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Question

if a, b, c are distinct +ve real numbers and a2+b2+c2=1 then ab + bc + ca is

A
less than 1
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B
equal to 1
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C
greater than 1
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D
any real no.
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Solution

The correct option is A less than 1
In such type of problem if sum of the squares of number is known

and you need product of numbers taken two at a time or needed range of the product of numbers taken two at a time.

Start with square of the sum of the numbers like

(a+b+c)2=a2+b2+c2+2(ab+bc+ca)

2(ab+bc+ca)=(a+b+c)2(a2+b2+c2)

(ab+bc+ca)=[(a+b+c)21]2

Since a2+b2+c2=1, all a,b and c are less than 1.

So, (ab+bc+ca) must be less than 1.

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