JEE MAIN - Mathematics (2020 - 4th September Morning Slot - No. 8)
The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations
are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :
5
3
7
9
Explanation
Let the two remaining observations be x and y.
$$ \because $$ $$\bar x = 10 = {{5 + 7 + 10 + 12 + 14 + 15 + x + y} \over 8}$$
$$ \Rightarrow x + y = 17$$ ....(1)
$$ \because $$ $${\mathop {\rm var}} (x) = 13.5 = {{25 + 49 + 100 + 144 + 196 + 225 + {x^2} + {y^2}} \over 8} - {(10)^2}$$
$$ \Rightarrow {x^2} + {y^2} = 169$$ ....(2)
From (1) and (2)
(x, y) = (12, 5) or (5, 12)
So $$\left| {x - y} \right| = 7$$
$$ \because $$ $$\bar x = 10 = {{5 + 7 + 10 + 12 + 14 + 15 + x + y} \over 8}$$
$$ \Rightarrow x + y = 17$$ ....(1)
$$ \because $$ $${\mathop {\rm var}} (x) = 13.5 = {{25 + 49 + 100 + 144 + 196 + 225 + {x^2} + {y^2}} \over 8} - {(10)^2}$$
$$ \Rightarrow {x^2} + {y^2} = 169$$ ....(2)
From (1) and (2)
(x, y) = (12, 5) or (5, 12)
So $$\left| {x - y} \right| = 7$$
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