YTSS A Math 2024 Prelim P1 QP

NAME: ( ) CLASS: YISHUN TOWN SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXPRESS / 5 NORMAL ACADEMIC ADDITIONAL MATHEMATICS PAPER 1 (4049/01) DATE : 21 August 2024 DAY : Wednesday DURATION : 2 h 15 min MARKS : 90 READ THESE INSTRUCTIONS FIRST Do not turn over the cover page until you are told to do so. Write your name, class and class index number in the spaces at the top of this page. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 90. MARKS OBTAINED FULL 1 4 2 4 3 5 4 6 5 6 6 6 7 8 8 7 9 8 10 9 11 10 12 8 13 9 TOTAL 90 This question paper consists of 21 printed pages and 1 blank page. 60

2 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation , Binomial Expansion where is a positive integer and 2. TRIGONOMETRY Identities Formulae for !" =++ !"#$# ! !"##$ ! "!!±!= ( ) !""!!!!! #$$$#%" !$$$#%!#%!%#% ++!! " # $$ % &++!! " # $$ % &+!! " # $$ % &+=+ ''' !!" !" !( ) ( )( ) ! "###" !$! ! ! !""" !"! " ! " +!!=!="" # $ %% & ' !! !! !! !! !! !! "#$%&""#' $()%'&" %"#''*) += += =+ ( ) ( ) ( ) ! !! !!!!! !!! "! "!"! "!"!"! "!"!"! ! !!!! "#$% "#$!!"#$&&&&&&&&&&&&&&&&&&& '($!%%)*'!'($)*'!)*' )*''($!!'($&&&&&&&& "#$"#$% "#$"#$"#$&&&&&&&& '($'($)*')*')*'&&&&&&&& '($)*')*''($'($&&&&&&&& ! = !=!=!= = ±=± =± ±=± ! ! !"#! !"# A#BB#" ! B C # A " !"#$ % &'!$ !"#!"#!"# $$$ =! "+= ==

3 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) [Turn over Answer all questions. 1 (a) Write down the turning point of the curve , stating clearly whether it is a maximum or minimum point. [2] (b) In a clearly labelled diagram, sketch the graph of , indicating clearly the coordinates of the turning point and axial intercepts. [2] ! "!"=! + ! "!"=! + Maximum point (0, 1) B1 maximum B1 (0, 1) x O y (0, 1) 1 B1 shape of graph B1 label

4 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) 2 A cylinder has radius cm and volume cm3. Find, without using a calculator, the height of the cylinder, in cm, in the form , where a and b are integers. [4] Volume, The height of the cylinder is cm. !" "#! !" # $#%! " !" #!"+ !" #$ % $& "! !& !!!"+ = " ! "# ! "$ %! !& ! %' ! (& )# ! '! ( * !! *! ! *! ! !" ! !$ !' "$ ! ($ "* ! ! != ! != ! !+=" !+ +!!= ! =+ !" # $ %+ B1 M1 for expansion of M1 for multiplying with conjugate A1 !"! !#!

5 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) [Turn over 3 (a) Explain why the principal value of cannot be . [1] , hence principal value cannot be . B1 (Do not accept as ) (b) Given that , where , and is an obtuse angle. Without using a calculator and leaving your answers in terms of p, find (i) , [2] (ii) . [2] ! "#$% " !"# $%$%&' ! !" !"# $ % !!"## ! !" ! "#$% "& !"#$ =%&%&'( ! "#$% ""& !! !"#$ =+%&%&'( !"# !!=" !!> !!"#$ !"#$%&'() ! "#$ $%&! '! !""#$%=&'() =% ! "#$! ! ! ! ! "#$% %&' " ! ! ! != ="

