DIRECTIONS for questions: Read the information given below and answer the question that follows.

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.

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.

The correct answer is Only II.

Let's break down the logic step-by-step based on the given rules for the employees and groups.

Step 1: Determine the Movements Between Groups

At the start of Quarter 1 (Q1), the Elite group consists of Asha (A), Bunty (B), and Chintu (C). Therefore, the Novice group consists of Dolly (D), Eklavya (E), and Falguni (F).

By the start of Quarter 4 (Q4), all three members of the original Elite group (A, B, and C) are in the Novice group. Because exactly one person is demoted from Elite to Novice at the end of each quarter, the three people demoted over Q1, Q2, and Q3 must be exactly A, B, and C in some order. Consequently, the three people promoted from Novice to Elite must be exactly D, E, and F in some order.

Step 2: Track Dolly (D) and Falguni (F)'s Constant Ratings

Condition 2 states that D and F received the exact same rating across all quarters.

Condition 3 tells us that in Q3, Lalu (who manages the Novice group) gave D a rating of 2. This reveals two things:

• D is in the Novice group during Q3. Since we established that D eventually moves to Elite, D must be promoted at the end of Q3.
• D's constant rating for all quarters is 2.

Since D and F are in the same group in Q1 and cannot receive the same rating, and D's rating is 2, F's constant rating must be either 1 or 3.

Step 3: Track Bunty (B) and Asha (A)'s Movements

Condition 3 states that B received a rating of 1 in Q2 from Lalu. This means B was in the Novice group in Q2, which implies B was the employee demoted from Elite at the end of Q1.

Condition 3 also states A received a rating of 1 in Q3 from Lalu. This means A was in the Novice group in Q3. Since B was already demoted in Q1, A must have been the one demoted at the end of Q2.

By elimination, C is demoted at the end of Q3.

Step 4: Determine Falguni (F)'s Constant Rating

In Q1, the Novice group has D, E, and F.

If F's constant rating is 1, then in Q1 Novice: F gets 1, D gets 2, so E must get 3. Since E has the highest score (3), E is promoted. Then in Q2, the Novice group consists of D, F, and B. Since D gets 2 and F gets 1 constantly, B would have to get 3. However, we are given that B received a rating of 1 in Q2! This creates a contradiction because both F and B would have a rating of 1 in the same group.

Therefore, F's constant rating cannot be 1. F's constant rating must be 3.

Step 5: Verify the Q2 Novice Promotions

In Q1 Novice: F gets 3, D gets 2, E gets 1. F has the highest score and is promoted to Elite at the end of Q1.

In Q2 Novice: The members are D, E, and B. We know D gets 2 and B gets 1. Thus, E gets 3.

Let's calculate their cumulative scores after Q2:

D's score is 2 + 2 = 4. E's score is 1 + 3 = 4.

E must be promoted at the end of Q2. Since D and E are tied with a score of 4, the tie-breaker rule applies: the employee with the higher rating in the latest quarter wins. E's Q2 rating (3) is greater than D's Q2 rating (2), so E is correctly promoted!

Step 6: Evaluate Q3 Novice and Bunty's Q1 Rating

In Q3 Novice: The members are A, B, and D. Lalu gives A a rating of 1 and D a rating of 2. Therefore, B gets 3.

Let's calculate the cumulative scores after Q3 for D and B (let B's Q1 rating be b₁):

D's score is 4 + 2 = 6. B's score is b₁ + 1 + 3 = b₁ + 4.

We know D is promoted at the end of Q3. If B's score was 6, they would tie, but B's Q3 rating (3) is higher than D's (2), meaning B would be promoted instead of D. For D to be promoted, B's score must be strictly less than 6.

$b_1+4<6$

This means b₁ < 2. Since ratings are integers from 1 to 3, b₁ must be 1. Bunty got a rating of 1 in Q1.

Step 7: Analyze Q2 Elite and Asha's Ratings

In Q1 Elite: The members are A, B, and C. Since B got 1, A and C must have received 2 and 3 in some order. This means Asha's Q1 rating is either 2 or 3.

In Q2 Elite: The members are A, C, and F. We know F's constant rating is 3. Therefore, A and C must receive 1 and 2 in some order.

Asha (A) is demoted at the end of Q2, meaning she must have the minimum score in the Elite group, or lose a tie-breaker. Let's test the possibilities for Asha's Q2 rating (let's call it a₂):

• If a₂ = 2, then C's Q2 rating is 1. Since C's Q1 rating was at least 2, C's total score would be at most 3 + 1 = 4. Asha's Q1 rating was at least 2, so her total score would be at least 2 + 2 = 4. If they tie at 4, Asha has a higher Q2 rating (2 vs 1), so C would be demoted. If C's score is lower (e.g., 3), C is strictly demoted. This contradicts the fact that Asha is the one demoted!
• If a₂ = 1, then C's Q2 rating is 2. If Asha's Q1 rating was 3, her score is 3 + 1 = 4 and C's is 2 + 2 = 4. They tie, but C has a higher Q2 rating (2 vs 1), so Asha is demoted (Valid). If Asha's Q1 rating was 2, her score is 2 + 1 = 3 and C's is 3 + 2 = 5, so Asha is directly demoted (Valid).

Therefore, Asha's Q2 rating MUST be 1. Statement II is necessarily true. However, Asha's Q1 rating could be 2 or 3, so Statement I is not necessarily true.