SCGS 2025 AM Prelim Paper 2 solutions

2025 A Math Prelim Paper 2 Solutions 1(a) (b) t= t = 15.3 min 2(a) Or !" #"" $ % $&'% $"""""" %"""""" &' $"""""" &'(%"""""" ) &' $"""""" &' %"""""" &' &' $"""""" &' %"""""" $&' $"""""" &' %"""""" &' % ! ! ! ! ! "" # # ! A% # # # ! ! = = = = = =+ =+ =!= !"#$%&&& $&&&&&& ! " !"#$%&= !" # # #$% $%&& # # # # # #!!" =#$%& !"#$%&& '"#% !"#$%& !"#$%& !"# !$ % &!!!! = ( )( )!" !# $!!!+ = !"#$%%%%&&!!== ! !"# $%&&&&# '%()*## !!== ! ! !! ! "#$%! #$% " #$% ! #$% " & != =! =!

2 (b) x = 5 or x = -1 (reject) 3(a) (b) x 1- 1 1+ + 0 + ! "! ! "#$ ! #$! "#$ #%$"! #%$ &&&&'#%$ (#$ ! ! #%$ #$) #$* +#$ ! " # " ! A!A # B&'() B ** = = !" #$%&== = '( ! " # !$%&' (! )* %&' ( !* %&' $!!!! + = ! " " " #$% & !' (#$% &! (' #$% ) ! !! +!! = ! ""! #$% &! '( #$% & !( '!!!! + = ! "" ! #! $ %&'( &'( "! ! ! ! =+ ! ! "! # $ %! ! ! ! =+!!"" # $%!! !!+ = + ( )( ) ! "# $ #% $ !! !! !! = !+ = !" #$ #" ! """=! + ! ! " #" $% $ # $C '( # ')*) % +,,-"./0123),4)152)3101.,/0-6)7,./1)0-2)C'* %(8 ! " "" " "! = !" + = !" = == " " ! ! ! "

is a point of inflexion. 4(a) (b) Max height = 42 m Time = 100 s (c) t = 50 or 150 \length of time = 100s Or !"# $%! !"#!$! $ %!" "=+ ! ( ) ( ) ( ) ! ! ! ! " !## !!$# " "## "## !!$# " "## %# !!$# !! ! ! =! ! + "#=! ! ! +$% =! ! + + ( )!"#! "$$ !%$ !=! ! ( ) ( ) ( ) ! ! ! "#! "$$ %!!&$ " "$$ "$!&$ "$$ !&$$ "$$ &$ ! ! ! ! !! = != != =± ( ) ( ) ( ) ( ) ( ) ! ! ! ! "#! "$$ %!!&$ "$$ !&$$ "$$ &$ $ "$$ &$ "$$ &$ $ D &$(D "&$( $ &$ "&$ )*+,-./01"$$2 ! ! ! !! !! ! !! " !# $%!! #&' !+ !! #$% $%&' &' !! # ##

5(a) (b) = 0 \ (x -2) is a factor Compare x2, - 5 = - 4 + b b = - 1 When f (x) = 0. x = 2 or !"# $ % %!! " !=+ + () ()! "# $ % $ $ $ $ !!= ! + !+ !"!=! !()! "# $ % $ & # $ % &!=+ + !"!=+!" # $$!!!! = + !!! " ! ! =! =! !"#$ % " & ! ' (!! ! !=!! + () () ()!" #$ " % " " & " ! " ' (=!! + ( )( )!" ""#! $ % " " #!!! ! ! " !!! + = ! + ! ( )( )!"# $ ! ! %!! ! !=! ! ! !!" # $ % % % ! &"!! !!= !!= ()()!! " #$% "! ±! != ! "!#$ " ±

