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Q3 of 202 Page 188

If (x + iy)³ = (u + iv) then prove that = 4 (x² – y²).

Given that, (x + iy)³ = (u + iv)

⇒ x³ + (iy)³ + 3x²iy + 3xi²y² = u + iv

⇒ x³ - iy³ + 3x²iy - 3xy² = u + iv

⇒ x³ - 3xy² + i(3x²y - y³) = u + iv

On equating real and imaginary parts, we get

U = x³ - 3xy² and v = 3x²y - y³

Now ,

= x² - 3y² + 3x² - y²

= 4x² - 4y²

= 4(x² - y²)

Hence,

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1

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2

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