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Question

If $2x=3+5i$, then the value of $2{x}^{3}+2{x}^{2}-7x+72$ is

A

$4$

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B

$-4$

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C

$8$

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D

$-8$

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Solution

The correct option is A $4$Explanation for the correct option:Step 1. Find the value of $2{x}^{3}+2{x}^{2}-7x+72$:Given, $2x=3+5i$$⇒$ $x=\frac{\left(3+5i\right)}{2}$Now, $\begin{array}{rcl}{x}^{3}& =& \frac{\left(27+135i-225-125i\right)}{8}\end{array}$ $\begin{array}{rcl}& =& \frac{\left(-198+10i\right)}{8}\end{array}$ ${x}^{2}=\frac{\left(9-25+30i\right)}{4}$ $=\frac{\left(-16+30i\right)}{4}$Step 2. Put the value of ${x}^{3}$, ${x}^{2}$ and $x$ in the given expression:$2{x}^{3}+2{x}^{2}–7x+72=\frac{2\left(-198+10i\right)}{8}+\frac{2\left(-16+30i\right)}{4}–\frac{7\left(3+5i\right)}{2}+72$ $=\left(\frac{-99}{2}-8-\frac{21}{2}+72\right)+\left(\frac{10}{4}+15-\frac{35}{2}\right)i$ $=\frac{\left(-99-16-21+144\right)}{2}+\left[\frac{\left(10+60-70\right)}{4}\right]i$ $=\frac{8}{2}$ $=\mathbf{4}$Hence, Option ‘A’ is Correct.

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