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

$4$

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$8$

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$- 8$

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Solution

The correct option is A
$4$

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

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Similar questions

Evaluate the following :

$(i) 2 x^{3} + 2 x^{2} - 7 x + 72, w h e n x = \frac{3 - 5 i}{2} (i i) x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 44, w h e n x = 3 + 2 i (i i i) x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 9, w h e n x = - 1 + i \sqrt{2} (i v) x^{6} + x^{4} + x^{2} + 1, w h e n x = \frac{1 + i}{\sqrt{2}} (v) 2 x^{4} + 5 x^{3} + 7 x^{2} - x + 41, w h e n x = - 2 - \sqrt{3} i$