Question

What was Rini's score in the project?

• A. 0.60
• B. 0.75
• C. 0.40
• D. 0.50
• A. 62
• B. 66
• C. 80
• D. 68
• A. Only (i)
• B. Only (ii)
• C. Both (i) and (ii)
• D. Neither (i) nor (ii)

Answer 1

The Correct Option is A

Given:

• 3 female students: Amala, Koli, Rini
• 3 male students: Biman, Mathew, Shyamal
• Total score is a weighted average: project weight = \( x \), test weight = \( 1 - x \)
• Projects are completed in male-female pairs; scores are shared: 40, 60, 80
• Test scores are multiples of 10; min = 40, max = 80, average = 60; 4 distinct scores, 2 are 60

From the problem statement:

• Amala's project score is twice Koli's. This implies Amala: 80, Koli: 40, Rini: 60.
• Koli scored 20 more than Amala on the test.
• Shyamal is second highest in test, scored 2 more than Koli and 2 less than Amala’s aggregate.
• Amala has the highest aggregate score.

Possible test scores: 40, 50, 60 (x2), 70, 80

Scenario 1: Amala test = 40, Koli = 60, Shyamal = 70

Aggregates:

• Amala: \( 40(1 - x) + 80x \)
• Koli: \( 60(1 - x) + 40x \)
• Condition: Amala = Koli + 4 →

\[ 40(1 - x) + 80x = 60(1 - x) + 40x + 4 \\ \Rightarrow 60x = 24 \Rightarrow x = 0.4 \] Amala's Aggregate: \( 40(0.6) + 80(0.4) = 24 + 32 = 56 \)
Shyamal's Aggregate (minimum): \( 70(0.6) + 40(0.4) = 42 + 16 = 58 \)

Contradiction: Shyamal's aggregate (58)>Amala's aggregate (56) → Scenario 1 is invalid.

Scenario 2: Amala test = 60, Koli = 80, Shyamal = 70

• Amala: \( 60(1 - x) + 80x \)
• Koli: \( 80(1 - x) + 40x \)

\[ 60(1 - x) + 80x = 80(1 - x) + 40x + 4 \\ \Rightarrow 60 + 20x = 84 - 40x \Rightarrow 60x = 24 \Rightarrow x = 0.4 \]

Amala's aggregate: \( 60(0.6) + 80(0.4) = 36 + 32 = 68 \)
Shyamal's aggregate: 66. Project score calculation: \[ \frac{66 - 70 \times 0.6}{0.4} = \frac{66 - 42}{0.4} = \frac{24}{0.4} = 60 \]

Further deductions:

• Biman: test = 50, project = 40 (lowest total)
• Mathew's project score>Rini's → Mathew = 80, Rini = 60
• Rini's test score>Mathew's → Rini = 60, Mathew = 40

Final Table:

Student	Test Score (T)	Project Score (P)	Aggregate (0.6×T + 0.4×P)	Project Pair
Amala	60	80	68	Amala, Mathew
Koli	80	40	64	Koli, Biman
Rini	60	60	60	Rini, Shyamal
Biman	50	40	46	Biman, Koli
Mathew	40	80	56	Mathew, Amala
Shyamal	70	60	66	Shyamal, Rini

Answer: Rini's project score is 60.

Answer 2

To determine the weight of the test component, follow these steps based on the provided information:

1. Let \( w_p \) represent the project weight and \( w_t \) represent the test weight. The sum of these weights is 1:

\[ w_p + w_t = 1 \]

2. Amala's project score is double Koli's. Koli's test score is 20 points higher than Amala's. Amala has the highest aggregate score. This indicates the test component carries significant weight.

3. Shyamal's test score is 2 points higher than Koli's. Since Shyamal ranks second in the test, Amala must be the highest scorer in the test. Amala's highest aggregate score, despite potentially lower test scores than others, suggests the test component has considerable weight.

4. Amala's aggregate score is 2 points higher than Shyamal's. This implies the test score is heavily weighted to maintain Amala's top aggregate position, even if her test score is not the highest.

