

15.5

T3 + T4 = 32, T3 = T4

16

4

16

16

5

17

T4 + T5 = 34

18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day

Total

Ranks (A, B, C)

Possible scores (Akhil, Bimal, Chatur)

1

15

1, 2, 3

(7, 5, 3) or (8, 4, 3)

2

15

2, 3, 1

(4, 2, 9), (5, 1, 9) or (5, 4, 6)

3

16

2, 2, 1

(5, 5, 6)

4

16

3, 1, 2

(1, 9, 6), (2, 8, 6) or (3, 7, 6)

5

18

3, 1, 2

(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case

Akhil (D1–D5)

Bimal (D1–D5)

Chatur (D1–D5)

Totals A / B / C

1

7, 4, 5, 3, 4

5, 2, 5, 7, 8

3, 9, 6, 6, 6

23 / 27 / 30

2

7, 4, 5, 3, 5

5, 2, 5, 7, 7

3, 9, 6, 6, 6

24 / 26 / 30

3

7, 5, 5, 3, 4

5, 1, 5, 7, 8

3, 9, 6, 6, 6

24 / 26 / 30

4

7, 5, 5, 3, 5

5, 1, 5, 7, 7

3, 9, 6, 6, 6

25 / 25 / 30

This question

Bimal's totals in the four cases are 27, 26, 26 and 25. Minimum = 25.

Hence, the answer is 25.

Question 14 · Caselet DI

If the total score of Bimal is a multiple of 3, what is the score of Akhil on Day 2?
  1. A
    Cannot be determined
  2. B
    5
  3. C
    6
  4. D
    4
Pick an option, then check.
Answer & solution

Answer: Option D

Solving the set

Step 1 – Day totals from the 2-day averages.

Day

2-day average

Equation

Day total

1

—

T1 + T2 = 30

15

2

15

T2 + T3 = 31

15

3

15.5

T3 + T4 = 32, T3 = T4

16

4

16

16

5

17

T4 + T5 = 34

18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day

Total

Ranks (A, B, C)

Possible scores (Akhil, Bimal, Chatur)

1

15

1, 2, 3

(7, 5, 3) or (8, 4, 3)

2

15

2, 3, 1

(4, 2, 9), (5, 1, 9) or (5, 4, 6)

3

16

2, 2, 1

(5, 5, 6)

4

16

3, 1, 2

(1, 9, 6), (2, 8, 6) or (3, 7, 6)

5

18

3, 1, 2

(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case

Akhil (D1–D5)

Bimal (D1–D5)

Chatur (D1–D5)

Totals A / B / C

1

7, 4, 5, 3, 4

5, 2, 5, 7, 8

3, 9, 6, 6, 6

23 / 27 / 30

2

7, 4, 5, 3, 5

5, 2, 5, 7, 7

3, 9, 6, 6, 6

24 / 26 / 30

3

7, 5, 5, 3, 4

5, 1, 5, 7, 8

3, 9, 6, 6, 6

24 / 26 / 30

4

7, 5, 5, 3, 5

5, 1, 5, 7, 7

3, 9, 6, 6, 6

25 / 25 / 30

This question

Only Case 1 gives Bimal a multiple of 3 (27). In Case 1, Akhil's Day 2 score is 4.

Hence, the answer is option 4.

Question 15 · Caselet DI

If Akhil attains a total score of 24, then what is the total score of Bimal?
Answer & solution

Answer: 26

Solving the set

Step 1 – Day totals from the 2-day averages.

Day

2-day average

Equation

Day total

1

—

T1 + T2 = 30

15

2

15

T2 + T3 = 31

15

3

15.5

T3 + T4 = 32, T3 = T4

16

4

16

16

5

17

T4 + T5 = 34

18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day

Total

Ranks (A, B, C)

Possible scores (Akhil, Bimal, Chatur)

1

15

1, 2, 3

(7, 5, 3) or (8, 4, 3)

2

15

2, 3, 1

(4, 2, 9), (5, 1, 9) or (5, 4, 6)

3

16

2, 2, 1

(5, 5, 6)

4

16

3, 1, 2

(1, 9, 6), (2, 8, 6) or (3, 7, 6)

5

18

3, 1, 2

(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case

Akhil (D1–D5)

Bimal (D1–D5)

Chatur (D1–D5)

Totals A / B / C

1

7, 4, 5, 3, 4

5, 2, 5, 7, 8

3, 9, 6, 6, 6

23 / 27 / 30

2

7, 4, 5, 3, 5

5, 2, 5, 7, 7

3, 9, 6, 6, 6

24 / 26 / 30

3

7, 5, 5, 3, 4

5, 1, 5, 7, 8

3, 9, 6, 6, 6

24 / 26 / 30

4

7, 5, 5, 3, 5

5, 1, 5, 7, 7

3, 9, 6, 6, 6

25 / 25 / 30

This question

Akhil's total is 24 in Cases 2 and 3. Bimal's total is 26 in both.

Hence, the answer is 26.

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