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Question

If $$a+b+c=11$$ and $$a^2+b^2+c^2=81$$, then find $$ab+bc+ca $$.


A
121
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B
40
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C
30
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D
20
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Solution

The correct option is D 20

Formula $${ (a+b+c) }^{ 2 } = {a}^{2}  + { b }^{ 2 } + { c }^{ 2 }+
2(ab + bc + ca)$$
Substituting  $$a+b+c = 11; {a}^{2}  + { b }^{ 2 } + { c }^{ 2 } = 81$$
$$ { 11 }^{ 2 } = 81 + 2(ab + bc + ca) $$
$$ 121 - 81 = 2(ab + bc + ca) $$
$$ ab + bc + ca = \frac{40}{2} = 20 $$


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