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Definite Integral as Limit of Sum

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Question

$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{a} & a^{2} & b c \frac{1}{b} & b^{2} & c a \frac{1}{c} & c^{2} & a b \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ =$

abc

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a+b+c

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4abc

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Solution

The correct option is C 0
Given,

$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{a} & a^{2} & b c \frac{1}{b} & b^{2} & c a \frac{1}{c} & c^{2} & a b \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$

$= \frac{1}{a} det (\begin{matrix} b^{2} & c a c^{2} & a b \end{matrix}) - a^{2} det (\begin{matrix} \frac{1}{b} & c a \frac{1}{c} & a b \end{matrix}) + b c det (\begin{matrix} \frac{1}{b} & b^{2} \frac{1}{c} & c^{2} \end{matrix})$

$= \frac{1}{a} (a b^{3} - a c^{3}) - a^{2} \cdot 0 + b c (\frac{c^{2}}{b} - \frac{b^{2}}{c})$

$= b^{3} - c^{3} - 0 + b c (\frac{c^{2}}{b} - \frac{b^{2}}{c})$

$= b^{3} + c^{3} - b^{3} - c^{3}$

$= 0$

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