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Let A(z1) and (z2) represent two compl...

Let `A(z_1) and (z_2)` represent two complex numbers on the complex plane. Suppose the complex slope of the line joining A and B is defined as `(z_1-z_2)/(bar z_1-bar z_2)`.If the line `l_1`, with complex slope `omega_1, and l_2`, with complex slope `omeg_2`, on the complex plane are perpendicular then prove that `omega_1+omega_2=0`.

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. Let A(z1) and (z2) represent two complex numbers on the complex plan...

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  2. Express (2+i)^2/(2+3i) in the form x+iy.

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  3. if z=(3-4i)/((4-2i)(1+i) then find Re(z).

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  4. IF frac(sinfrac(x)(2)+cosfrac(x)(2)+itanfrac(x)(2))(1+2isinfrac(x)(2) ...

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  5. Simplify 20/(sqrt3-sqrt(-2))+30/(3sqrt(-2)-2sqrt3)-14/(2sqrt3-sqrt(-2)...

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  6. Find all values of theta for which (1-isintheta)/(1+icostheta) is pur...

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  7. Find the real values of x\ a n d\ y ,\ if:(3m-2i)(2+i)^2=10(1+i)

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  8. (x+i y)(2-3i)=4+i

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  9. Find all complex numbers z for which (z-2)/(z+2) is purely imaginary.

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  10. Find the real value of x and y if ((1+i)x-2i)/(3+i)+((2-3i)y+i)/(3-i)...

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  11. If z1=-3+ixy,z2=x+y+4i and z1=barz2 then find x and y .

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  12. Evaluate: [i^19+(1/i)^25]^2

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  13. Evaluate (i^17-1/i^34)^3 .

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

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  15. Prove that (1+i)^(2n)+(1-i)^(2n)= 0 if n is an odd integer and, equal ...

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  16. If x=sqrt(-2)-1 find he value of x^4+4x^3+6x^2+4x+9

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  17. If a=(1+i)/sqrt(2) find the value of a^6+a^4+a^2+1

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  18. If sqrt(a-ib)= x-iy , prove that sqrt(a+ib)=x+iy.

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  19. If root(3)(a+ib) = x + iy, prove that : root(3)(a-ib) = x- iy .

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  20. (x+iy)^(1/3) =(a+ib) then prove that (x/a+y/b)=4(a^2-b^2)

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  21. Express (1+a^2)(1+b^2) as the sum of two squares.

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