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Five real numbers $$x_1, x_2, x_3, x_4, x_5$$ are such that: $$\sqrt {x_1-1}+2\sqrt {x_2-4}+3\sqrt {x_3-9}+4\sqrt {x_4-16}+5\sqrt {x_5-25}=\dfrac {x_1+x_2+x_3+x_4+x_5}{2}$$
The value of $$\dfrac {x_1+x_2+x_3+x_4+x_5}{2}$$ is


A
Not uniquely determined
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B
55
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C
110
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D
210
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Solution

The correct option is D $$55$$
$$(x_1-2\sqrt {x_1-1})+(x_2-4\sqrt {x_2-4})+(x_3-6\sqrt {x_3-9})+(x_4-8\sqrt {x_4-16})+(x_5-8\sqrt {x_5-25})=0$$
$$\Rightarrow (\sqrt {x_1-1}-1)^2+(\sqrt {x_2-4}-2)^2+(\sqrt {x_3-9}-3)^2+(\sqrt {x_4-16}-4)^2+(\sqrt {x_5-25}-5)^2$$
Now, $$\sqrt {x_1-1}-1=0, \sqrt {x_2-4}-2=0, \sqrt {x_3-9}-3=0, \sqrt {x_4-16}-4, \sqrt {x_5-25}-5=0$$
$$x_1=2, x_2=8, x_3=18, x_4=32, x_5=50$$
$$\therefore \dfrac {x_1+x_2+x_3+x_4+x_5}{2}=55$$

Mathematics

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