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  • If <img alt="\begin{vmatrix} a &b &c \\ c &a &b \\ b &c &a \end{vmatrix}=0" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cbegin%7Bvmatrix%7D%20a%20%26b%20%26c%20%5

If \begin{vmatrix} a &b &c \\ c &a &b \\ b &c &a \end{vmatrix}=0 where a,b,c are non-zero real such that a\neq b\neq c. Three statements A,B,C are given below

A. \left ( a+b+c \right )=0

B. a+b\omega +c\omega ^{2}=0

C.  a+b\omega^{2} +c\omega=0

Which of the following must be correct? (Where \omega is a non-real cube root of unity)

Option: 1

Only A


Option: 2

Only B and C


Option: 3

Only C

 


Option: 4

All A, B, and C


Answers (1)

best_answer

\begin{vmatrix} a &b &c \\ c & a & b\\ b & c &a \end{vmatrix}=0

\Rightarrow a\left ( a^{2}-bc \right )-b\left ( ac-b^{2} \right )+c\left ( c^{2}-ab \right )=0

a^{3}+b^{3}+c^{3}-3abc=0

a^{3}+b^{3}+c^{3}-3abc=\left ( a+b+c \right )\left ( a^{2}+b^{2}+c^{2}-ab-bc-ca \right )=0

\Rightarrow a+b+c=0\; or\; a=b=c

Posted by

Rakesh

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