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: 

• There are 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 across pairs and are: 40, 60, and 80
• Test scores are multiples of 10; min = 40, max = 80, average = 60; 4 scores are distinct, 2 are 60

From the problem:

• Amala’s project score is twice that of Koli → Amala: 80, Koli: 40 → Rini: 60
• Koli scored 20 more than Amala in 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

Test score options: 40, 50, 60 (x2), 70, 80

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

Aggregate:

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

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

Contradiction: Shyamal > Amala → Case 1 invalid.

Case 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: \[ \frac{66 - 70 \times 0.6}{0.4} = \frac{66 - 42}{0.4} = \frac{24}{0.4} = 60 \]

Other deductions:

• Biman: test = 50, project = 40 (lowest total)
• Mathew > Rini in project → Mathew = 80, Rini = 60
• Rini > Mathew in test → 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: The score obtained by Rini in the project is 60.

Answer 2

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

1. Let's denote the weight of the project as \( w_p \) and the weight of the test as \( w_t \). Since the total weight must add up to 1, we have:

\[ w_p + w_t = 1 \] 

2. Amala's project score is double Koli's, and Koli scores 20 more than Amala in the test, yet Amala has the highest aggregate score. This implies the test score has a significant weight.

3. Shyamal scores 2 more points than Koli in the test. Since Shyamal is second highest in the test, Amala must be the highest. If Amala is also highest in aggregate, the test component must have considerable weight.

4. Amala's aggregate is 2 more than Shyamal's. So the test score must be heavily weighted to keep Amala's aggregate on top, despite her lower test score than Koli and Shyamal.

5. Given test scores range from 40 to 80, with an average of 60, we assume: \[ \text{Amala's test score} = 80,\quad \text{Koli's} = 60,\quad \text{Shyamal's} = 62 \]

6. Try various weights to match the constraints. The combination that fits all logic best is when: \[ w_t = 0.60 \quad \text{and} \quad w_p = 0.40 \]

Therefore, the weight of the test component is \( \boxed{0.60} \).

Answer 3

To determine the maximum aggregate score, we analyze the information step by step:

1. Project scores have minimum = 40, maximum = 80, and average = 60. 
2. Test scores are:
   • Multiples of 10
   • Four distinct scores
   • Two scores are equal to the average, i.e., 60
3. Amala's project score is double of Koli's, and Koli scored 20 more than Amala in the test.
4. Amala has the highest aggregate score.
5. Shyamal is second highest in test score.
6. Shyamal scored 2 more than Koli in the test.
7. Shyamal's aggregate score is 2 less than Amala's aggregate.
8. Biman is second lowest in test score and lowest in aggregate score.
9. Mathew scored more than Rini in project, but less in test.

Let us assume Amala’s project score is 80 (maximum), then Koli's is 40 (since Amala’s project = 2 × Koli’s). 

Let Koli’s test score be \( x \), so Amala’s test score is \( x - 20 \).
Shyamal’s test score is \( x + 2 \).

We know that:

• Biman's test score = 50 (second lowest, as 40 is lowest)
• There are two students with test score 60 (average): likely Koli and Rini

So we can take \( x = 60 \Rightarrow \) Koli's test = 60, Amala's test = 40, Shyamal's test = 62 Now, calculate their aggregates assuming equal weights:
Aggregate = \( 0.5 \times \text{Project} + 0.5 \times \text{Test} \)

• Amala: \( 0.5 \times 80 + 0.5 \times 40 = 60 \)
• Koli: \( 0.5 \times 40 + 0.5 \times 60 = 50 \)
• Shyamal: \( 0.5 \times 72 + 0.5 \times 62 = 67 \) if project = 72
• Then Amala’s aggregate would be 2 more than Shyamal = 69

By trying valid combinations within constraints, we find: Maximum possible aggregate score = 68 Hence, the correct answer is: 68.

Answer 4

Given: 
In a course, there are three female students: Amala, Koli, and Rini, and three male students: Biman, Mathew, and Shyamal.

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

Each male-female pair did one project together. Project scores are: 40, 60, and 80. Each student is in one unique pair, and both members of a pair get the same project score.

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

Given: Amala's project score is twice Koli's, and Koli scored 20 more than Amala in the test.

Thus:
Amala’s project score = 80,
Koli’s project score = 40,
So Rini’s project score = 60.

Also: Koli’s test score = Amala’s test score + 20

Let’s consider possible test scores for Amala: 40, 50, or 60 ⇒ then Koli’s test score = 60, 70, or 80

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

Given: Amala had highest overall score, Shyamal was second-highest in test (score = 70), and his overall score was 2 less than Amala’s and 2 more than Koli’s.

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

Case 1: Amala's test score = 40

Then:
Aggregate of Amala = $40(1 - x) + 80x = 40 + 40x$
Aggregate of Koli = $60(1 - x) + 40x = 60 - 20x$
Setting: $40 + 40x = 60 - 20x + 4$
$\Rightarrow 60x = 24 \Rightarrow x = 0.4$
Aggregate of Amala = $40 + 40(0.4) = 56$
Aggregate of Shyamal = 58 → contradicts "Amala highest", so reject this case.

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

Aggregate of Amala = $60(1 - x) + 80x = 60 + 20x$
Aggregate of Koli = $80(1 - x) + 40x = 80 - 40x$
Set: $60 + 20x = 80 - 40x + 4$
$\Rightarrow 60x = 24 \Rightarrow x = 0.4$
So: Aggregate of Amala = $60 + 20(0.4) = 68$
Shyamal’s aggregate = $66$, so find his project score:

Let Shyamal’s project score be $P$:
$66 = 70(1 - 0.4) + 0.4P$
$\Rightarrow 66 = 42 + 0.4P$
$\Rightarrow P = \frac{24}{0.4} = 60$

So Shyamal's project score = 60. Since Rini’s project score is also 60, Shyamal-Rini must be a project pair.

Remaining project pairs:
Amala (80) → Mathew (80)
Koli (40) → Biman (40)

Other info: Biman has second-lowest test score → 50
He has lowest aggregate.

Mathew’s project = 80 (Rini has 60)
Rini’s test > Mathew’s test → Rini = 60, Mathew = 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 got 40 in the test.

Final Answer: 40

Answer 5

To determine which pairs of students were part of the same project team, analyze the given data about scores and composition of project teams consisting of one female and one male student. 

• Project scores are identical for group members. Score range is 40 (min), 80 (max), average 60.
• Amala scored double Koli in the project, i.e., Amala = 80 and Koli = 40.
• Koli scored 20 more than Amala in the test.

Steps:

1. Identify test scores: Amala(x)+Koli(x+20)=60×2 because they contribute equally to group scores.
2. Average score constraints give possible test set: 40, 50, 60, 60, 70, 80.
3. Amala(test score)+Koli(test score)= 60, Amala had highest aggregate. If Test score Amala = 60, Koli = 60+20, not possible.
4. Shyamal scored second-highest test, two more than Koli. If Koli's test 60, Shyamal 70, Amala 80 (highest aggregate).
5. Re-evaluate Biman (second lowest test and lowest aggregate) and Mathew.
6. Biman (test score) < Mathew (though Mathew less in test than Rini).

Based on constraints and deductions:

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

Neither pair (i) Amala & Biman nor (ii) Koli & Mathew were in groups together as per facts given.

Result: Neither (i) nor (ii).