6(a) Powers of x = 24 - 6r Since 24 and 6r are even numbers, subtraction between even numbers will give an even number. \ there are only even powers of x in this expansion. (b) k = 3 (c) Coefficient of = For the expansion Term independent of x = ()!" # $ ! #% ! ! ! "! #" ! + "# "#=!$% $%&'&' ( )!" " !# $! !!""! # !!"#"#=!$%$%&'&' !" ## $! !"! # !"#"#=!$%$%&'&' !" # $ " ! ! != = ! " #$ ! %&! !"!"#=$%$%&'&' ! "" #"! !"#=$%&' !" # # $ ! ! != ! = !!! ! " #" $ ! %$ & % ' # !"!"#= # = #$%$%&'&' ( ) ! !" # $#%!! "! !"## $%&' !"#$ #% #&$ !"! "##$%$ %&'& ' !" # #$ #$ !# #$ % & =+ = =

7(a) When t = 0, (b) When v = 0, (c) When t = 0 s = m = 0.31830 m When t = s =m or 0.95492 m When t = 4 s = Total distance travelled = !"#$ %&'$()! !== !"#$ % &'( )**"+**'( , '(, '- (**"+** -.))*"+*/.)))) ! !! ! !! !! ! !! ! += += += == !"#$ % &'( )**"+**'( , '(, '- (**"+** -.))*"+*/.)))) ! !! ! !! !! ! !! ! += += += == !"#$ %&'() (#*+$ %() ,-./+& 01 & 1 ,( #*+) ,( , 2 ,2 , (,#*+$ %() !" " !" # "! # # # !" !! !! ! ! ! !! !! !! ! !! !! =+ =+ + == "#=+$%&' "#=+$%&' =(= ( )*+= + (,-./ 0 !!!"# !" #$%&' ! !! ! "# $=%&'( !" # $%&' ()*( +,-.!-/(01 ! !! ! "# $= $ $%&'( !""# $%&$$!'(($)!!! !++=

8(a) (b) ( ) !" ! " " " " " #$% &'( &)* &'( H,,,,,,,,- &'( &'(*.& ,,,,,,,,- L0( &'( &'( ,,,,,,,,-, &'( L0( H ,,,,,,,,- &'( &)* H,,,,,,,,- &'(*.& ,,,,,,,,- L0( &)* ,,,,,,,,-,1$%,2&3.4(R !! ! !! ! !! ! !! !! ! ! !! =+ !" +#$%& + + !" #$%& ( ) ! ! ! ! " "" #AB &'( ) &*B + )("& ("& &*B )("& &*B ) + &*B )&*B &*B ) , + -+. /-).- ).&*B !-). &*B ,0N,/N233333"43333 +0+,/N23-56. 7A&*(3AB89'3:3;/02!+) ;/02 < 3++)0! !! ! !! ! ! !! !! ! ! ! = !" =#$%& = =' +' = '± ' '= =' =

9(a)(i) When the curve meets the y-axis, x = 0 9(a)(ii) Since >0 for x >2, y is an increasing function. ! ! !" ! # $ !%$!% $! "%$&% # $ !% !'''''('$ !% !" ! "! ! !! ! != ! !! != ! ! ! "! # "$ % ! &! 'E)"G+,-./0.,/E1)2.3. ! %4 5 678)-G/,./0.,/E1)2.G9. 5! $ % & !5 ! " "! !" !" ==! ! == != ! ! =! + ! "#$% !& % ! '& (! F' !! ! > > " > ! "! "# ! $ ! ""= !

9b ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ! ! ! ! " " " " # # $ "%$ $ "% $$ "%&&&&&'&(")(") "%&&&&&' * $ #+, & !,$ "#"#+ * -! * "%$- ! $* * "% -!$** "% -! %./ * !#.% %!#"- %./ * % %./ "% -! % %./ * %./ ! "" ! """ " # " # ! "" # # "! " "!" "!" $ $ $ "!" =+ =+ + + + + == =+ = +=+ !" +=+#$ #$%& +=+ + =+ + =' +(= + ' ) )

Solution 1 10(a) Draw a straight line of against t. t 2 4 6 8 10 lg n 3.51 3.54 3.57 3.60 3.63 Best fit line (b) (c) (d) ()!"!" ##= ( )!"# "# "# $!! " #=+ !"! !"#$ %& !"#$ '( !()( ! ! " " = = = !"#$% !"%$&' ( ) *"*%+% ! ! != = ( )!"#$%& '"() '* !")$ +",----------------------------.+"+-/-,"(0! !" =#$ =