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Evaluate the Following: 2 X 3 + 2 X 2 − 7 X + 72 , When X = 3 − 5 I 2 - Mathematics

Evaluate the following:

\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]

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Solution

\[ x = \frac{3 - 5i}{2}\]

\[ \Rightarrow x^2 = \left( \frac{3 - 5i}{2} \right)^2 \]

\[ = \frac{9 + 25 i^2 - 30i}{4}\]

\[ = \frac{- 16 - 30i}{4}\]

\[ \Rightarrow x^3 = \frac{- 16 - 30i}{4} \times \frac{3 - 5i}{2}\]

\[ = \frac{- 48 + 80i - 90i + 150 i^2}{8}\]

\[ = \frac{- 198 - 10i}{8}\]

\[ \therefore 2 x^3 + 2 x^2 - 7x + 72 = 2\left( \frac{- 198 - 10i}{8} \right) + 2\left( \frac{- 16 - 30i}{4} \right) - 7\left( \frac{3 - 5i}{2} \right) + 72\]

\[ = \frac{- 198 - 10i - 32 - 60i - 42 + 70i + 288}{4}\]

\[ = \frac{16}{4}\]

\[ = 4\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.2 | Q 16.1 | Page 32
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