ข้อสอบ PAT 1 - ตุลาคม 2558

ข้อ 16

กำหนดให้ stack A space with rightwards arrow on top และ B with rightwards arrow on top เป็นเวกเตอร์ในระนาบ โดยที่ A with rightwards arrow on top equals 16 i with bar on top space plus a top enclose j และ B with rightwards arrow on top equals 8 italic i with bar on top space plus italic b top enclose italic j เมื่อ a และ b เป็นจำนวนจริง ถ้า open vertical bar A with rightwards arrow on top close vertical bar equals open vertical bar B with rightwards arrow on top close vertical bar space และเวกเตอร์ B with rightwards arrow on top ทำมุม 60 degree กับเวกเตอร์ stack A space with rightwards arrow on top​แล้วค่าของ open parentheses a plus b close parentheses squared เท่ากับข้อใดต่อไปนี้

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เฉลยข้อสอบ

begin mathsize 12px style ส ู ตรเวกเตอร ์ space straight u with rightwards harpoon with barb upwards on top equals ai plus bj space จะได ้ space open vertical bar straight u with rightwards harpoon with barb upwards on top close vertical bar equals square root of straight a squared plus straight b squared end root
space space space space space space space space space space space space space space space space space space space space space space ถ ้ า space straight u with rightwards harpoon with barb upwards on top equals ai plus bj space และ space straight v with rightwards harpoon with barb upwards on top equals ci plus dj
จะได ้ space space thin space thin space thin space thin space thin space thin space thin space thin space straight u with rightwards harpoon with barb upwards on top times straight v with rightwards harpoon with barb upwards on top equals ac plus bd
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 space straight u with rightwards harpoon with barb upwards on top times straight v with rightwards harpoon with barb upwards on top equals open vertical bar straight u with rightwards harpoon with barb upwards on top close vertical bar open vertical bar straight v with rightwards harpoon with barb upwards on top close vertical bar cos space straight theta

ด ั งน ั้ น space thin space thin space thin space thin space thin space thin space thin space thin space ac plus bd equals open vertical bar straight u with rightwards harpoon with barb upwards on top close vertical bar open vertical bar straight v with rightwards harpoon with barb upwards on top close vertical bar cos space straight theta rightwards arrow box enclose straight a end style

              
begin mathsize 12px style จากโจทย ์ thin space thin space thin space thin space thin space space straight A with rightwards harpoon with barb upwards on top equals 16 top enclose straight i plus straight a top enclose straight j space จะได ้ space open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar equals square root of open parentheses 16 close parentheses squared plus straight a squared 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 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 equals square root of 256 plus straight a squared end root

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 straight B with rightwards harpoon with barb upwards on top equals 8 top enclose straight i plus straight b top enclose straight j space จะได ้ space space space open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar equals square root of 8 squared plus straight b squared 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 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 equals square root of 64 plus straight b squared end root end style

begin mathsize 12px style จากโจทย ์ space thin space thin space thin space thin space thin space open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar space space space space space space space space space space space semicolon space แทน space open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar comma open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar
space space space thin space thin space thin space thin space space space square root of 256 plus straight a squared end root equals square root of 64 plus straight b squared end root space thin space thin space thin space thin space thin space semicolon space ยกกำล ั ง space 2 space ท ั้ ง space 2 space ข ้ าง
space space space space space thin space thin space thin space thin space thin space space space space space 256 plus straight a squared equals 64 plus straight b squared
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 straight b squared equals straight a squared plus 192 rightwards arrow box enclose 1 end style

begin mathsize 12px style จากโจทย ์ thin space เวกเตอร ์ space straight B with rightwards arrow on top space ทำม ุ ม space 60 degree space ก ั บเวกเตอร ์ space straight A with rightwards arrow on top space space space thin space thin space thin space thin space thin space semicolon space จาก space box enclose straight a

จะได ้ space space space space space space space space space space space space space space straight A with rightwards arrow on top times straight B with rightwards arrow on top equals open vertical bar straight A with rightwards arrow on top close vertical bar open vertical bar straight B with rightwards arrow on top close vertical bar cos space 60 degree space space space space space space semicolon space จากโจทย ์ thin space open vertical bar straight A with rightwards arrow on top close vertical bar equals open vertical bar straight B with rightwards arrow on top close vertical bar
open parentheses 16 top enclose straight i plus straight a top enclose straight j close parentheses times open parentheses 8 top enclose straight i plus straight b top enclose straight j close parentheses equals left parenthesis square root of 256 plus straight a squared end root right parenthesis left parenthesis square root of 256 plus straight a squared end root right parenthesis open parentheses 1 half close parentheses space
space space space space space space space space space space space space space 16 open parentheses 8 close parentheses plus ab equals left parenthesis 256 plus straight a squared right parenthesis open parentheses 1 half close parentheses space
space space space space space space space space space space space space space space space space 128 plus ab equals fraction numerator 256 plus straight a squared over denominator 2 end fraction
space space space space space space space space space space space space space up diagonal strike space 256 end strike plus 2 ab equals up diagonal strike 256 plus straight a squared 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 2 ab equals straight a squared
space space space space space space space space space space space space space space space space 2 ab minus straight a squared equals 0
space space space space space space space space space space space space space space space open parentheses 2 straight b minus straight a close parentheses straight a equals 0
จะได ้ space thin space thin space thin space thin space thin space 2 straight b minus straight a equals 0 space space space space หร ื อ space space space space straight a equals 0
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 straight a equals 2 straight b

ถ ้ า space straight a equals 2 straight b space จาก space box enclose 1 space
จะได ้ space space space space straight b squared equals straight a squared plus 192 space space space space space semicolon space แทน space straight a equals 2 straight b
space space space space space space space space space space space space space straight b squared equals open parentheses 2 straight b close parentheses squared plus 192 end style
begin mathsize 12px style space space space space space space space space space space space space space space straight b squared equals 4 straight b squared plus 192
space space space space space space space space space space space space 3 straight b squared equals negative 192
space space space space space space space space space space thin space thin space เป ็ นไปไม ่ ได ้ space เพราะ space straight b squared greater or equal than 0
ด ั งน ั้ น space thin space thin space thin space thin space thin space straight a not equal to 2 straight b

ถ ้ า space straight a equals 0 space จาก space box enclose 1 space
จะได ้ space space space space straight b squared equals straight a squared plus 192 space space space thin space thin space thin space thin space semicolon space แทน space straight a equals 0 end style
begin mathsize 12px style space space space space space space space space space space space space space straight b squared equals open parentheses 0 close parentheses squared plus 192
space space space space space space space space space space space space space straight b squared equals 192

หาค ่ าของ space open parentheses straight a plus straight b close parentheses squared space space space space space space space space space semicolon space แทนค ่ า space straight a equals 0
space space space space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space open parentheses straight a plus straight b close parentheses squared equals open parentheses 0 plus straight b close parentheses squared
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 thin space thin space thin space thin space thin space thin space thin space thin space thin space equals straight b squared space space space space space space space space space space space space space space semicolon space แทนค ่ า space straight b squared
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 thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space equals 192 end style

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