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Find the value of: x3 – 5x2 + 4x + 8, if x = i103-i. - Mathematics and Statistics

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प्रश्न

Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.

योग
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उत्तर

x = `10/(3 - "i")`

∴ x = `(10(3 + "i"))/((3 - "i")(3 + "i")`

= `(10(3 + "i"))/(9 - "i"^2)`

= `(10(3 + "i"))/(9 - (- 1)`        ...[∵ i2 = –1]

= `(10(3 + "i"))/10`

∴ x = 3 + i

∴  x –  3 = i

∴ (x –  3)2 = i2

∴ x2 –  6x + 9 = –1         ...[∵ i2 = –1]

∴ x2 –  6x + 10 = 0          ...(i)

                       x + 1
`x^2 – 6x + 10")"overline(x^3 - 5x^2 + 4x + 8)"`
                        x3 – 6x2 + 10x
                        –      +       –              
                                x2 – 6x + 8
                                x2 – 6x + 10
                              –      +       –         
                                               –  2

∴ x3 – 5x2 + 4x + 8

= (x2 – 6x + 10)(x + 1) – 2

= 0 (x + 1) – 2             ...[From (i)]

= 0 – 2

∴ x3 – 5x2 + 4x + 8 = – 2.

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अध्याय 3: Complex Numbers - MISCELLANEOUS EXERCISE - 3 [पृष्ठ ४३]

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बालभारती Mathematics and Statistics 1 (Commerce) [English] Standard 11 Maharashtra State Board
अध्याय 3 Complex Numbers
MISCELLANEOUS EXERCISE - 3 | Q 5) ii) | पृष्ठ ४३

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