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If (x+i y)^3=u+i v ,then show that u/x+...

If `(x+i y)^3=u+i v ,`then show that `u/x+v/y=4(x^2-y^2)`.

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`" "(x+iy)^(3)=u+iv`
`rArr" "x^(3)+ 3x^(2) iy+ 3x i^(2)y^(2)+i^(3)y^(3)= u+iv`
`rArr" "x+3x^(2)iy-3 xy^(2) -iy^(3) =u+iv`
Equate real and imaginary parts, we get
`" "x^( 3)- 3xy^(2)=u and 3x^(2)y-y^(3)=upsilon`
`rArr" "(u)/(x)=x^(2) -3y^(2) and (upsilon)/(y)=3x^(2)-y^(2)`
Now, ` (u)/( x)+( upsilon)/(y)=x^(2)- 3y^(2)+ 3x^(2)-y^(2)`
`" "=4x^(2)- 4y^(2) `
`" "=4(x^(2)-y^(2))`
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NAGEEN PRAKASHAN-COMPLEX NUMBERS AND QUADRATIC EQUATION -MISCELLANEOUS EXERCISE
  1. Evaluate : [i^(18)+(1/i)^(25)]^3

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  2. Prove that Re (z1 z2)= Re z1 Re z2 - Imz1 Imz2 ,

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  3. Reduce (1/(1-4i)-2/(1+i))((3-4i)/(5+i))to the standard form.

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  4. If (a+i b)/(c+i d)=x+i y , prove that (a-i b)/(c-i d)=x-i ya n d(a^2+b...

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  5. Convert the following in the polar form : (i) (1+7i)/((2-i)^2) (ii) (...

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  6. Solve the equation : 3x^2-4x+(20)/3=0

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  7. x^(2)-2x+(3)/(2)=0

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  8. Solve the equation :27 x^2-10 x+1=0

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  9. Solve the following quadratic: 21 x^2-28 x+10=0

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  10. If z1=2-i ,z2=1+i ,find |(z1+z2+1)/(z1-z2+i)|

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  11. If a + i b =((x+i)^2)/(2x^2+1),prove that a^2+b^2=((x^2+1)^2)/((2x^2+...

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  12. Let z1=2-i ,z2=-2+i. Find (i) Re ((z1z2)/( bar z1)) (ii) Im(1/(z1 bar...

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  13. Find the modulus and argument of the complex number (1+2i)/(1-3i).

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  14. Find the real numbers x\ a n d\ y , if (x-i y)(3+5i) is the conjugate ...

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  15. Find the modulus of (1+i)/(1-i)-(1-i)/(1+i).

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  16. If (x+i y)^3=u+i v ,then show that u/x+v/y=4(x^2-y^2).

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  17. If alpha and beta are different complex number with |beta|=1, then f...

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  18. Find the number of non-zero integral solutions of the equation |1-i|^...

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  19. If (a + i b) (c + i d) (e + i f) (g + i h) = A + i B, then show that (...

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  20. If ((1+i)/(1-i))^m=1, then find the least positive integral value of m...

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