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

- 3a
^{2}– 32a + 84 = 0 - 3a
^{2}– 34a + 91 = 0 - 3a
^{2}– 23a + 44 = 0 - 3a
^{2}– 26a + 55 = 0

**Solution**

Numbers are 2, 3, a and 11

N = 4

standard Deviation, σ = 3.5

Mean of numbers, μ = (2+3+11+a)/4 = (16+a)/4

\( \sigma^2 = \dfrac{1}{N} \sum (x_i - \mu)^2 \)

3.5 × 3.5 × 4 × 16 = (8+a)^{2} + (4+a)^{2} + (16-3a)^{2} + (a-28)^{2}

= 64 + a^{2} + 16a + 16 + a^{2} + 8a + 256 + 9a^{2} - 96a + a^{2} + 784 - 56a

= 12a^{2} - 128a + 1120

196 = 3a^{2} - 32a + 280

3a^{2} + 16a + 84 = 0

**The correct answer is A.**