ข้อสอบ A-Level คณิตประยุกต์ 1 ปี 2566 (หลักสูตรใหม่)

ข้อ 13

ให้จำนวนเชิงซ้อน begin mathsize 16px style u equals cos straight pi over 3 plus i sin straight pi over 3 end style

และ begin mathsize 16px style v end style เป็นรากที่ begin mathsize 16px style 3 end style ของจำนวนเชิงซ้อน begin mathsize 16px style cos straight pi over 2 plus i sin straight pi over 2 end style

ถ้าส่วนจริงของ begin mathsize 16px style u over v end style เป็นจำนวนจริงลบ แล้วส่วนจริงของ begin mathsize 16px style v end style เท่ากับเท่าใด

รีวิว - เสียงตอบรับจากผู้เรียน

เฉลยข้อสอบ

begin mathsize 12px style จากส ู ตร space space space รากท ี่ space n space ของจำนวนเช ิ งซ ้ อน space r open parentheses cos theta plus i sin theta close parentheses space ค ื อ
space space space space space space space space space space space space space space space space space z subscript k space equals space r to the power of 1 over n end exponent open parentheses cos open parentheses fraction numerator theta plus 2 k pi over denominator n end fraction close parentheses plus i sin open parentheses fraction numerator theta plus 2 k pi over denominator n end fraction close parentheses close parentheses rightwards arrow box enclose 1
space space space space space space space space space space space space space space space space เม ื่ อ space k equals 0 comma space 1 comma space horizontal ellipsis comma space n minus 1
space space space space space space space space space space space space space space space space fraction numerator c i s theta subscript 1 over denominator c i s theta subscript 2 end fraction space equals space c i s open parentheses theta subscript 1 minus theta subscript 2 close parentheses space rightwards arrow box enclose 2 end style

begin mathsize 12px style จากโจทย ์ space space space space v space เป ็ นรากท ี่ space 3 space ของจำนวนเช ิ งซ ้ อน space cos straight pi over 2 plus i sin straight pi over 2
จะได ้ space space space space space space space space space space space รากท ี่ space 3 space ของ space cos straight pi over 2 plus i sin straight pi over 2 space equals space รากท ี่ space 3 space ของ space 1 open parentheses cos straight pi over 2 plus i sin straight pi over 2 close parentheses
space space space space space space space space space space space space space space space space space space space โดย space r equals 1 comma space n equals 3 comma space theta equals straight pi over 2 end style

begin mathsize 12px style จาก space box enclose 1 space space space space space space space space space space z subscript k space equals space r to the power of 1 over n end exponent open parentheses cos open parentheses fraction numerator straight theta plus 2 straight k straight pi over denominator 2 end fraction close parentheses plus i sin open parentheses fraction numerator straight theta plus 2 straight k straight pi over denominator 2 end fraction close parentheses close parentheses
จะได ้ space space space space space space space space space space space space รากท ี่ space 3 space ของ space cos straight pi over 2 plus i sin straight pi over 2 space ค ื อ end style

size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px z subscript size 12px k size 12px space size 12px equals size 12px space size 12px r to the power of size 12px 1 over size 12px 3 end exponent begin mathsize 12px style stretchy left parenthesis cos open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 straight k straight pi end style over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 straight k straight pi end style over denominator 3 end fraction close parentheses stretchy right parenthesis end style
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px z subscript size 12px k size 12px space size 12px equals size 12px space end subscript size 12px cos begin mathsize 12px style stretchy left parenthesis fraction numerator begin display style bevelled straight pi over 2 plus 2 straight k straight pi end style over denominator 3 end fraction stretchy right parenthesis end style size 12px plus size 12px i size 12px sin begin mathsize 12px style stretchy left parenthesis fraction numerator begin display style bevelled straight pi over 2 plus 2 straight k straight pi end style over denominator 3 end fraction stretchy right parenthesis end style
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px โดย size 12px space size 12px k size 12px equals size 12px 0 size 12px comma size 12px space size 12px 1 size 12px comma size 12px space size 12px 2 size 12px space size 12px space size 12px left parenthesis size 12px รากท ี่ size 12px space size 12px 3 size 12px space size 12px เป ็ นไปได ้ size 12px space size 12px 3 size 12px space size 12px ค ่ า size 12px right parenthesis

