Home
Class 12
MATHS
Prove that there exists no complex numbe...

Prove that there exists no complex number z such that `|z| < 1/3 and sum_(n=1)^n a_r z^r =1`, where `| a_r|< 2`.

A

0

B

1

C

4

D

infinite

Text Solution

Verified by Experts

The correct Answer is:
A
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COMPLEX NUMBERS

    MCGROW HILL PUBLICATION|Exercise SOLVED EXAMPLES LEVEL 9|1 Videos

  • COMPLEX NUMBERS

    MCGROW HILL PUBLICATION|Exercise SOLVED EXAMPLES LEVEL 10|1 Videos

  • COMPLEX NUMBERS

    MCGROW HILL PUBLICATION|Exercise SOLVED EXAMPLES LEVEL 7|1 Videos

  • CIRCLES AND SYSTEMS OF CIRCLES

    MCGROW HILL PUBLICATION|Exercise QUESTIONS FROM PREVIOUS YEARS. B-ARCHITECTURE ENTRANCE EXAMINATION PAPERS|17 Videos

  • DEFINITE INTEGRALS

    MCGROW HILL PUBLICATION|Exercise Questions from Previous Years. B-Architecture Entrance Examination Papers|18 Videos

Similar Questions

Explore conceptually related problems

Prove that there exists no complex number z such that |z|<(1)/(3) and sum_(n=1)^(n)a_(r)z^(r)=1, where |a_(r)|<2

Prove that |(z-1)/(1-barz)|=1 where z is as complex number.

Knowledge Check

  • Let z be a complex number such that | z + (1)/(z)| = 2 If |z| = r_(1) and r_(2) " for arg " z = (pi)/(4) than As arg z varies | r_(1) - r_(2)| =

    A
    [0,2]
    B
    [0,1]
    C
    `[(1)/( sqrt(2)) , sqrt(2)]`
    D
    [1,2]

  • Let z be a complex number such that | z + (1)/(z)| = 2 If |z| = r_(1) and r_(2) " for arg " z = (pi)/(4) than | r_(1) - r_(2)| =

    A
    `(1)/(sqrt(2))`
    B
    1
    C
    `sqrt(2)`
    D
    2

  • If z_(1) , z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 and iz_(1)=Kz_(2) , where K in R , then the angle between z_(1)-z_(2) and z_(1)+z_(2) is

    A
    `tan^(-1)((2K)/(K^(2)+1))`
    B
    `tan^(-1)((2K)/(1-K^(2)))`
    C
    `-2tan^(-1)K`
    D
    `2tan^(-1)K`

    • Similar Questions

    Explore conceptually related problems

    If z_(1) and z_(2) are two complex numbers such that |z_(1)| lt 1 lt |z_(2)| , then prove that |(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1

    If z_1 and z_2 are two complex numbers such that |z_1|lt1lt|z_2| then prove that |(1-z_1barz_2)/(z_1-z_2)|lt1

    If z_(1),z_(2),z_(3) are three nonzero complex numbers such that z_(3)=(1-lambda)z_(1)+lambda z_(2) where lambda in R-{0} then prove that points corresponding to z_(1),z_(2) and z_(3) are collinear.

    If z_(1),z_(2),z_(3) are three non-zero complex numbers such that z_(3)=(1-lambda)z_(1)+lambda z_(2) where lambda in R-{0}, then points corresponding to z_(1),z_(2) and z_(3) is

    If | beta _(k)| lt 3, 1 le k le n then all the complex number z satisfying the equation 1 + beta_(1) z + beta_(2) z^(2) + . . . beta_(n) z^(n) = 0 , | z| lt 1

    Home
    Profile