$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ is not equal to
$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ is not equal to
- $\left|\begin{array}{ccc}a+1 & b+1 & c+1 \\ a^2+1 & b^2+1 & c^2+1 \\ 1 & 1 & 1\end{array}\right|$
- $\left|\begin{array}{ccc}a-b & b-c & c \\ a^2-b^2 & b^2-c^2 & c^2 \\ 0 & 0 & 1\end{array}\right|$
- $\left|\begin{array}{ccc}a(a+1) & b(b+1) & c(c+1) \\ a+1 & b+1 & c+1 \\ -1 & -1 & -1\end{array}\right|$
- $\left|\begin{array}{ccc}a+b & b+c & c+a \\ a^2+b^2 & b^2+c^2 & c^2+a^2 \\ 2 & 2 & 2\end{array}\right|$
Solution
$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|=\left|\begin{array}{ccc}a+1 & b+1 & c+1 \\ a^2+1 & b^2+1 & c^2+1 \\ 1 & 1 & 1\end{array}\right|$
$\begin{aligned}
& \mathrm{R}_1 \rightarrow \mathrm{R}_1+\mathrm{R}_3, \mathrm{R}_2 \rightarrow \mathrm{R}_2+\mathrm{R}_3 \\
& \left|\begin{array}{ccc}
a & b & c \\
a^2 & b^2 & c^2 \\
1 & 1 & 1
\end{array}\right|=\left|\begin{array}{ccc}
a-b & b-c & c \\
a^2-b^2 & b^2-c^2 & c^2 \\
0 & 0 & 1
\end{array}\right| \\
& \mathrm{C}_1 \rightarrow \mathrm{C}_1-\mathrm{C}_2, \mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_3 \\
& \left|\begin{array}{ccc}
a & b & c \\
a^2 & b^2 & c^2 \\
1 & 1 & 1
\end{array}\right| \\
& =-\left|\begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
1 & 1 & 1
\end{array}\right|=-\left|\begin{array}{ccc}
a^2+a & b^2+b & c^2+c \\
a+1 & b+1 & c+1 \\
1 & 1 & 1
\end{array}\right| \\
& =\left|\begin{array}{ccc}
a(a+1) & b(b+1) & c(c+1) \\
a+1 & b+1 & c+1 \\
-1 & -1 & -1
\end{array}\right|
\end{aligned}$
Only option (d) is left out.
$\left|\begin{array}{ccc}
a & b & c \\
a^2 & b^2 & c^2 \\
1 & 1 & 1
\end{array}\right| \neq\left|\begin{array}{ccc}
a+b & b+c & c+a \\
a^2+b^2 & b^2+c^2 & c^2+a^2 \\
2 & 2 & 2
\end{array}\right|$
Asked in: AP EAMCET 2024 (20 May Shift 2)
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