Q153 / 270 · NDA 2022 (Phase 1) · Mathematics

Under which of the following conditions does the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ vanish? 1. $a+b+c=0$ 2. $a^3+b^3+c^3=3 a b c$ 3. $a^2+b^2+c^2-a b-b c-c a=0$ Select the correct answer using the code given below :

  • A
    1 and 2 only
  • B
    2 and 3 only
  • C
    1 and 3 only
  • Correct answer
    D
    1, 2 and 3

Solution

Let $\Delta=\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$ Applying $\quad R_1 \rightarrow R_1+R_2+R_3$ $=\left|\begin{array}{ccc}a+b+c & a+b+c & a+b+c \\ b & c & a \\ c & a & b\end{array}\right|=0$ $=(a+b+c)\left|\begin{array}{ccc}1 & 1 & 1 \\ b & c & a \\ c & a & b\end{array}\right|=0$ Applying $\quad c_2 \rightarrow c_2-c_1$ and $c_3 \rightarrow c_3-c_1$ $=(a+b+c)\left|\begin{array}{ccc}1 & 0 & 0 \\ b & c-b & a-b \\ c & a-c & b-c\end{array}\right|=0$ $\begin{aligned} & =(a+b+c)\left[-a^2-b^2-c^2+a b+b c+c a\right]=0 \\ & =-(a+b+1)\left[a^2+b^2+c^2-a b-b c-c a\right]=0 \\ & =-\left(a^3+b^3+c^3-3 a b c\right)=0 \\ & \Rightarrow a+b+\mathrm{c}=0 \text { or } a^2+b^2+c^2-a b-b c-c a=0 \\ & \text {or } a^3+b^3+c^3=3 a b c\end{aligned}$ Hence all statement 1,2 and 3 are correct.

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

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