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`(c )` Let `Z=a+ib`, `b ne 0` where `Im Z=b` <br> `Z^(5)=(a+ib)^(5)` <br> `=a^(5)+^(5)C_(1)a^(4)bi+^(5)C_(2)a^(3)b^(2)i^(2)+^(5)C_(3)a^(2)b^(3)i^(3)+^(5)C_(4)ab^(4)i^(4)+i^(5)b^(5)` <br> `ImZ^(5)=5a^(4)b-10a^(2)b^(3)+b^(5)` <br> `y=(ImZ^(5))/(Im^(5)Z)=5((a)/(b))^(4)-10((a)/(b))^(2)+1` <br> Let `((a)/(b))^(2)=x("say")`, `x in R^(+)` <br> `y=5x^(2)-10x+1` <br> `=5[x^(2)-2x]+1` <br> `=5[(x-1)^(2)]-4` <br> Hence,`y_(min)=-4`Transcript

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00:00 - 00:59 | the question says that if Z is a known liya complex number then where to find the minimum value of imaginary part of Z power 5.3 part of Z whole to the power 5 now Z is a complex number so we can write Z is equal to X + y by now imaginary part of Z to the power 5 upon imaginary part of Z ho to the power 5 is equal to imaginary part of X + iota why 5 upon imaginary part of X + y whole to the power 5 now |

01:00 - 01:59 | this is this would be a purely imaginary term in this early image data in place of this imaginary of X + iota why we can write why so this will be equal to imaginary part of X + iota b to the power 5 upon Y to the power 5 now we will open this expansion with Binomial expansion will get imaginary part of 1 upon wife 558 to see only weather report festival expanded 50 x to the power 5 + 5 C1 |

02:00 - 02:59 | 5C 1 x to the power 4 invite to the Toyota Y + 5 Si 2x cube into iOS device whole square + 53 x square y cube + 5 c4x into iota Y to the power 4 and 5C 5 to the power 5 power 0 now for purely imaginary terms this time he doesn't have any item that is not a positively return this is are purely many time now here I trust |

03:00 - 03:59 | where is -1 so it is also not a purely imaginary term it is a purely imaginary term here I to power 4 is one so it is also not a pure leave any time and it is a purely imaginary term will get power 550 is 5 we can calculate used this formula that is NCR and factorial upon our factory in 2 and minus factorial so it is 5575 x ^ 42 by noting out of the cause we are only taking the only major return that is coefficient of iota - for this 533 Eastern x square y cube + this is 535 is one and Y to the power 5 for this |

04:00 - 04:59 | no after dividing Y to the power 5 inside we get 5 X X upon b ka whole to the power 4 minus 10 X upon Y to the whole square + 1 not taking 5 coming from this we get X upon Y to the power 4 from these two terms - 2 X upon y whole square 9 vi add one hair if the bracket open this will come plus 5 so we will subtract - 5 and this one is as it is here now |

05:00 - 05:59 | this will equal to this is equal to a minus b whole square that is x y y - 12 the whole square minus 4 this term is X my wife whole square minus 1 whole square this term is having a whole square outside so it is definitely not equal to zero show the minimum of this term imaginative is -4 so we have to find the minimum of this imaginary term is minus 4 is the correct answer you to find the minimum value |