JEE MAIN - Mathematics (2019 - 10th January Morning Slot - No. 2)
The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is -
6 : 7
10 : 3
4 : 9
5 : 8
Explanation
Let two observations are x1 & x2
mean = $${{\sum {{x_i}} } \over 5} = 5 $$
$$\Rightarrow 1 + 3 + 8 + {x_1} + {x_2} = 25$$
$$ \Rightarrow {x_1} + {x_2} = 13$$ . . . . (1)
variance $$\left( {{\sigma ^2}} \right)$$ = $${{\sum {x_i^2} } \over 5} - 25 = 9.20$$
$$ \Rightarrow $$ $${\sum {x_i^2 = 171} }$$
$$ \Rightarrow $$ $$x_1^2 + x_2^2 = 97$$ . . . . . (2)
$$ \Rightarrow $$(x1 + x2)2 $$-$$ 2x1x2 = 97
$$ \Rightarrow $$ 169 - 2x1x2 = 97
or x1x2 = 36
$$ \therefore $$ x1 : x2 = 4 : 9
mean = $${{\sum {{x_i}} } \over 5} = 5 $$
$$\Rightarrow 1 + 3 + 8 + {x_1} + {x_2} = 25$$
$$ \Rightarrow {x_1} + {x_2} = 13$$ . . . . (1)
variance $$\left( {{\sigma ^2}} \right)$$ = $${{\sum {x_i^2} } \over 5} - 25 = 9.20$$
$$ \Rightarrow $$ $${\sum {x_i^2 = 171} }$$
$$ \Rightarrow $$ $$x_1^2 + x_2^2 = 97$$ . . . . . (2)
$$ \Rightarrow $$(x1 + x2)2 $$-$$ 2x1x2 = 97
$$ \Rightarrow $$ 169 - 2x1x2 = 97
or x1x2 = 36
$$ \therefore $$ x1 : x2 = 4 : 9
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