begin mathsize 12px style k equals 0 space space จะได ้ space space space space space space space z subscript 0 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 end style plus 2 k straight pi over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 k straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 0 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 0 close parentheses straight pi end style over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 0 close parentheses straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 0 space equals space cos fraction numerator begin display style straight pi end style over denominator 6 end fraction plus i sin fraction numerator begin display style straight pi end style over denominator 6 end fraction space equals space c i s straight pi over 6 space equals space c i s 30 degree end style

begin mathsize 12px style k equals 1 space space จะได ้ space space space space space space space z subscript 1 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 end style plus 2 k straight pi over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 k straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 1 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 1 close parentheses straight pi end style over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 1 close parentheses straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 1 space equals space cos fraction numerator begin display style 5 straight pi end style over denominator 6 end fraction plus i sin fraction numerator begin display style 5 straight pi end style over denominator 6 end fraction space equals space c i s fraction numerator 5 straight pi over denominator 6 end fraction space equals space c i s 150 degree end style

begin mathsize 12px style k equals 2 space space จะได ้ space space space space space space space z subscript 2 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 end style plus 2 k straight pi over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 k straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 2 space equals space cos open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 2 close parentheses straight pi end style over denominator 3 end fraction close parentheses plus i sin open parentheses fraction numerator begin display style bevelled straight pi over 2 plus 2 open parentheses 2 close parentheses straight pi end style over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 2 space equals space cos fraction numerator begin display style 9 straight pi end style over denominator 6 end fraction plus i sin fraction numerator begin display style 9 straight pi end style over denominator 6 end fraction space equals space cos fraction numerator 3 straight pi over denominator 2 end fraction plus i sin space fraction numerator 3 straight pi over denominator 2 end fraction rightwards arrow box enclose 3
space space space space space space space space space space space space space space space space space space space space space space space space space space z subscript 2 space equals space c i s fraction numerator 3 straight pi over denominator 2 end fraction equals c i s 270 degree end style

            
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 begin mathsize 12px style stretchy left parenthesis 1 stretchy right parenthesis end style size 12px minus size 12px 3 begin mathsize 12px style stretchy left parenthesis 8 stretchy right parenthesis end style size 12px plus size 12px a begin mathsize 12px style stretchy left parenthesis negative 5 stretchy right parenthesis end style size 12px space size 12px equals size 12px space size 12px 0
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 size 12px minus size 12px 24 size 12px minus size 12px 5 size 12px a size 12px space size 12px equals size 12px space size 12px 0
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size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px a size 12px b size 12px equals size 12px space size 12px minus size 12px 4
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size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 begin mathsize 12px style stretchy left parenthesis 1 stretchy right parenthesis end style size 12px minus size 12px 3 begin mathsize 12px style stretchy left parenthesis 8 stretchy right parenthesis end style size 12px plus size 12px a begin mathsize 12px style stretchy left parenthesis negative 5 stretchy right parenthesis end style size 12px space size 12px equals size 12px space size 12px 0
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 size 12px minus size 12px 24 size 12px minus size 12px 5 size 12px a size 12px space size 12px equals size 12px space size 12px 0
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size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px a size 12px b size 12px equals size 12px space size 12px minus size 12px 4
size 12px ด ั งน ั้ น size 12px space size 12px space size 12px space size 12px ค ่ าของ size 12px space size 12px a size 12px space size 12px เท ่ าก ั บ size 12px space size 12px minus size 12px 4

size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 begin mathsize 12px style stretchy left parenthesis 1 stretchy right parenthesis end style size 12px minus size 12px 3 begin mathsize 12px style stretchy left parenthesis 8 stretchy right parenthesis end style size 12px plus size 12px a begin mathsize 12px style stretchy left parenthesis negative 5 stretchy right parenthesis end style size 12px space size 12px equals size 12px space size 12px 0
size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 size 12px minus size 12px 24 size 12px minus size 12px 5 size 12px a size 12px space size 12px equals size 12px space size 12px 0
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size 12px ด ั งน ั้ น size 12px space size 12px space size 12px space size 12px ค ่ าของ size 12px space size 12px a size 12px space size 12px เท ่ าก ั บ size 12px space size 12px minus size 12px 4

size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px 4 begin mathsize 12px style stretchy left parenthesis 1 stretchy right parenthesis end style size 12px minus size 12px 3 begin mathsize 12px style stretchy left parenthesis 8 stretchy right parenthesis end style size 12px plus size 12px a begin mathsize 12px style stretchy left parenthesis negative 5 stretchy right parenthesis end style size 12px space size 12px equals size 12px space size 12px 0
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size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px space size 12px a size 12px b size 12px equals size 12px space size 12px minus size 12px 4
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