5. Given test scores range from 40 to 80, with an average of 60, we infer the following test scores:

\[ \text{Amala's test score} = 80,\quad \text{Koli's} = 60,\quad \text{Shyamal's} = 62 \]

6. Experiment with different weight combinations to satisfy the given conditions. The weights that best align with all logical constraints are:

\[ w_t = 0.60 \quad \text{and} \quad w_p = 0.40 \]

The weight of the test component is therefore \( \boxed{0.60} \).

Answer 3

To determine the maximum aggregate score, the following information is analyzed systematically:

1. Project scores range from a minimum of 40 to a maximum of 80, with an average of 60.
2. Test scores meet the following criteria:
   • They are multiples of 10.
   • There are four distinct scores.
   • Two scores are equal to the average, which is 60.
3. Amala's project score is twice Koli's project score. Koli's test score is 20 points higher than Amala's test score.
4. Amala achieved the highest aggregate score.
5. Shyamal has the second highest test score.
6. Shyamal's test score is 2 points higher than Koli's test score.
7. Shyamal's aggregate score is 2 points lower than Amala's aggregate score.
8. Biman has the second lowest test score and the lowest aggregate score.
9. Mathew's project score is higher than Rini's, but his test score is lower than Rini's.

Assuming Amala's project score is the maximum, 80, then Koli's project score is 40 (as Amala's project score = 2 × Koli's project score).

Let Koli's test score be represented by \( x \). Consequently, Amala's test score is \( x - 20 \).
Shyamal's test score is \( x + 2 \).

The following is established:

• Biman's test score is 50 (as 40 is the lowest, making 50 the second lowest).
• Two students achieved a test score of 60 (the average). Koli and Rini are the likely candidates.

Setting \( x = 60 \) yields: Koli's test score = 60, Amala's test score = 40, Shyamal's test score = 62. Now, calculating their aggregates with equal weighting:
Aggregate = \( 0.5 \times \text{Project Score} + 0.5 \times \text{Test Score} \)

• Amala's aggregate: \( 0.5 \times 80 + 0.5 \times 40 = 60 \)
• Koli's aggregate: \( 0.5 \times 40 + 0.5 \times 60 = 50 \)
• Shyamal's aggregate: \( 0.5 \times 72 + 0.5 \times 62 = 67 \) (assuming project score = 72)
• This implies Amala's aggregate should be 69, which is 2 more than Shyamal's.

Through the examination of valid score combinations within the given constraints, the maximum possible aggregate score is determined to be 68. Therefore, the correct answer is 68.

Answer 4

Given:

A course includes three female students (Amala, Koli, Rini) and three male students (Biman, Mathew, Shyamal).

The total score is a weighted average of project and test scores, with project weight $x$ and test weight $(1 - x)$, where $0<x<1$.

Each male-female pair completed one project, earning scores of 40, 60, or 80. Each student participated in one unique pair, receiving the same project score as their partner.

Test scores are: 40, 50, 60 (twice), 70, and 80. The unique scores are 40, 50, 60, 70, and 80.

Amala's project score is double Koli's. Koli's test score is 20 points higher than Amala's.

Derived project scores:

• Amala's project score = 80
• Koli's project score = 40
• Rini's project score = 60

Relationship between Koli's and Amala's test scores:

• Koli’s test score = Amala’s test score + 20

Possible test scores for Amala and Koli:

• If Amala's test score is 40, Koli's is 60.
• If Amala's test score is 50, Koli's is 70.
• If Amala's test score is 60, Koli's is 80.

Students	Test Scores	Project Scores
Amala	40 / 50 / 60	80
Koli	60 / 70 / 80	40
Rini	?	60

Additional information:

• Amala achieved the highest overall score.
• Shyamal had the second-highest test score (70).
• Shyamal's overall score was 2 less than Amala's and 2 more than Koli's.

