ข้อสอบ PAT 1 - มีนาคม 2559

ข้อ 41

ให้ straight real numbers แทนเซตของจำนวนจริง ให้ r subscript 1 equals open curly brackets open parentheses x comma y close parentheses element of straight real numbers cross times straight real numbers right enclose blank end enclose space y greater or equal than 0 space และ space x squared minus y squared minus 2 x plus 4 y less or equal than 3 close curly brackets

และ r subscript 2 equals open curly brackets open parentheses x comma y close parentheses element of straight real numbers cross times straight real numbers right enclose blank end enclose space y greater or equal than 0 space และ space x squared plus y squared minus 2 x less or equal than 33 close curly brackets

ถ้าโดเมนของเซต r subscript 1 intersection r subscript 2 คือช่วงปิด open square brackets a comma b close square brackets เมื่อ a และ b เป็นจำนวนจริง โดยที่ a less than b แล้วค่าของ a squared plus b squared เท่ากับเท่าใด

รีวิว - เสียงตอบรับจากผู้เรียน

เฉลยข้อสอบ

begin mathsize 12px style bottom enclose พ ิ จารณา space r subscript 1 end enclose
จากโจทย ์ space space space space space x squared minus y squared minus 2 x plus 4 y space less or equal than space 3
space space space space space space space space space space space space space space space space space space space space space open parentheses x minus 1 close parentheses squared minus open parentheses y minus 2 close parentheses squared space less or equal than space 3 plus 1 minus 4
space space space space space space space space space space space space space space space space space space space space space open parentheses x minus 1 close parentheses squared minus open parentheses y minus 2 close parentheses squared space less or equal than space 0
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 space space space space space space space space space space space space open parentheses x minus 1 close parentheses squared space less or equal than space open parentheses y minus 2 close parentheses squared 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 space space space space space space space space square root of open parentheses x minus 1 close parentheses squared end root space less or equal than space square root of open parentheses y minus 2 close parentheses squared space end root
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 space space space space space space space space space space space space space open vertical bar x plus 1 close vertical bar space less or equal than space open vertical bar y minus 2 close vertical bar end style

begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style
begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style

begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style

หลังจากนั้นนำกรณี 1 และกรณี 2 มารวมกัน
 begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style

begin mathsize 12px style bottom enclose พ ิ จารณา space r subscript 2 end enclose
จาก space space space space space space space space space space space space space space x squared plus y squared minus 2 x space less or equal than space 33
space space space space space space space space space space space space space space x squared minus 2 x plus 1 plus y squared space less or equal than space 33 plus 1
space space space space space space space space space space space space space space space space space space space space open parentheses x minus 1 close parentheses squared plus y squared space less or equal than space 33 plus 1
space space space space space space space space space space space space space space space space space space space space open parentheses x minus 1 close parentheses squared plus y squared space less or equal than space open parentheses square root of 34 close parentheses squared
แสดงว ่ า space space space space space space เป ็ นสมการวงกลม
space space space space space space space space space space space space space space space space space space space แต ่ space y greater or equal than 0 space จะได ้ กราฟของ space r subscript 2 end style

begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style

begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style
begin mathsize 12px style space space space space space space space space space space space space space space space space space space bottom enclose หาค ่ า space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals straight f left parenthesis 1 over 6 right parenthesis
space space space space space space space space space space space space space space space space space space โดย space space space open vertical bar 1 over 6 close vertical bar equals 1 over 6 less than 1 half space เล ื อกสมการ space box enclose 1
จะได ้ space space space space space space space space space space space space space space space space space space space space space space straight f open parentheses straight x close parentheses equals 1 over straight x
แทน space straight x equals 1 over 6 space space space space space space space straight f open parentheses 1 over 6 close parentheses equals fraction numerator 1 over denominator begin display style 1 over 6 end style end fraction equals 6
ด ั งน ั้ น space space straight f open parentheses straight f open parentheses straight f open parentheses negative 1 third close parentheses close parentheses close parentheses equals 6 end style

begin mathsize 12px style จากโจทย ์ space space space space space โดเมนของเซต space r subscript 1 intersection r subscript 2 space ค ื อช ่ วงป ิ ด space open square brackets a comma space b close square brackets
จะได ้ space space space space space space space space space space space space space open square brackets a comma b close square brackets space equals space open square brackets negative 2 comma 4 close square brackets
แสดงว ่ า space space space space space space space space space a equals negative 2 space comma space b equals 4

ด ั งน ั้ น space space space space space space space space space space space space space a squared plus b squared space equals space 4 plus 16
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 space space space space space space space space space space equals space 20

bold ค ่ าของ bold space bold space bold italic a to the power of bold 2 bold plus bold italic b to the power of bold 2 bold italic space bold italic equals bold italic space bold italic 20 end style

ปิด
ทดลองเรียน