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If $[\mathbf{a} \mathbf{b} \mathbf{c}]=3$, then the volume (in cubic units) of the parallelopiped with $2 \mathbf{a}+\mathbf{b}, 2 \mathbf{b}+\mathbf{c}$ and $2 \mathbf{c}+\mathbf{a}$ as edges, is
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Verified Answer
The correct answer is:
$27$
Given that,
$$
\text { [abc] }=3
$$
Volume of the parallelopiped
$$
=[2 \mathbf{a}+\mathbf{b} \mathbf{2} \mathbf{b}+\mathbf{c}+2 \mathbf{c}+\mathbf{a}]
$$
$$
\begin{aligned}
& =[2 \mathrm{a} 2 \mathrm{~b}+\mathrm{c} 2 \mathrm{c}+\mathbf{a}]+[\mathrm{b} \mathbf{2 b}+\mathrm{c} 2 \mathrm{c}+\mathrm{a}] \\
& =[2 \mathrm{a} 2 \mathrm{~b} 2 \mathrm{c} 2 \mathrm{a}]+[2 \mathrm{ac} 2 \mathrm{c}+\mathrm{a}] \\
& +[b 2 b 2 c+a]+[b c 2 c+a] \\
& =[2 \mathbf{a} 2 \mathbf{b ~} 2 \mathbf{c}]+[2 \mathbf{a} 2 \mathbf{b} \mathbf{a}]+0+0+[\mathbf{b} \mathbf{c} 2 \mathbf{c}]+[\mathrm{b} \mathbf{c} a] \\
& =8[a b c]+0+0+[a b c] \\
& =9[\mathbf{a} \mathbf{b} \mathbf{c}]=9 \cdot 3=27 \\
&
\end{aligned}
$$
$$
\text { [abc] }=3
$$
Volume of the parallelopiped
$$
=[2 \mathbf{a}+\mathbf{b} \mathbf{2} \mathbf{b}+\mathbf{c}+2 \mathbf{c}+\mathbf{a}]
$$
$$
\begin{aligned}
& =[2 \mathrm{a} 2 \mathrm{~b}+\mathrm{c} 2 \mathrm{c}+\mathbf{a}]+[\mathrm{b} \mathbf{2 b}+\mathrm{c} 2 \mathrm{c}+\mathrm{a}] \\
& =[2 \mathrm{a} 2 \mathrm{~b} 2 \mathrm{c} 2 \mathrm{a}]+[2 \mathrm{ac} 2 \mathrm{c}+\mathrm{a}] \\
& +[b 2 b 2 c+a]+[b c 2 c+a] \\
& =[2 \mathbf{a} 2 \mathbf{b ~} 2 \mathbf{c}]+[2 \mathbf{a} 2 \mathbf{b} \mathbf{a}]+0+0+[\mathbf{b} \mathbf{c} 2 \mathbf{c}]+[\mathrm{b} \mathbf{c} a] \\
& =8[a b c]+0+0+[a b c] \\
& =9[\mathbf{a} \mathbf{b} \mathbf{c}]=9 \cdot 3=27 \\
&
\end{aligned}
$$
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