Q68 / 105 · JEE Main 2008 · Mathematics

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=$ $c y+b z, y=a z+c x$ and $z=b x+a y$. Then $a^2+b^2+c^2+2 a b c$ is equal to

  • A
    2
  • B
    $-1$
  • C
    0
  • Correct answer
    D
    1

Solution

The system of equations $x-c y-b z=0, c x-y+a z=0$ and $b x+a y-z=0$ have non-trivial solution if $\left|\begin{array}{ccc}1 & -c & -b \\ c & -1 & a \\ b & a & -1\end{array}\right|=0 \Rightarrow 1\left(1-a^2\right)+c(-c-a b)-b(c a+b)=0$ $\Rightarrow a^2+b^2+c^2+2 a b c=1$.

Hence, the correct answer is option: 'D'.

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