Test Index

NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

© examsnet.com
Question : 48 of 52
Marks: +1, -0
If (x+iy)3(x + iy)^3 = u + iv, then show that ux+vy\frac{u}{x}+\frac{v}{y} = 4 (x2+y2)(x^2+y^2)
Solution:  
We have, (x+iy)3(x + iy)^3 = u + iv
(x+iy)3(x + iy)^3 = x3+(iy)3+3x2(iy)+3x(iy)2x^3 + (iy)^3 + 3x^2(iy) + 3x(iy)^2 = x3−y3i+3x2yi−3xy2x^3 - y^3 i + 3x^2 y i - 3x y^2
= (x3−3xy2)+i(3x2y−y3)(x^3 - 3xy^2) + i(3x^2 y - y^3)
⇒ (x3−3xy2)+i(3x2y−y3)(x^3 - 3xy^2) + i(3x^2 y - y^3) = u + iv
Comparing the real and imaginary parts, we get
u = x3−3xy2x^3 - 3xy^2, v = 3x2y−y33x^2 y - y^3
u = x(x2−3y2)x(x^2 - 3y^2), v = y(3x2−y2)y(3x^2 - y^2)
⇒ ux\frac{u}{x} = (x2−3y2)(x^2-3y^2) , vy\frac{v}{y} = 3x2−y23x^2 - y^2
∴ ux+vy\frac{u}{x}+\frac{v}{y} = 4x2−4y24x^2 - 4y^2 = 4(x2−y2)4(x^2 - y^2)
© examsnet.com
Go to Question:
Prev Question
Next Question