6 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) 4 (a) Find the remainder when is divided by . [2] Let The remainder is . (b) When a polynomial is divided by and , the remainders are 1 and 2 respectively. Find the remainder when is divided by . [4] Degree of remainder is less than degree of divider. Let the remainder be . Then Therefore the remainder is . !"#!$ $ !!! !++! !!+!" #$ % & ! '' !!! ! !=++ ! ! " #$ % " &$ '"%$ ((" #$ ' %) != !+ + !! =! !"!!" #! !!!!"!!!" #! !" # ! $" #!!!! !" #+!" # " $ #"% $ # &" #!! ! ! " ! #=! ! ++ !" # $ # #" # $ #!%% # % "%$% #"#$ "%$ & #% ''''''''''''''''''''' % ()*+',-./'"#$&'' 0 !" !" ! ! " = += !"=#$%& += '= ' =' = !! !! !"!!+ M1 A1 M1 M1 M1 A1

7 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) [Turn over 5 (a) Prove the identity . [4] Proof 1 Proof 2 (b) Hence find the value of A for such that . [2] Reference angle !! ! ! "#$ "#$ $%& $%& $%& ' ( $%& ' ( "#$ !!" ! " ! " ! " !!! + ! = ( )( ) ( ) ( ) !! ! ! !! ! ! !! ! ! ! ! !! !! ! ! ! ! !! "#$ %&' %&' '() '() '() * H '() * H %&' %&' '() '() '() %&' %&' '() '() %&' %&' '() %&' %&' '() '() '() %&' %&' '() %&' %&' '() '() '() %&' %&' '() %&' ! ,#$- !" ! " ! " ! " !" ! " ! " ! " ! " ! " !" ! " !" !" !" " ! " " !! ! =! ! + ! =! ! + ! =! ! + =+ ! + =! = = -*.L&01)H ( ) ( ) ( )( ) !! "#$ %&' %&' '() '() '() * H '() * H %&' %&' '() '() %&' %&' '() '() '() * H '() * H %&' * H %&' * H '() * H '() * H %&'* H %&' ! ,#$--*.L&01)H !" ! " ! " ! " !" ! " !" ! " ! " ! " !" !" !" !" !" !" ! =! ! + ! =+ ! ! + ! =! + ! + ! =+ + ! = = !"!!"## !! ! ! "#$ "#$ $%& $%& $%& ' ( $%& ' ( )*!+,!" ! " ! " ! "!! + ! = ! !! ! !"#$ "#$ $%& $%& $%& ' ( $%& ' ( )*!+, "#$ ! )*!+, !" ! " ! " ! " ! !! + ! = ! =! !"#$%&=! " #$%!&'( )$*)&++"%+,$-$( ! ! !=" " =" M1 M1 M1 A1 M1 M1 A1 B1 B1 M1

8 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) 6 (a) The curve and the line intersect at the points A and B. Find the x-coordinate of A and of B. [3] Sub. (2) into (1), !!""+= !" #!"!+ = !" ! #!""+= !! !" # !" $ ! % !" !" !+ = =! !! ! ! !" # !" "$ % !&% '& $ !((((()*(((( ' !(((((((((( #! ! !! !! ! !! !! !! "" !+= !+ = +!= !+ = == ! =! =! M1 for substitution. M1 for A1 for both x-coordinates !" # !$ # %!!!+ =

9 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) [Turn over (b) Hence solve for the value(s) of x in the simultaneous equations [3] From (a), ! "# "# $ %"# "# &' ! "" ! " += =! ! "# "# $ %$&"# "# ' "# ! "# ' ( %!& ! "" ! " !" += =! !+ = !! !! ! "# !$$$$%&$$$$$$ '( ! ! ! =! = !" #$ !" !% ! ! ! =! = M1 M1 A1 (Accept ) ! ""# $ % $ %#&#""## !

10 2024 YTSS Preliminary Examination Secondary 4E5N Additional Mathematics EXP (4049/01) 7 The equation of a curve is , where . (a) Show that the gradient of the curve can be written as , where p, q and r are integers. [2] (b) Find the range of values of for the graph to be decreasing. [3] For decreasing graph, Since . Hence ! ! "#!" ! = !!>!" ! "#$ $ + ! !" #!" != ! " " # !$%" & '( ( !" & ' !" & '))$*+,-'% !! !" ! !! !! ! ! ! ! ! = != != !"!! "! ! "< ! "# $ % &! ! ! < ! "# "!!>> !" # $ % "# $ ! &#$ ! ! ! ! < > > M1 A1 M1 M1 A1