Home
Class 11
MATHS
If z=x+i y and w=(1-i z)/(z-i) , show t...

If `z=x+i y and w=(1-i z)/(z-i)` , show that `|w|=1 z` is purely real.

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • COMBINATIONS

    RD SHARMA ENGLISH|Exercise All Questions|117 Videos

  • DERIVATIVES

    RD SHARMA ENGLISH|Exercise All Questions|230 Videos

Similar Questions

Explore conceptually related problems

If z=x+i y and w=(1-i z)//(z-i) and |w| = 1 , then show that z is purely real.

If z=x + yi and omega= (1-zi)/(z-i) show that |omega|=1 rArr z is purely real

If z=x+iy and w=(1-iz)/(z-i) , then |w|=1 implies that in the complex plane (A) z lies on imaginary axis (B) z lies on real axis (C) z lies on unit circle (D) None of these

Let z be a complex number such that |(z-5i)/(z+5i)|=1 , then show that z is purely real

If z= x + yi and omega = ((1- zi))/(z-i) , then |omega|=1 implies that in the complex plane

If z= x+ yi and (|z-1 -i|+4)/(3|z-1-i|-2)=1 , show that x^(2) + y^(2)- 2x-2y-7=0

If z= x +yi and (|z-1-i|+4)/(3|z-1-i|-2)=1 , show that x^(2) + y^(2) -2x-2y- 7=0

If z is a unimodular number (!=+-i) then (z+i)/(z-i) is (A) purely real (B) purely imaginary (C) an imaginary number which is not purely imaginary (D) both purely real and purely imaginary

Prove that z=i^i, where i=sqrt-1, is purely real.

If z = x + yi, omega = (2-iz)/(2z-i) and |omega|=1 , find the locus of z in the complex plane

RD SHARMA ENGLISH-COMPLEX NUMBERS-All Questions
  1. If x=-5+2sqrt(-4) , find the value of x^4+9x^3+35 x^2-x+4.

    Text Solution

    |

  2. If z is a complex number such that |z|=1, prove that (z-1)/(z+1) is pu...

    Text Solution

    |

  3. If z=x+i y and w=(1-i z)/(z-i) , show that |w|=1 z is purely real.

    Text Solution

    |

  4. If z=2-3i show that z^2-4z+13=0 and hence find the value of 4z^3-3z^...

    Text Solution

    |

  5. If |z1|=|z2|=.......=|zn|=1, prove that |z1+z2+z3++zn|=1/(z1)+1/(z2)+...

    Text Solution

    |

  6. Find the non-zero complex numbers z satisfying barz =i z^2dot

    Text Solution

    |

  7. Show that if iz^3+z^2-z+i=0, then |z|=1

    Text Solution

    |

  8. Solve the equation z^2 +|z|=0 , where z is a complex number.

    Text Solution

    |

  9. If z1, z2a n dz3, z4 are two pairs of conjugate complex numbers, find ...

    Text Solution

    |

  10. If zr=cos(pi/(3^r))+isin(pi/(3^r)),r=1,2,3, , prove that z1z2z3 zoo=i...

    Text Solution

    |

  11. If xn=cospi/(2^n)+isinpi/(2^n) , prove that x1x2x3 xoo=-1.

    Text Solution

    |

  12. Find the square root of the following: 5+12 i

    Text Solution

    |

  13. Find the square roots of -15-8idot

    Text Solution

    |

  14. Find the number of solutions of z^2+|z|^2=0.

    Text Solution

    |

  15. If ((a^2+1)/(2a-i))^2=x+i y , find the value of x^2+y^2dot

    Text Solution

    |

  16. Let z1a n dz2 be two complex numbers such that ( z )1+i( z )2=0 and ...

    Text Solution

    |

  17. If |z-5i|=|z+5i| , then the locus of zdot

    Text Solution

    |

  18. If z1a n dz2 both satisfy z+barz r =2|z-1| and a r g(z1-z2)=pi/4 , t...

    Text Solution

    |

  19. Show that the complex number z, satisfying the condition arg ((z -1)/(...

    Text Solution

    |

  20. If z=(1+7i)/((2-i)^2) , then find |z|?

    Text Solution

    |