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CAT 2023 Slot 3 DILR · Set 2 of 4Logical ReasoningHard DifficultyQ6–Q10 · 3 MCQ + 2 TITA

CAT 2023 Slot 3 DILR Set 2: Math Puzzles (5 Questions) with Solutions

The Math Puzzles set from the official CAT 2023 Slot 3 DILR paper: the caselet, all 5 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2023 Slot 3 DILR · Questions 6–10
There are only three female students- Amala, Koli and Rini and only three male students Biman, Mathew and Shyamal in a course. The course has two evaluation components, a project and a test. The aggregate score in the course is a weighted average of the two components, with the weights being positive and adding to 1.
The projects are done in groups of two, with each group consisting of a female and a male student. Both the group members obtain the same score in the project.
The following additional facts are known about the scores in the project and the test.
1
The minimum, maximum and the average of both project and test scores were identical, 40, 80 and 60, respectively.
2
The test scores of the students were all multiples of 10; four of them were distinct and the remaining two were equal to the average test scores.
3
Amala's score in the project was double that of Koli in the same, but Koli scored 20 more than Amala in the test. Yet Amala had the highest aggregate score.
4
Shyamal scored the second highest in the test. He scored two more than Koli, but two less than Amala in the aggregate.
5
Biman scored the second lowest in the test and the lowest in the aggregate.
6
Mathew scored more than Rini in the project, but less than her in the test.

In the exam this set competed with 3 others for 40 minutes. Budget about 10 minutes if you attempt it.

Q6. Math Puzzles

TITA · +3 / 0
What was Rini's score in the project?
Correct Answer
60
Step-by-step solution
📌 Core Concept
Project scores come in equal pairs (one female, one male per group), so the totals and the min/max/average fix the three group scores.
🔢 Step-by-Step Solution
1
Fact 1 gives a project total of 6×60=3606 \times 60 = 360, so the three group scores add to 180180.
2
The minimum is 40 and the maximum is 80, so the three group scores are 40,60,8040, 60, 80.
3
Clue 3 says Amala's project score is double Koli's. Among 40, 60 and 80 the only such pair is Amala =80=80 and Koli =40=40.
4
Amala and Koli are both female, so they are in different groups. The remaining female, Rini, must be in the group scoring 6060.
Quick Check
The groups score 80,60,4080, 60, 40, which add to 180 and match the minimum, maximum and average.
Correct Answer: 60

Q7. Math Puzzles

MCQ · +3 / −1
What was the weight of the test component?
  1. A) 0.500.50
  2. B) 0.600.60
  3. C) 0.400.40
  4. D) 0.750.75
Correct Answer
B) 0.600.60
Step-by-step solution
📌 Core Concept
Fix the test scores first, then use the aggregate relations in clues 3 and 4 to solve for the weight ww of the test.
🔢 Step-by-Step Solution
1
Project scores: Amala 8080, Koli 4040, Rini's group 6060. Clue 6 makes Mathew's project score more than Rini's 6060, so Mathew has 8080 and is Amala's partner.
2
Test scores total 360360. Two students score 6060, and the other four are distinct and include 40 and 80. The remaining two add to 240−120=120240-120=120, so they are 5050 and 7070. The scores are 40,50,60,60,70,8040, 50, 60, 60, 70, 80.
3
Clue 4 gives Shyamal 7070 (second highest). Clue 5 gives Biman 5050 (second lowest).
4
Amala, Koli, Mathew and Rini share 40,60,60,8040, 60, 60, 80. Koli is 20 above Amala in the test, so either Amala 4040 and Koli 6060, or Amala 6060 and Koli 8080.
5
Clue 4 gives Amala's aggregate == Koli's aggregate +4+4. Write the test weight as ww, so the project weight is 1−w1-w.
6
Case 1 (Amala 40, Koli 60): 80(1−w)+40w−[40(1−w)+60w]=40−60w=480(1-w)+40w-[40(1-w)+60w]=40-60w=4, so w=0.6w=0.6.
7
Case 2 (Amala 60, Koli 80): 80(1−w)+60w−[40(1−w)+80w]=40−60w=480(1-w)+60w-[40(1-w)+80w]=40-60w=4, so w=0.6w=0.6.
8
Both cases give the same weight.
Quick Check
With w=0.6w=0.6, the project weight is 0.40.4, and the two weights add to 1.
Correct Answer: B

