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CAT 2025Slot 1DILRQues & Sol

Logical ReasoningHard

Data Set

At InnovateX, six employees, Asha, Bunty, Chintu, Dolly, Eklavya, and Falguni, were split into two groups of three each: Elite led by Manager Kuku, and Novice led by Manager Lalu.

At the end of each quarter, Kuku and Lalu handed out ratings to all members in their respective groups. In each group, each employee received a distinct integer rating from 1 to 3.

The score for an employee at the end of a quarter is defined as their cumulative rating from the beginning of the year. At the end of each quarter the employee in Novice with the highest score was promoted to Elite, and the employee in Elite with the minimum score was demoted to Novice. If there was a tie in scores, the employee with a higher rating in the latest quarter was ranked higher.

1. Asha, Bunty, and Chintu were in Elite at the beginning of Quarter 1. All of them were in Novice at the beginning of Quarter 4.
2. Dolly and Falguni were the only employees who got the same rating across all the quarters.
3. The following is known about ratings given by Lalu:
• Bunty received a rating of 1 in Quarter 2.
• Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.

Question 1

What was Eklavya's score at the end of Quarter 2?

Solution

Setup: We use the group movement rules (promotion/demotion based on cumulative scores), the constraint that A, B, C all end up in Novice by Q4, and the fixed-rating constraints for D and F to reconstruct the full rating table quarter by quarter.

Steps:

  1. Initial groups: Elite = {A, B, C}, Novice = {D, E, F}. By Q4, A, B, C are all in Novice, meaning each must have been demoted exactly once (and none re-promoted, since no employee changes groups more than once as confirmed by the logic).
  2. Bunty's demotion: B gets demoted after Q1 (since B receives rating 1 in Q2 from Lalu, meaning B is in Novice by Q2). For B to be demoted from Elite after Q1, B must have received the lowest rating in Elite in Q1 → B gets rating 1 in Q1.
  3. Dolly's fixed rating: D is in Novice all along. Since D gets rating 2 in Q3 (given), and D has the same rating every quarter → D gets 2 in Q1, Q2, Q3.
  4. Falguni's fixed rating: F is in Novice initially. F's rating must be consistent across all quarters. Since D takes rating 2, F must get 1 or 3. If F gets 1 every quarter, F's cumulative after Q1 = 1, after Q2 = 2 — but then E or F must be promoted to Elite after Q1, and the highest scorer in Novice after Q1 would be E (since F=1, D=2, E=3 or similar). If F=3 every quarter, F's cumulative after Q1 = 3, which is the highest in Novice → F gets promoted to Elite after Q1. This is consistent: F joins Elite after Q1 and stays (F has rating 3 every quarter, always competitive).
  5. Eklavya's promotion: With F promoted after Q1, E remains in Novice for Q2. In Novice during Q2: {B(demoted), D, E}. Ratings are 1, 2, 3 distinct. B gets 1 (given), D gets 2 (fixed) → E gets 3 in Q2. E's cumulative after Q2 = Q1 rating + 3. In Q1 Novice: {D=2, E=?, F=3} → E gets 1 in Q1. So E's cumulative after Q2 = 1 + 3 = 4.
  6. Verification: E has the highest cumulative in Novice after Q2 (E=4, B=1+1=2, D=2+2=4). Tie between E and D: E's Q2 rating (3) > D's Q2 rating (2) → E is promoted to Elite after Q2. ✓

Final Answer: 4

Question 2

How many employees changed groups more than once up to the beginning of Quarter 4?

Solution

Setup: We trace each employee's group membership across all four quarters using the promotion/demotion logic established in the full solution, checking if any employee moved groups (Elite↔Novice) more than once before Q4 begins.

Steps:

  1. Reconstruct group memberships:
    • Q1 start: Elite = {A, B, C}, Novice = {D, E, F}
    • After Q1: B demoted (lowest in Elite), F promoted (highest in Novice). Q2 start: Elite = {A, C, F}, Novice = {B, D, E}
    • After Q2: A demoted (lowest in Elite), E promoted (highest in Novice). Q3 start: Elite = {C, E, F}, Novice = {A, B, D}
    • After Q3: C demoted (lowest in Elite), and highest scorer in Novice joins Elite. But all of A, B, C must be in Novice at start of Q4. A and B are already in Novice. C gets demoted after Q3 → Q4 start: Elite = {E, F, ?}, Novice = {A, B, C}. The promoted Novice member after Q3 must be someone already accounted for — since A gets rating 1 in Q3 (given, Lalu gives A rating 1), B gets 3 (only remaining), D gets 2. Scores: A = prior + 1, B = prior + 3, D = prior + 2. B has highest Q3 rating among Novice but check cumulative — B is promoted back? No: all of A, B, C are in Novice at Q4 start, so no one from {A,B,C} is promoted after Q3. The promoted member must be D (or whoever has highest cumulative in Novice after Q3 that isn't A/B/C — but Novice in Q3 only contains A, B, D). Since A, B, C must all be in Novice at Q4 start, B must NOT be promoted → D gets promoted after Q3.
  2. Track each employee's moves:
    • A: Elite→Elite→Novice→Novice (changed once, after Q2)
    • B: Elite→Novice→Novice→Novice (changed once, after Q1)
    • C: Elite→Elite→Elite→Novice (changed once, after Q3)
    • D: Novice→Novice→Novice→Elite (changed once, after Q3)
    • E: Novice→Novice→Elite→Elite (changed once, after Q2)
    • F: Novice→Elite→Elite→Elite (changed once, after Q1)
  3. Count employees with more than one group change: Every employee changed groups exactly once. No employee changed groups more than once.

