If for some `x in R` the frequency distribution of the marks obtained by 20 statement in a test is :
`{:("Marks",2,3,5,7),("Frequency",(x + 1^(2)),(2x - 5),(x^(2) - 3x),x):}` then the mean of the marks is :

Options

A. 3.2
B. 3
C. 2.5
D. 2.8

Answer

To find the mean of the marks obtained by 20 students based on the given frequency distribution, we will follow these steps:

### Step 1: Set up the frequency distribution
The frequency distribution is given as:
- Marks: 2, 3, 5, 7
- Frequency: \( (x + 1^2), (2x - 5), (x^2 - 3x), x \)

### Step 2: Write the equation for total frequency
The total frequency must equal the total number of students, which is 20. Therefore, we set up the equation:
\[ (x + 1^2) + (2x - 5) + (x^2 - 3x) + x = 20 \]

### Step 3: Simplify the equation
Now, we simplify the left side:
\[ x + 1 + 2x - 5 + x^2 - 3x + x = 20 \]
Combining like terms:
\[ x^2 + (x + 2x - 3x + x) + (1 - 5) = 20 \]
This simplifies to:
\[ x^2 + 0x - 4 = 20 \]
Thus, we have:
\[ x^2 - 4 = 20 \]
Rearranging gives:
\[ x^2 = 24 \]

### Step 4: Solve for x
Now we solve for \( x \):
\[ x^2 - 24 = 0 \]
Factoring gives:
\[ (x - 4)(x + 4) = 0 \]
So, \( x = 4 \) or \( x = -4 \). Since frequency cannot be negative, we take \( x = 4 \).

### Step 5: Substitute x back into the frequency distribution
Now substitute \( x = 4 \) back into the frequency expressions:
- Frequency for 2: \( 4 + 1^2 = 5 \)
- Frequency for 3: \( 2(4) - 5 = 3 \)
- Frequency for 5: \( 4^2 - 3(4) = 4 \)
- Frequency for 7: \( 4 = 4 \)

So the frequencies are:
- Marks: 2, 3, 5, 7
- Frequencies: 5, 3, 4, 4

### Step 6: Calculate the mean
The mean is given by the formula:
\[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \]
Calculating \( \sum f_i \):
\[ 5 + 3 + 4 + 4 = 16 \]
Calculating \( \sum (x_i \cdot f_i) \):
\[ (2 \cdot 5) + (3 \cdot 3) + (5 \cdot 4) + (7 \cdot 4) = 10 + 9 + 20 + 28 = 67 \]

Now, we can find the mean:
\[ \text{Mean} = \frac{67}{16} = 4.1875 \]

### Final Answer
The mean of the marks is \( 4.1875 \).

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