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Question: \[\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ (a + 1)^{2} & (b + 1)^{2} & (c + 1)^{2} \\ (a - 1)^...

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{matrix} \right| =$$
A

4∣a2b2c2abc111∣4\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{matrix} \right|

B

3∣a2b2c2abc111∣3\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{matrix} \right|

C

2∣a2b2c2abc111∣2\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{matrix} \right|

D

None of these

Answer

4∣a2b2c2abc111∣4\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{matrix} \right|

Explanation

Solution

Apply R2−R3R_{2} - R_{3} and note that

(x+y)2−(x−y)2=4xy(x + y)^{2} - (x - y)^{2} = 4xy

∴\therefore Δ=4∣a2b2c2abc(a−1)2(b−1)2(c−1)2∣\Delta = 4\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ (a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2} \end{matrix} \right|

= 4∣a2b2c2abc111∣4\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{matrix} \right| {Applying R3−(R1−2R2)}R_{3} - (R_{1} - 2R_{2})\}.