Q8. Math Puzzles

MCQ · +3 / −1
What was the maximum aggregate score obtained by the students?
  1. A) 6868
  2. B) 8080
  3. C) 6666
  4. D) 6262
Correct Answer
A) 6868
Step-by-step solution
📌 Core Concept
With test weight 0.60.6 and project weight 0.40.4, compute each student's aggregate. Shyamal's aggregate is used to decide between the two test-score cases.
🔢 Step-by-Step Solution
1
From the weight question, w=0.6w=0.6 in both cases, with test scores 40,50,60,60,70,8040, 50, 60, 60, 70, 80. Shyamal has test 7070, Biman has 5050.
2
Case 1 (Amala test 40, Koli test 60): Koli's aggregate is 0.4(40)+0.6(60)=520.4(40)+0.6(60)=52 and Amala's is 5656. Clue 4 then makes Shyamal's aggregate 5454.
3
Shyamal's test is 7070, so his project score would have to satisfy 0.4p+42=540.4p+42=54, giving p=30p=30. This is impossible because the project scores are 40, 60 or 80.
4
Case 2 (Amala test 60, Koli test 80): Koli's aggregate is 0.4(40)+0.6(80)=640.4(40)+0.6(80)=64 and Amala's is 0.4(80)+0.6(60)=680.4(80)+0.6(60)=68. Shyamal's aggregate is 6666.
5
Then 0.4p+42=660.4p+42=66 gives p=60p=60, which is valid. Shyamal's project score is 60.
6
The other aggregates are Rini 0.4(60)+0.6(60)=600.4(60)+0.6(60)=60, Mathew 0.4(80)+0.6(40)=560.4(80)+0.6(40)=56 and Biman 0.4(40)+0.6(50)=460.4(40)+0.6(50)=46.
7
Biman is the lowest, as clue 5 requires, and Amala's 6868 is the highest.
Why Not the Others
80: this is only the maximum raw score, not an aggregate.
66: this is Shyamal's aggregate, which is below Amala's.
62: no student has this aggregate.
Correct Answer: A

Q9. Math Puzzles

TITA · +3 / 0
What was Mathew's score in the test?
Correct Answer
40
Step-by-step solution
📌 Core Concept
Once the multiset of test scores is known, the clues on ranks and on the aggregate fix which case holds and so decide Mathew's score.
🔢 Step-by-Step Solution
1
The test scores are 40,50,60,60,70,8040, 50, 60, 60, 70, 80. Shyamal has 7070 (second highest) and Biman has 5050 (second lowest).
2
Amala, Koli, Mathew and Rini share 40,60,60,8040, 60, 60, 80. Koli's test score is 20 above Amala's, so the pair is either (Amala 40, Koli 60) or (Amala 60, Koli 80).
3
If Amala has 40 and Koli has 60, Mathew and Rini get 60 and 80. Clue 6 then gives Mathew 6060 and Rini 8080.
4
This case fails. With weight 0.60.6 for the test, Shyamal's aggregate must be Koli's 5252 plus 2, which is 5454. His aggregate with test 70 would be 0.4p+420.4p+42, which gives p=30p=30, not a possible project score.
5
So Amala has 6060 and Koli has 8080. Mathew and Rini share 4040 and 6060, and clue 6 (Mathew lower than Rini in the test) gives Mathew 4040 and Rini 6060.
Quick Check
Mathew's aggregate is 0.4(80)+0.6(40)=560.4(80)+0.6(40)=56. This is above Biman's 4646, so Biman stays lowest, and below Amala's 6868.
Correct Answer: 40

Q10. Math Puzzles

MCQ · +3 / −1
Which of the following pairs of students were part of the same project team?
i
Amala and Biman ii) Koli and Mathew
  1. A) Both (i) and (ii)
  2. B) Only (ii)
  3. C) Neither (i) nor (ii)
  4. D) Only (i)
Correct Answer
C) Neither (i) nor (ii)
Step-by-step solution
📌 Core Concept
Each project group has one female and one male student, and the group scores are 80, 60 and 40. Matching the males to those scores gives the teams.
🔢 Step-by-Step Solution
1
The group scores are Amala's group 8080, Rini's group 6060 and Koli's group 4040.
2
Clue 6 says Mathew's project score is higher than Rini's 6060, so Mathew scored 8080. He is therefore Amala's partner.
3
Shyamal's aggregate is 6666 and his test score is 7070. This gives 0.4p+42=660.4p+42=66, so his project score is 6060 and he is Rini's partner.
4
Biman is left with Koli, and their project score is 4040.
5
Statement (i): Amala and Biman are not teammates, because Amala's partner is Mathew.
6
Statement (ii): Koli and Mathew are not teammates, because Koli's partner is Biman.
Why Not the Others
A, B and D each accept at least one of the pairs, but neither pair is a team.
Correct Answer: C
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CAT 2023 Slot 3 DILR at a Glance

Sets in this section4
Questions in this section20
Section time40 minutes
Convenor IIMIIM Lucknow

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