Final Answer: 0

Question 3

What was Bunty's score at the end of Quarter 3?

Solution

Setup: Bunty is in Novice from Q2 onward. We use Bunty's known ratings (1 in Q1 from Elite, 1 in Q2 from Lalu) and determine Q3 rating by elimination within the Novice group {A, B, D} in Q3.

Steps:

  1. Bunty's Q1 rating: B is in Elite in Q1 with A and C. B gets demoted after Q1, meaning B had the lowest cumulative score in Elite after Q1 (which equals Q1 rating since it's the first quarter). So B got rating 1 in Q1.
  2. Bunty's Q2 rating: Given directly — Lalu gives B a rating of 1 in Q2. B's cumulative after Q2 = 1 + 1 = 2.
  3. Bunty's Q3 rating: In Q3, Novice = {A, B, D}. Lalu gives ratings 1, 2, 3 (distinct). Given: A gets 1, D gets 2 (fixed rating) → B gets 3 in Q3.
  4. Bunty's score at end of Q3: Cumulative = Q1 + Q2 + Q3 = 1 + 1 + 3 = 5.

Final Answer: 5

Question 4

For how many employees can the scores at the end of Quarter 3 be determined with certainty?

Solution

Setup: We check whether each employee's complete rating sequence (Q1, Q2, Q3) can be uniquely determined. Some employees have ambiguity in Q1 ratings (specifically A and C in Elite during Q1 and Q2), while others are fully pinned.

Steps:

  1. Employees with certain scores by end of Q3:
    • Bunty (B): Q1=1, Q2=1, Q3=3 → Score = 5. ✓ Certain.
    • Dolly (D): Fixed rating 2 every quarter → Score = 6. ✓ Certain.
    • Falguni (F): Fixed rating 3 every quarter → Score = 9. ✓ Certain.
    • Eklavya (E): Q1=1 (Novice, by elimination with D=2, F=3), Q2=3 (Novice, by elimination with B=1, D=2), Q3=2 (Elite, determined by the need for C to be demoted — see below) → Score = 6. ✓ Certain.
  2. Why E's Q3 is certain: After Q2, Elite = {C, E, F}. Ratings given by Kuku: 1, 2, 3. C must be demoted after Q3 (C must be in Novice at Q4 start), so C gets the lowest cumulative. F has fixed rating 3, so F gets 3 in Q3. C and E share ratings 1 and 2. For C to be demoted, C must have lower cumulative than E after Q3. C's cumulative after Q2 and E's cumulative after Q2 differ by at most 1 point (depending on their Q1 and Q2 ratings). Analysis shows C must get 1 and E must get 2 in Q3 for C to be definitively demoted → E's Q3 rating = 2. ✓
  3. Employees with uncertainty:
    • Asha (A): In Elite during Q1 (gets 2 or 3) and Q2 (gets 1 necessarily). Two possible Q1 ratings → Q3 score (A is in Novice Q3, gets rating 1 given) = (2 or 3) + 1 + 1 = 4 or 5. ✗ Not certain.
    • Chintu (C): In Elite Q1 (gets 3 or 2, opposite of A) and Q2 (gets 2 or 1). Q3 gets 1 → Score = uncertain. ✗ Not certain.
  4. Count of employees with certain Q3 scores: B, D, F, E → 4 employees.

Final Answer: 4

Question 5

Which of the following statements is/are NECESSARILY true?
I. Asha received a rating of 2 in Quarter 1.
II. Asha received a rating of 1 in Quarter 2.

Solution

Setup: We evaluate both statements about Asha's ratings using the two valid scenarios that emerge from the Elite group analysis in Q1 and Q2, checking which statement holds in ALL valid scenarios (necessary truth) vs. only some.

Steps:

  1. Elite group in Q1: A, B, C are in Elite. B gets rating 1 (established by demotion rule). A and C share ratings 2 and 3 in some order. This gives two possible scenarios:
    • Scenario 1: A gets 2 in Q1, C gets 3 in Q1.
    • Scenario 2: A gets 3 in Q1, C gets 2 in Q1.
  2. Asha's Q2 rating — checking necessity: A must be demoted after Q2 (A must be in Novice by Q3). In Elite Q2: {A, C, F}. For A to be demoted, A must have the lowest cumulative score, OR tie with someone but have a lower Q2 rating.
    • Scenario 1: A's cumulative after Q1 = 2. For A to be demoted, A needs cumulative ≤ C's cumulative after Q2. C had 3 in Q1. If A gets 1 in Q2 → A cumulative = 3. C gets 2 in Q2 → C cumulative = 5. A < C → A demoted. ✓ If A gets 2 in Q2 → A cumulative = 4 = C's cumulative if C gets 1 → tie broken by Q2 rating: A(2) > C(1) → C demoted, not A. ✗ So A must get 1 in Q2 in Scenario 1.
    • Scenario 2: A's cumulative after Q1 = 3. C's cumulative after Q1 = 2. If A gets 1 in Q2 → A cumulative = 4. C gets 2 in Q2 → C cumulative = 4. Tie: A's Q2 rating (1) < C's Q2 rating (2) → A demoted. ✓ If A gets 2 → A cumulative = 5, C gets 1 → C cumulative = 3. A > C → C demoted, not A. ✗ So A must get 1 in Q2 in Scenario 2 as well.
  3. Evaluate Statement I (A gets 2 in Q1): True only in Scenario 1, not in Scenario 2 (where A gets 3). → NOT necessarily true.
  4. Evaluate Statement II (A gets 1 in Q2): True in BOTH Scenario 1 and Scenario 2. → Necessarily true.

Final Answer: Only II (Option D)