# If the standard deviation of the numbers 2, 3, a and 11 is 3.5,then which of the following is true ? Option 1) 3a2 − 26 a + 55 = 0   Option 2)  3a2 − 32a + 84 = 0   Option 3) 3a2 − 34a + 91 = 0   Option 4) 3a2 − 23a + 44 = 0

As we learnt in

Standard Deviation -

If x1, x2...xn are n observations then square root of the arithmetic mean of

$\dpi{100} \sigma = \sqrt{\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n}}$

$\dpi{100} \bar{}$

- wherein

where $\bar{x}$ is mean

$SD=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left (\frac{\sum x_{i}}{n} \right )^{2}}$

$\Rightarrow \frac{49}{4}=\frac{4+9+a^{2}+121}{4}\:-\left (\frac{16+a}{4} \right )^{2}$

$\Rightarrow 3a^{2}-32a+84=0$

Option 1)

3a2 − 26 a + 55 = 0

This option is incorrect.

Option 2)

3a2 − 32a + 84 = 0

This option is correct.

Option 3)

3a2 − 34a + 91 = 0

This option is incorrect.

Option 4)

3a2 − 23a + 44 = 0

This option is incorrect.

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