If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true ?

(1) `3a^2-26 a+55=0`
(2) `3a^2-32 a+84=0`
(3) `3a^2-34 a+91=0`
(4) `3a^2-23 a+44=0`

Options

A. `3a^(2)-32a+84=0`
B. `3a^(2)-34a+91=0`
C. `3a^(2)-23a+44=0`
D. `3a^(2)-26a+55=0`

Explanation

To solve the problem, we need to find the value of \( a \) such that the standard deviation of the numbers 2, 3, \( a \), and 11 is equal to 3.5.

### Step 1: Write the formula for standard deviation
The formula for standard deviation \( \sigma \) of a set of numbers \( x_1, x_2, \ldots, x_n \) is given by:

\[ \sigma = \sqrt{\frac{\sum (x_i^2)}{n} - \left(\frac{\sum x_i}{n}\right)^2} \]

where \( n \) is the number of observations.

### Step 2: Calculate the mean and the sum of squares
In our case, we have the numbers 2, 3, \( a \), and 11.

- The number of observations \( n = 4 \).
- The sum of the numbers is:

\[ 2 + 3 + a + 11 = 16 + a \]

- The sum of the squares of the numbers is:

\[ 2^2 + 3^2 + a^2 + 11^2 = 4 + 9 + a^2 + 121 = 134 + a^2 \]

### Step 3: Set up the equation for standard deviation
According to the problem, the standard deviation is given as 3.5. Thus, we can set up the equation:

\[ 3.5 = \sqrt{\frac{134 + a^2}{4} - \left(\frac{16 + a}{4}\right)^2} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides, we get:

\[ (3.5)^2 = \frac{134 + a^2}{4} - \left(\frac{16 + a}{4}\right)^2 \]

Calculating \( (3.5)^2 \):

\[ 12.25 = \frac{134 + a^2}{4} - \frac{(16 + a)^2}{16} \]

### Step 5: Simplify the equation
Now, we need to simplify the right-hand side:

\[ \frac{(16 + a)^2}{16} = \frac{256 + 32a + a^2}{16} = 16 + 2a + \frac{a^2}{16} \]

Substituting this back into the equation gives:

\[ 12.25 = \frac{134 + a^2}{4} - \left(16 + 2a + \frac{a^2}{16}\right) \]

### Step 6: Clear the fractions
Multiply through by 16 to eliminate the denominators:

\[ 16 \times 12.25 = 4(134 + a^2) - 16(16 + 2a + \frac{a^2}{16}) \]

Calculating \( 16 \times 12.25 = 196 \):

\[ 196 = 4(134 + a^2) - (256 + 32a + a^2) \]

### Step 7: Expand and simplify
Expanding the left side:

\[ 196 = 536 + 4a^2 - 256 - 32a - a^2 \]

Combining like terms:

\[ 196 = 280 + 3a^2 - 32a \]

### Step 8: Rearranging the equation
Rearranging gives us:

\[ 3a^2 - 32a + 84 = 0 \]

### Conclusion
Thus, the correct option is:

**(2) \( 3a^2 - 32a + 84 = 0 \)**