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Question

# Let a,b,c be non-zero real numbers such that a+b+c=0; let q=a2+b2+c2 and r=a4+b4+c4 Then

A
q2<2r always
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B
q2=2r always
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C
q2>2r always
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D
q22r can take both positive and negative value
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Solution

## The correct option is A q2<2r alwaysGiven (a+b+c)=0& q=a2+b2+c2 & r=a4+b4+c4We know, ⇒(a+b+c)2=a2+b2+c2+2(ab+bc+ca)Therefore 02=q+2(ab+bc+ca)⇒ab+bc+ca=−q2⟶(1)Now, (a2+b2+c2)2=a4+b4+c4+2(a2b2+b2c2+c2a2) (all squared )⇒q2=r+2 ( some positive number )⇒(ab+bc+ca)2=a2b2+b2c2+c2a2+2(ab2c+a2bc+abc2)⇒q24−2=a2b2+b2c2+c2a2∴a2b2+b2c2+c2a2<q24∴q2<r+2(q24)⇒q2<r+q22⇒q22<r⇒q2<2rHence, the answer is q2<2r.

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