Q61 / 200 · Manipal (MET) 2011 · Mathematics

The value of $\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2\end{array}\right|$ is

  • A
    $3\left|\begin{array}{lll}a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1\end{array}\right|$
  • Correct answer
    B
    $4\left|\begin{array}{lll}a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1\end{array}\right|$
  • C
    $2\left|\begin{array}{lll}a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1\end{array}\right|$
  • D
    None of these

Solution

Let $\Delta=\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2\end{array}\right|$ Applying $R_2 \rightarrow R_2-R_3$ $=4\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ a & b & c \\ (a-1)^2 & (b-1)^2 & (c-1)^2\end{array}\right|$ Applying $R_3 \rightarrow R_3-\left(R_1-2 R_2\right)$ $=4\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1\end{array}\right|$

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

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