Home
Class 12
MATHS
If z=((sqrt(3))/2+i/2)^5+((sqrt(3))/2-i/...

If `z=((sqrt(3))/2+i/2)^5+((sqrt(3))/2-i/2)^5` , then prove that `I m(z)=0.`

Text Solution

Verified by Experts

The correct Answer is:
0
`z=((sqrt(3))/(2)+(i)/(2))^(5)+((sqrt(3))/(2)-(i)/(2))^(5)`
`z=(e^((i pi)/(6)))^(5)+(e^((-ipi)/(6)))^(5)=e^(i5pi//6)+e^(-i5pi//6)=-2"cos"(pi)/(6)=-sqrt(3)`
`R(z)=-sqrt(3)" " I(z)=0`
Promotional Banner

Topper's Solved these Questions

  • JEE MAIN REVISION TEST-3 (2020)

    VMC MODULES ENGLISH|Exercise MATHEMATICS (SECTION 2)|5 Videos

  • JEE Main Revision Test-20 | JEE-2020

    VMC MODULES ENGLISH|Exercise MATHEMATICS|25 Videos

  • JEE Main Revision Test-6 | JEE-2020

    VMC MODULES ENGLISH|Exercise MATHEMATICS|25 Videos

Similar Questions

Explore conceptually related problems

If z=(sqrt(3)/2+i/2)^5+(sqrt(3)/2-i/2)^5 , then prove that Im(z)=0

If z=[(sqrt(3)/2)+i/2]^5+[((sqrt(3))/2)-i/2]^5 , then a. R e(z)=0 b. I m(z)=0 c. R e(z)>0 d. R e(z)>0,I m(z)<0

If z=((sqrt3)/(2)+(1)/(2)i)^5+((sqrt3)/(2)-(i)/(2))^5 , then (a) im(z)=0 (b) Re(z)gt0 , Im(z)gt0 (c) Re(z)gt0 , Im(z)lt0 (d) Re(z)=3

If z= ((sqrt3)/(2) + (i)/(2))^(107) + ((sqrt3)/(2)-(i)/(2))^(107) , then show that Im(z)=0

If z(2-2sqrt(3i))^2=i(sqrt(3)+i)^4, then a r g(z)=

If z=i log(2-sqrt(3)) then cosz

For z_(1)=""^(6)sqrt((1-i)//(1+isqrt(3))),z_(2)=""^(6)sqrt((1-i)//(sqrt(3)+i)) , z_(3)= ""^(6)sqrt((1+i) //(sqrt(3)-i)) , prove that |z_(1)|=|z_(2)|=|z_(3)|

If z = (-4 +2 sqrt3i)/(5 +sqrt3i) , then the value of arg(z) is

if z = (sqrt 3 ) /(2) + (i)/(2) ( i=sqrt ( -1) ) , then ( 1 + iz + z^5 + iz^8)^9 is equal to:

If z=(1)/(2)(sqrt(3)-i) , then the least possible integral value of m such that (z^(101)+i^(109))^(106)=z^(m+1) is