ข้อสอบ PAT 1 - ตุลาคม 2558

ข้อ 42

ให้ A เป็นเซตของจำนวนจริง x ทั้งหมด ที่สอดคล้องกับอสมการ fraction numerator square root of 2 x over denominator square root of 1 plus x end root space plus space square root of 1 minus x end root end fraction less than square root of 1 minus x end root ถ้า a เป็นขอบเขตบนน้อยสุดของเซต A และ b เป็นขอบเขตล่างมากสุดของเซต A แล้วค่าของ a squared plus b squared เท่ากับเท่าใด

รีวิว - เสียงตอบรับจากผู้เรียน

เฉลยข้อสอบ

begin mathsize 12px style จากโจทย ์ thin space thin space thin space space fraction numerator square root of 2 straight x over denominator square root of 1 plus straight x end root space plus space square root of 1 minus straight x end root end fraction space less than space square root of 1 minus straight x end root rightwards arrow box enclose @

กำหนดให ้ space thin space thin space thin space straight A equals square root of 1 plus straight x end root space space space space space space space space space space space space space space space space straight B equals square root of 1 minus straight x end root
space space space space space space space space space space space space space space space space thin space thin space thin space straight A squared equals 1 plus straight x rightwards arrow box enclose 1 space space space space space space space space space space straight B squared equals 1 minus straight x rightwards arrow box enclose 2 end style

begin mathsize 12px style นำ space box enclose 1 minus box enclose 2
จะได ้ space space straight A squared minus straight B squared equals 2 straight x
space space space space space space space space space space space space space space space space space space space space space space thin space thin space straight x equals fraction numerator straight A squared minus straight B squared over denominator 2 end fraction
จาก space box enclose @ space thin space thin space thin space thin space fraction numerator square root of 2 straight x over denominator square root of 1 plus straight x end root space plus space square root of 1 minus straight x end root end fraction less than square root of 1 minus straight x end root space thin space thin space thin space thin space thin space thin space thin space thin space thin space semicolon space แทน space straight A comma space straight B end style

begin mathsize 12px style จะได ้ space space space space space space space space space space space space space space fraction numerator square root of 2 open parentheses begin display style fraction numerator straight A squared minus straight B squared over denominator 2 end fraction end style close parentheses over denominator straight A plus straight B end fraction less than straight B

space space space space space space space space space space space space space space space space space space space space space space space fraction numerator square root of 2 over denominator 2 end fraction open parentheses fraction numerator straight A squared minus straight B squared over denominator straight A plus straight B end fraction close parentheses less than straight B thin space thin space thin space thin space thin space thin space thin space thin space thin space semicolon space 2 equals square root of 2 cross times square root of 2

space space space fraction numerator up diagonal strike square root of 2 end strike over denominator square root of 2 left parenthesis up diagonal strike square root of 2 right parenthesis end strike end fraction open parentheses fraction numerator open parentheses straight A minus straight B close parentheses open parentheses up diagonal strike straight A plus straight B end strike close parentheses over denominator up diagonal strike straight A plus straight B end strike end fraction close parentheses less than straight B

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 fraction numerator open parentheses straight A minus straight B close parentheses over denominator square root of 2 end fraction less than straight B

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 space space straight A minus straight B less than square root of 2 straight B end style

begin mathsize 12px style 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 space space space space space space space space space straight A less than square root of 2 straight B plus straight B

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 space space space space space space space space space straight A less than open parentheses square root of 2 plus 1 close parentheses straight B
space space space space space space space space space space space space space
แทน space straight A comma straight B space กล ั บค ื น space semicolon space space space space space space space space space square root of 1 plus straight x end root less than open parentheses square root of 2 plus 1 close parentheses open parentheses square root of 1 minus straight x end root close parentheses

ยกกำล ั ง space 2 space ท ั้ ง space 2 space ข ้ าง space semicolon thin space thin space thin space space space space space space space 1 plus straight x less than open parentheses 2 plus 2 square root of 2 plus 1 close parentheses open parentheses 1 minus straight x 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 space space space space space space space space space space space space space space space space space 1 plus straight x less than open parentheses 3 plus 2 square root of 2 close parentheses open parentheses 1 minus straight x 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 space space space space space space space space space space space space space space space space space 1 plus straight x less than 3 minus 3 straight x plus 2 square root of 2 minus 2 square root of 2 straight x

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 4 straight x plus 2 square root of 2 straight x less than 2 plus 2 square root of 2

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 2 straight x plus square root of 2 straight x less than 1 plus square root of 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 space space space space space space space space space space space space space space space space open parentheses 2 plus square root of 2 close parentheses straight x less than 1 plus square root of 2

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 space space space space space space space space space straight x less than fraction numerator 1 plus square root of 2 over denominator 2 plus square root of 2 end fraction space space space semicolon space นำมาเข ี ยนเส ้ นจำนวน end style
                    

begin mathsize 12px style จาก space box enclose @ space เง ื่ อนไข space square root of square rightwards arrow square greater or equal than 0
จะได ้ space thin space thin space thin space thin space thin space 1 plus x greater or equal than 0 space space space space space space space space thin space thin space thin space thin space thin space และ space space space space space space space space space thin space thin space thin space thin space thin space 1 minus sign x greater or equal than 0
space space space space space space space space thin space thin space thin space thin space thin space space space space space space space space space x greater or equal than minus sign 1 thin 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 thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space 1 greater or equal than x
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 space space thin space thin space thin space thin space thin space space space space space space space space space space space space space space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space x less-than or slanted equal to 1 end style

size 12px นำเส ้ นจำนวนท ั้ ง size 12px space size 12px 3 size 12px space size 12px มาเข ี ยนรวมก ั น
    
begin mathsize 12px style จะได ้ space open vertical bar top enclose straight a close vertical bar squared minus open vertical bar top enclose straight b close vertical bar squared equals 0
space space space space space space space space space space space space space space space space space space space space space open vertical bar top enclose straight a close vertical bar squared equals open vertical bar top enclose straight b close vertical bar squared
จะได ้ space open vertical bar top enclose straight a close vertical bar equals open vertical bar top enclose straight b close vertical bar space space space space หร ื อ space space space space open vertical bar top enclose straight a close vertical bar equals negative open vertical bar top enclose straight b close vertical bar
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 space เป ็ นไปไม ่ ได ้ space เพราะ space open vertical bar space close vertical bar greater than 0 space เสมอ
ด ั งน ั้ น space thin space thin space thin space thin space thin space open vertical bar top enclose straight a close vertical bar equals open vertical bar top enclose straight b close vertical bar space space space thin space thin space thin space ข ้ อ space open parentheses straight ค close parentheses space ถ ู ก end style

begin mathsize 12px style bold หาค ่ าของ bold space straight a squared plus straight b squared equals open parentheses fraction numerator 1 plus square root of 2 over denominator 2 plus square root of 2 end fraction close parentheses squared plus open parentheses negative 1 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 equals open parentheses fraction numerator 1 plus 2 square root of 2 plus 2 over denominator 4 plus 4 square root of 2 plus 2 end fraction close parentheses plus 1

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 fraction numerator open parentheses up diagonal strike 3 plus 2 square root of 2 end strike close parentheses over denominator 2 open parentheses up diagonal strike 3 plus 2 square root of 2 end strike close parentheses end fraction plus 1

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 1 half plus 1
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 1.5 end style

ปิด
ทดลองเรียน