If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to : [2025]

Question

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to : [2025]

Smart Achievers · Accepted Answer

(2)

Mean, $\bar{x} = 6 + 4 + a + 8 + b + 12 + 10 + 13$

$\Rightarrow 9 = \frac{53 + a + b}{8} \Rightarrow a + b = 72 - 53 = 19$

$∴$ a + b = 19 ... (i)

Variance, $σ^{2} = \frac{{(6 - 9)}^{2} + {(4 - 9)}^{2} + {(a - 9)}^{2} + {(8 - 9)}^{2} + {(b - 9)}^{2} + {(12 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{8}$

$\Rightarrow 9.25 \times 8 = 9 + 25 + 1 + 9 + 1 + 16 + {(a - 9)}^{2} + {(b - 9)}^{2}$

$\Rightarrow 74 - 61 = {(a - 9)}^{2} + {(19 - a - 9)}^{2}$ [Using (i)]

$\Rightarrow 13 = {(a - 9)}^{2} + {(10 - a)}^{2}$

$\Rightarrow a^{2} + 81 - 18 a + a^{2} + 100 - 20 a = 13$

$\Rightarrow 2 a^{2} - 38 a = 13 - 181$

$\Rightarrow a^{2} - 19 a + 84 = 0$

$\Rightarrow (a - 12) (a - 7) = 0$

$\Rightarrow a = 12 or 7 \Rightarrow b = 7 or 12$

$∴$ a + b + ab = 12 + 7 + 84 = 103.

Solution

Q.
If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to : [2025]

1 106

2 103

3 100

4 105

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