Log1 Contest Round 2 Theta Complex Numbers. 4 points each. 5 points each

Transcription

1 01 Log1 Cotest Roud Theta Complex Numbers 1 Wrte a b Wrte a b form: 1 5 form: pots each Wrte a b form: Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may just wrte always, sometmes, or ever as your aswer): The sum of two real umbers s a complex umber. 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. For complex umber z, fd the value of zz. 10 Fd the dstace betwee the complex umbers ad 5 the Argad plae Wrte a b form: 1 Fd the sum of all postve tegers pots each such that 1 Fd all complex solutos to the equato x 15 x 1 0. s a real umber. Wrte a b form: Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f.

2 01 Log1 Cotest Roud Alpha Complex Numbers 4 pots each 1 Wrte a b Wrte a b form: 1 5 form: 1 5 Wrte a b form: 4 Evaluate: Determe f the followg statemet s always, sometmes, or ever true (you may just wrte always, sometmes, or ever as your aswer): The sum of two real umbers s a complex umber. 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal 10 f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. Fd the rectagular form of the complex umber wth polar form 1 Wrte a b form: 1 c s. 11 Fd the sum of all postve tegers pots each such that s a real umber. 1 Cosder the set 1, 1,,. The recprocal of oe elemet of the set equals the cojugate of a dfferet elemet of the set for how may elemets the set? 1 Fd all complex solutos to the equato Wrte a b form: x 15 x Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f.

3 01 Log1 Cotest Roud Mu Complex Numbers 4 pots each 1 Wrte a b Wrte a b form: 1 5 form: 1 5 Wrte a b form: 4 Evaluate: Usg Euler s Formula, fd the value of cos s ut., where s the magary 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal 10 f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. Fd the rectagular form of the complex umber wth polar form 1 Wrte a b form: 1 c s.

4 11 Fd the sum of all postve tegers pots each such that s a real umber. 1 Fd all complex solutos to the equato x 15 x Cosder the complex umbers that are solutos to the equato x 1, where s a postve teger satsfyg. If the solutos are plotted the Argad plae, ad f those solutos are the vertces of a regular polygo whose eclosed area s A, fd the value of lm A. Wrte a b form: Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f.

5 01 Log1 Cotest Roud Theta Complex Numbers 1 Wrte a b Wrte a b form: pots each 1 4 form: 1 5 Wrte a b form: Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may just wrte always, sometmes, or ever as your aswer): The sum of two real umbers s a complex umber. always 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. For complex umber z, fd the value of zz Fd the dstace betwee the complex umbers ad 5 the Argad plae Wrte a b form: 1 Fd the sum of all postve tegers pots each such that 1 Fd all complex solutos to the equato s a real umber. 1 x 15 x 1 0., Wrte a b form: Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f. 1

6 01 Log1 Cotest Roud Alpha Complex Numbers 1 Wrte a b Wrte a b form: pots each 1 4 form: 1 5 Wrte a b form: Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may just wrte always, sometmes, or ever as your aswer): The sum of two real umbers s a complex umber. always 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal 10 f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. Fd the rectagular form of the complex umber wth polar form 1 Wrte a b form: 1 c s Fd the sum of all postve tegers pots each such that s a real umber. 1 Cosder the set 1, 1,,. The recprocal of oe elemet of the set equals the cojugate of a dfferet elemet of the set for how may elemets the set? 1 Fd all complex solutos to the equato x 15 x 1 0., 0 Wrte a b form: Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f. 1

7 01 Log1 Cotest Roud Mu Complex Numbers 1 Wrte a b Wrte a b form: pots each 1 4 form: 1 5 Wrte a b form: Evaluate: 65 5 Usg Euler s Formula, fd the value of cos s ut., where s the magary e 5 pots each 6 What s the sum of the magtudes of the eght complex solutos of the equato x 1? Smplfy: For the polyomal 10 f x x 1x bx 60, where b s real, oe of the roots equals the sum of the other two roots. Fd the complex root of f whose magary part s postve. Fd the rectagular form of the complex umber wth polar form 1 Wrte a b form: 1 c s. 1 1

8 11 Fd the sum of all postve tegers pots each such that s a real umber. 1 Fd all complex solutos to the equato x 15 x 1 0., 1 Cosder the complex umbers that are solutos to the equato x 1, where s a postve teger satsfyg. If the solutos are plotted the Argad plae, ad f those solutos are the vertces of a regular polygo whose eclosed area s A, fd the value of lm A. Wrte a b form: Cosder polyomal 5 4 f x x ax bx cx dx 44 wth tegral coeffcets. f has oe real root ad four magary roots. The product of two of the magary roots s Fd the value of the real root of f. 1

9 01 Log1 Cotest Roud Complex Numbers Solutos Mu Al Th Soluto or comparable aswer s ot a b form Sce a complex umber s a umber of the form a b, where a ad b are real, all real umbers are complex umbers (just make b 0 ). Sce two real umbers sum s always real, ther sum s also always complex. 5 cos s cos s 1 e e e The solutos to x 1 are all eghth roots of uty, so each soluto s magtude s 1. Therefore, sce there are eght dstct solutos, the sum of the eght solutos magtudes s Because the sum of all roots s 1, ad because oe of the roots equals the 1 sum of the other two, oe of the roots must be 6 whle the other two sum to 6. Therefore, b 60 6b 6 b 46. Thus, 60 x 6x 6x 10, ad the other two roots are 1 the three roots, the oly oe wth postve magary part s x 1x 46x. Of zz c s cos s Sce 1 1, cs 0 cs 0, ad ths wll be real f 0 s cotermal wth 0 or 10. Therefore, must be a multple of 6, ad the desred sum s

10 1 I 1, 1,, 1 1 1, the recprocals of the elemets are 1, 1,, ad, respectvely, ad the cojugates of these elemets are the orgal elemets the set, respectvely. Therefore, o elemet equals the cojugate of a dfferet elemet. I actualty, each elemet s recprocal s the cojugate of the exact same elemet Usg the quadratc formula, x 5 1, ad the two solutos are ad. 1 The vertces are equally spread o the ut crcle, so as, the polygo approaches the ut crcle, the eclosed area of whch s cs15 cs 15 cs010 cs Sce the coeffcets of f are tegers, the product of the other two roots must be , makg the product of all four magary roots 1. Sce the product of all fve roots s 44 44, the real root must be. 1