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Show that all the roots of the equation `a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)=3,`
`(where|a_(i)|le1,i=1,2,3,4,)` lie
outside the circle with centre at orihin and radius `2//3.`

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All the roots of the equation a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)=3 where |a_(i)| lt 1, i=1,2,3,4, lie outside the circle with center at the origin and radius equal to

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ARIHANT MATHS-COMPLEX NUMBERS-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Show that all the roots of the equation a(1)z^(3)+a(2)z^(2)+a(3)z+a(4)...

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  2. If omega is a cube root of unity but not equal to 1, then minimum valu...

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  3. If one of the vertices of the square circumscribing the circle |z - 1|...

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  4. If z1 and z2, are two non-zero complex numbers such tha |z1+z2|=|z1...

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  5. If 1,omega,omega^(2) are the cube roots of unity, then the roots of t...

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  6. If omega = z//[z-(1//3)i] and |omega| = 1, then find the locus of z.

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  7. If w=alpha+ibeta where Beta 0 and z ne 1 satisfies the condition that...

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  8. Find the value of sum(k=1)^10[sin((2pik)/(11))-icos((2pik)/(11))],wher...

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  9. If z^2+z+1=0 where z is a complex number, then the value of (z+1/z)^2+...

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  10. A man walks a distance of 3 units from the origin towards north-east t...

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  11. If |z|=1 and z!=+-1, then all the values of z/(1-z^2) lie on

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  12. If abs(z+4) le 3, the maximum value of abs(z+1) is

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  13. Let A, B, C be three sets of complex number as defined below: A={z:Img...

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  14. Let A,B and C be three sets of complex numbers as defined below: {:(,A...

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  15. Let A,B and C be three sets of complex numbers as defined below: {:(,A...

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  16. A particle P starts from the point z0=1+2i , where i=sqrt(-1) . It mov...

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  17. If the conjugate of a complex numbers is 1/(i-1), where i=sqrt(-1). Th...

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  18. Let z=x+i y be a complex number where xa n dy are integers. Then, the ...

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  19. Let z=costheta+isintheta. Then the value of sum(m-&gt;1-15)Img(z^(2m-1...

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  20. If |Z-4/Z|=2 then maximum value of |Z| is equal to

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  21. Let z(1) and z(2) be two distinct complex numbers and z=(1-t)z(1)+iz(2...

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