(b) Show that the x and y coordinates of P can be written as...Who killed Mr. Potato? [36 marks] 1....
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Copyright Andrew Chambers 2020. Licensed for non-commercial use only. Visit ibmathsresources.com to download the full worked mark-scheme and for 300 exploration ideas. 7 (b) Show that the x and y coordinates of P can be written as: ! = !"#$% − ! ! !"#$ (3) ! = !"#$% + ! ! !"#$ (4) [4] (c) By multiplying equation (3) by !"# ! and equation (4) by !"# ! show that this can be written as: !"#$% − !"#$% = !(!!"#$ + !"#$%) [6] (d) By taking ! ! = 2! + 3, show that the equation when ! = 2! + 3 is rotated ! ! radians anticlockwise around (0,0) is given by ! = −3! − 3 2 . [5] For 2 straight lines ! ! ! + ! ! ! + ! ! = 0 and ! ! ! + ! ! ! + ! ! = 0 , the acute angle between them is given by: !"#$ = ! ! ! ! − ! ! ! ! ! ! ! ! + ! ! ! ! (e) Show that the angle between ! = 2! + 3 and ! = −3! − 3 2 is indeed ! ! . [3] (f) By taking ! ! = ! ! find the equation of y = ! ! when it is rotated ! ! radians anticlockwise around (0,0). Leave your answer in the form: !! ! + !" + !! ! + !" + !"# = 0 [5]
Transcript of (b) Show that the x and y coordinates of P can be written as...Who killed Mr. Potato? [36 marks] 1....
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CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visitibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.
7(b) ShowthatthexandycoordinatesofPcanbewrittenas:
! = !"#$% − ! ! !"#$ (3)
! = !"#$% + ! ! !"#$ (4)[4]
(c) By multiplying equation (3) by!"# ! and equation (4) by!"# ! show that
thiscanbewrittenas:
!"#$% − !"#$% = !(!!"#$ + !"#$%)[6]
(d) By taking! ! = 2! + 3, showthat theequationwhen! = 2! + 3is rotated!!
radiansanticlockwisearound(0,0)isgivenby! = −3! − 3 2. [5]
For 2 straight lines!!! + !!! + !! = 0 and !!! + !!! + !! = 0 , the acute anglebetweenthemisgivenby:
!"#$ = !!!! − !!!!!!!! + !!!!
(e) Showthattheanglebetween! = 2! + 3and! = −3! − 3 2isindeed!!.
[3](f) By taking! ! = !!find the equation of y =!! when it is rotated!!radians
anticlockwisearound(0,0).
Leaveyouranswerintheform:
!!! + !" + !!! + !" + !"# = 0[5]
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CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visitibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.
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WhokilledMr.Potato?[36marks]1.[Maximummarks:12]Coolingratesareanessentialtoolinforensicsindeterminingtimeofdeath. Wecanuse differential equations to model the cooling of a body using Newton’s Law ofCooling. The rate of change in the body temperature with respect to time isproportionaltothedifferenceintemperaturebetweenthebodyandtheambientroomtemperature.
Inthefollowinginvestigationwewilldeterminewhenapotatowasremovedfromanoven. If the ambient room temperature is 75 degrees Fahrenheit and! is thetemperatureofthepotatoattime!,thenwehave:
!"!" = −!(! − 75)
a) Showthatthesolutiontothisdifferentialequationis:
! − 75 = !!!!"[4]
Wearriveataroomatmidday12:00todiscoverapotatoonthekitchencounter.Theambient room temperature is 75 degrees Fahrenheit and we know the initialtemperature of the potato before being taken out of the oven was 194 degreesFahrenheit.Wetakethefollowingmeasurements:Timeafter12:00(tmins) Temperature of potato (T
degreesFahrenheit)Difference from ambient roomtemperature!! (T–75)
0 133 5830 116 4160 104 2990 98 23120 91 16(b) By using the measurements when! = 0 and! = 120, find!and!. Use your
modeltopredictwhenthepotatowasfirstremovedfromtheoven.[8]
