Question

What is the weight of the faculty quality parameter?

• A. 0.3
• B. 0.2
• C. 0.4
• D. 0.1
• A. 1
• B. 3
• C. 0
• D. 2

Answer 1

The Correct Option is D

Let \( a, b, c, d \) be the weights of parameters Faculty (F), Research (R), Placements (P), and Infrastructure (I) respectively.

Given inequalities:

1. \( 30a + 20b + 20c + 40d > 40a + 30b + 20c + 20d \)
2. \( 40a + 30b + 20c + 20d > 40a + 20b + 20c + 30d \)
3. \( 50a + 50b + 40c + 50d > 50a + 50b + 50c + 40d \)

From these, we derive:

• From (i): \( 2d > a + b \)
• From (ii): \( b > d \)
• From (iii): \( d > c \)

This implies the order: \( b > d > c \)

The weights \( a, b, c, d \) are from the set \( \{0.1, 0.2, 0.3, 0.4\} \).

Let’s consider possibilities. From \( b > d > c \), \( b = 0.4 \), \( d = 0.3 \), and \( c = 0.2 \). The remaining value \( a = 0.1 \).

Now check inequality (i):
\[ 2d = 0.6 > a + b = 0.1 + 0.4 = 0.5 \quad \text{✓} \]

Hence, the correct assignment is:

• \( a = 0.1 \)
• \( b = 0.4 \)
• \( c = 0.2 \)
• \( d = 0.3 \)

 

Answer: Weight of Faculty parameter (a) is 0.1

Answer 2

Let \( a, b, c, d \) be the weights of parameters F, R, P, and I respectively.

Given inequalities:

1. \( 30a + 20b + 20c + 40d > 40a + 30b + 20c + 20d \)
2. \( 40a + 30b + 20c + 20d > 40a + 20b + 20c + 30d \)
3. \( 50a + 50b + 40c + 50d > 50a + 50b + 50c + 40d \)

From the above, we deduce:

• From (i): \( 2d > a + b \)
• From (ii): \( b > d \)
• From (iii): \( d > c \)

This gives us the ranking: \( b > d > c \)

The weights \( a, b, c, d \) are all from the set \( \{0.1, 0.2, 0.3, 0.4\} \).

If \( d = 0.1 \) or \( 0.2 \), then \( 2d \) cannot be greater than \( a + b \). So the only valid values for \( d \) are 0.3 or 0.4. But since \( b > d \), we must have:

• \( b = 0.4 \)
• \( d = 0.3 \)

Now, using inequality (i):
\[ 2d > a + b \Rightarrow 0.6 > a + 0.4 \Rightarrow a < 0.2 \Rightarrow a = 0.1 \]

Remaining weight: \( c = 0.2 \)

Final weights:

• \( a = 0.1 \)
• \( b = 0.4 \)
• \( c = 0.2 \)
• \( d = 0.3 \)

According to the weighted scoring system, the number of colleges that received an AAA rating (presumably above a certain threshold score) is:

Answer: 3

Answer 3

Let \( a, b, c, d \) be the weights of parameters F, R, P, and I respectively.

Given constraints:

1. \( 30a + 20b + 20c + 40d > 40a + 30b + 20c + 20d \)
2. \( 40a + 30b + 20c + 20d > 40a + 20b + 20c + 30d \)
3. \( 50a + 50b + 40c + 50d > 50a + 50b + 50c + 40d \)

From these equations:

1. From (i): \( 2d > a + b \)
2. From (ii): \( b > d \)
3. From (iii): \( d > c \)

This implies the following weight ranking: \[ b > d > c \]

Weights are one of: \( \{0.1, 0.2, 0.3, 0.4\} \).

If \( d = 0.1 \) or \( d = 0.2 \), then \( 2d \) cannot exceed \( a + b \) unless both \( a \) and \( b \) are very small, which is not possible. So: \[ d = 0.3 \text{ or } 0.4 \] But since \( b > d \), we must have: \[ b = 0.4, \quad d = 0.3 \] Using inequality (i): \[ 2d > a + b \Rightarrow 0.6 > a + 0.4 \Rightarrow a < 0.2 \Rightarrow a = 0.1 \] Remaining weight is: \[ c = 0.2 \]

Final assigned weights:

• \( a = 0.1 \)
• \( b = 0.4 \)
• \( c = 0.2 \)
• \( d = 0.3 \)

Using these weights, the maximum weighted score among the eight colleges turns out to be:

Answer: 48

Answer 4

The correct answer is (C): 

Let \( a, b, c, d \) be the weights of parameters F, R, P, and I respectively.

Given inequalities:

1. \( 30a + 20b + 20c + 40d > 40a + 30b + 20c + 20d \)
2. \( 40a + 30b + 20c + 20d > 40a + 20b + 20c + 30d \)
3. \( 50a + 50b + 40c + 50d > 50a + 50b + 50c + 40d \)

Simplifying the inequalities:

1. \( 2d > a + b \)
2. \( b > d \)
3. \( d > c \)

So, the relative order of weights is: \[ b > d > c \quad \text{and from (i)} \quad 2d > a + b \]

Given possible weights: \( 0.1, 0.2, 0.3, 0.4 \) in some order.

Now, eliminate inconsistent combinations:

• If \( d = 0.1 \) or \( d = 0.2 \), then \( 2d \) can’t be greater than \( a + b \) unless both \( a \) and \( b \) are very small — not possible.
• So, \( d = 0.3 \) or \( 0.4 \). But from inequality (ii): \( b > d \), so \( d \) cannot be the highest.

Thus:

• \( b = 0.4 \)
• \( d = 0.3 \)

From inequality (i): \( 2d > a + b \Rightarrow 0.6 > a + 0.4 \Rightarrow a < 0.2 \Rightarrow a = 0.1 \)

Remaining weight: \( c = 0.2 \)

Final weights:

• \( a = 0.1 \)
• \( b = 0.4 \)
• \( c = 0.2 \)
• \( d = 0.3 \)

 

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Final Insight:

No college has a total score between 31 and 40 (inclusive), based on these weight combinations and the scoring data (assumed from context).

Answer: 0