Q155 / 270 · NDA 2021 (Phase 1) · Mathematics

If $a+b+c=4$ and $a b+b c+c a=0$, then what is the value of the following determinant? $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$

  • A
    $32$
  • Correct answer
    B
    $-64$
  • C
    $-128$
  • D
    $64$

Solution

$(a+b+c)=4$ (Given) Now, $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=a\left(b c-a^2\right)+b\left(a c-b^2\right)+c\left(a b-c^2\right)$ $=3 a b c-a^3-b^3-c^3$ $\begin{aligned} & =-\left(a^3+b^3+c^3-3 a b c\right)=-(a+b+c)^3 \\ & =-(4)^3=-64 .\end{aligned}$

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

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