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.

Which of the following pairs of students were part of the same project team?

i) Amala and Biman

ii) Koli and Mathew

To determine which students were paired together, let us analyze the given information step-by-step.

Step 1: Determine the project and test scores
1. From Fact 1, the minimum, maximum, and average of both project and test scores are 40, 80, and 60, respectively.
2. There are 3 pairs of students, with each pair consisting of one female and one male student. Since both members of a group obtain the same project score, let the three group project scores be $P_1$, $P_2$, and $P_3$. The sum of all project scores is:

$2(P\_1\; +\; P\_2\; +\; P\_3)\; =\; 6\; \backslash times\; 60\; =\; 360\; \backslash implies\; P\_1\; +\; P\_2\; +\; P\_3\; =\; 180$

Given that the minimum is 40 and the maximum is 80, we must have:

$P\_1\; =\; 40,\; P\_2\; =\; 60,\; P\_3\; =\; 80$

Thus, the project scores of the three teams are exactly 40, 60, and 80.
3. From Fact 2, the test scores are all multiples of 10. Four of them are distinct, and two are equal to the average test score (60). The sum of all test scores is:

$6\; \backslash times\; 60\; =\; 360$

Since two scores are 60, and the rest are distinct multiples of 10 between 40 and 80, the set of test scores must be:

$\backslash \{40,\; 50,\; 60,\; 60,\; 70,\; 80\backslash \}$

Step 2: Assign project scores to the female students
From Fact 3, Amala's project score is double that of Koli. Since the project scores are 40, 60, and 80, we have:

$P(\backslash text\{Koli\})\; =\; 40$

$P(\backslash text\{Amala\})\; =\; 80$

This leaves Rini with the remaining project score:

$P(\backslash text\{Rini\})\; =\; 60$

Step 3: Analyze the aggregate scores and weights
Let $w_p$ be the weight of the project and $w_t$ be the weight of the test, where $w_p+w_t=\; 1$.
From Fact 3, Koli scored 20 more than Amala in the test:

$T(\backslash text\{Koli\})\; =\; T(\backslash text\{Amala\})\; +\; 20$

From Fact 4, Shyamal scored the second highest in the test. The second highest test score in the set $\{40,50,60,60,70,80\}$ is 70. Thus:

$T(\backslash text\{Shyamal\})\; =\; 70$

Additionally, Shyamal's aggregate score was 2 more than Koli's, and 2 less than Amala's:

$A(\backslash text\{Amala\})\; =\; A(\backslash text\{Shyamal\})\; +\; 2$

$A(\backslash text\{Shyamal\})\; =\; A(\backslash text\{Koli\})\; +\; 2$

Therefore:

$A(\backslash text\{Amala\})\; =\; A(\backslash text\{Koli\})\; +\; 4$

We substitute the aggregate formula $A=w_{p}P+w_{t}T$:

$w\_p\; P(\backslash text\{Amala\})\; +\; w\_t\; T(\backslash text\{Amala\})\; =\; w\_p\; P(\backslash text\{Koli\})\; +\; w\_t\; T(\backslash text\{Koli\})\; +\; 4$

Substitute the known values: $P\left(\text{Amala}\right)=80$, $P\left(\text{Koli}\right)=40$, and $T\left(\text{Koli}\right)=T\left(\text{Amala}\right)+20$:

$80\; w\_p\; +\; T(\backslash text\{Amala\})\; w\_t\; =\; 40\; w\_p\; +\; (T(\backslash text\{Amala\})\; +\; 20)\; w\_t\; +\; 4$

$40\; w\_p\; =\; 20\; w\_t\; +\; 4$

Since $w_t=1-w_p$:

$40\; w\_p\; =\; 20\; (1\; -\; w\_p)\; +\; 4\; \backslash implies\; 60\; w\_p\; =\; 24\; \backslash implies\; w\_p\; =\; 0.4\; \backslash text\{\; and\; \}\; w\_t\; =\; 0.6$

Step 4: Find the test and project scores of the remaining students
Using $w_p=0.4$ and $w_t=0.6$:

$A(\backslash text\{Amala\})\; =\; 0.4(80)\; +\; 0.6\; T(\backslash text\{Amala\})\; =\; 32\; +\; 0.6\; T(\backslash text\{Amala\})$

$A(\backslash text\{Koli\})\; =\; 0.4(40)\; +\; 0.6\; T(\backslash text\{Koli\})\; =\; 16\; +\; 0.6\; (T(\backslash text\{Amala\})\; +\; 20)\; =\; 28\; +\; 0.6\; T(\backslash text\{Amala\})$

$A(\backslash text\{Shyamal\})\; =\; A(\backslash text\{Koli\})\; +\; 2\; =\; 30\; +\; 0.6\; T(\backslash text\{Amala\})$

Using Shyamal's test score of 70:

$A(\backslash text\{Shyamal\})\; =\; 0.4\; P(\backslash text\{Shyamal\})\; +\; 0.6\; (70)\; =\; 0.4\; P(\backslash text\{Shyamal\})\; +\; 42$

Equating both expressions for Shyamal's aggregate:

$0.4\; P(\backslash text\{Shyamal\})\; +\; 42\; =\; 30\; +\; 0.6\; T(\backslash text\{Amala\})\; \backslash implies\; P(\backslash text\{Shyamal\})\; =\; 1.5\; T(\backslash text\{Amala\})\; -\; 30$

Since $P\left(\text{Shyamal}\right)$ must be one of the project scores (40, 60, or 80), and $T\left(\text{Amala}\right)$ is one of the test scores, we test the values:
- If $T\left(\text{Amala}\right)=60$, then $P\left(\text{Shyamal}\right)=1.5\left(60\right)-30=60$, which is valid.
Thus:

$T(\backslash text\{Amala\})\; =\; 60,\; \backslash quad\; T(\backslash text\{Koli\})\; =\; 80,\; \backslash quad\; P(\backslash text\{Shyamal\})\; =\; 60$

Step 5: Identify the project teams
- Since $P\left(\text{Shyamal}\right)=60$ and Rini is the female student with a project score of 60 ($P\left(\text{Rini}\right)=60$), Rini and Shyamal must be on the same project team.
- From Fact 6, Mathew scored more than Rini in the project:

$P(\backslash text\{Mathew\})\; >\; P(\backslash text\{Rini\})\; =\; 60\; \backslash implies\; P(\backslash text\{Mathew\})\; =\; 80$

Since Amala is the female student with a project score of 80 ($P\left(\text{Amala}\right)=80$), Amala and Mathew must be on the same project team.
- By elimination, the third team must be Koli and Biman (project score of 40).

Conclusion:
- Pair i) Amala and Biman: Incorrect (Amala is paired with Mathew, Biman is paired with Koli).
- Pair ii) Koli and Mathew: Incorrect (Koli is paired with Biman, Mathew is paired with Amala).
Therefore, the correct option is Neither i) nor ii).