This implies the following aggregate score relationships:

• Amala's aggregate = $A$
• Shyamal's aggregate = $A - 2$
• Koli's aggregate = $A - 4$

Case 1: Amala's test score = 40

• Amala's aggregate = $40(1 - x) + 80x = 40 + 40x$
• Koli's aggregate = $60(1 - x) + 40x = 60 - 20x$
• Equating Koli's aggregate to Amala's aggregate minus 4: $40 + 40x = (60 - 20x) + 4$
• Solving for $x$: $60x = 24 \Rightarrow x = 0.4$
• Amala's aggregate = $40 + 40(0.4) = 56$
• Shyamal's aggregate would be 58. This contradicts Amala having the highest overall score, so this case is rejected.

Case 2: Amala's test score = 60, Koli's test score = 80

• Amala's aggregate = $60(1 - x) + 80x = 60 + 20x$
• Koli's aggregate = $80(1 - x) + 40x = 80 - 40x$
• Equating Koli's aggregate to Amala's aggregate minus 4: $60 + 20x = (80 - 40x) + 4$
• Solving for $x$: $60x = 24 \Rightarrow x = 0.4$
• Amala's aggregate = $60 + 20(0.4) = 68$
• Shyamal's aggregate = $68 - 2 = 66$
• Calculate Shyamal's project score ($P$): $66 = 70(1 - 0.4) + 0.4P$
• $66 = 42 + 0.4P$
• $0.4P = 24 \Rightarrow P = 60$

Shyamal's project score is 60. Since Rini's project score is also 60, Shyamal and Rini form a project pair.

Remaining project pairs:

• Amala (project score 80) pairs with Mathew (project score 80).
• Koli (project score 40) pairs with Biman (project score 40).

Further information:

• Biman had the second-lowest test score (50).
• Biman had the lowest aggregate score.

Determining Rini's and Mathew's test scores:

• Mathew's project score is 80.
• Rini's project score is 60.
• Rini's test score is higher than Mathew's test score.
• Test scores available for Rini and Mathew are 40 and 60 (since 50 and 70 are assigned to Biman and Shyamal respectively, and 60 is also assigned to Amala in this case).
• Therefore, Rini's test score = 60, and Mathew's test score = 40.

Student	Test Score (T)	Project Score (P)	Aggregate ($0.6T + 0.4P$)	Project Pair
Amala	60	80	68	Amala & Mathew
Koli	80	40	64	Koli & Biman
Rini	60	60	60	Rini & Shyamal
Biman	50	40	46	Biman & Koli
Mathew	40	80	56	Mathew & Amala
Shyamal	70	60	66	Shyamal & Rini

Conclusion:

Mathew's test score is 40.

Final Answer: 40

Answer 5

To identify students on the same project team, analyze the provided data concerning scores and project team composition, where each team consists of one female and one male student.

• Project scores are uniform among team members. The score range is 40 (minimum) to 80 (maximum), with an average of 60.
• Amala's project score is twice Koli's, meaning Amala = 80 and Koli = 40.
• In the test, Koli's score was 20 points higher than Amala's.

Analysis Steps:

1. Determine test scores: Amala's test score (x) + Koli's test score (x+20) = 2 * 60 (average project score), assuming equal contribution to group scores.
2. Possible test score combinations satisfying the average score constraint: 40, 50, 60, 60, 70, 80.
3. Amala's test score + Koli's test score = 60. Amala achieved the highest overall score. If Amala's test score is 60, Koli's would be 60+20, which is not feasible given the possible scores.
4. Shyamal achieved the second-highest test score, 2 points more than Koli. If Koli's test score is 60, Shyamal's is 70, and Amala's is 80 (highest overall).
5. Re-evaluate Biman (second lowest test score and lowest overall score) and Mathew.
6. Biman's test score is lower than Mathew's, though Mathew's test score is lower than Rini's.

Based on the constraints and deductions:

Female	Male	Project
Amala	Mathew	80
Koli	Shyamal	40
Rini	Biman	60

Neither the pair (i) Amala & Biman nor (ii) Koli & Mathew were in the same groups, according to the provided information.

Conclusion: Neither pair (i) nor (ii) were on the same team.