ACSI 2023 Y3 Core Mathematics Paper 1 Solution

ACS(Independent)MathDept/Y3IPCoreMathPaper1/2021/FinalExamination Core Maths Paper 1 2023 (Solutions) 1. (a) 312 42 11 4 − − 11 1 42 3 4 11 2 3 44 9 3 3 − = −= ÷ = = (b) ( ) 342 26 164 2 1 14 7 ab bc b − − ÷ ( ) ( ) ( ) 12 6 4 26 64 2 612 7 12 4 6 8 14 7 4 4 a bc b b ab c ab c − − +− −− −− = × = =

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 2 2. (a) ( ) 2 2 3 4,yx m= − +− minimum point is ( )3, 8− , 48 4 m m ∴−= − =− (b) ( ) 2 2 38yx= −− ( ) ( ) ( ) ( )( ) 22 2 2 Let 0, Let 0, 2 3 8 0 20 3 8 2 6 9 8 0 10 6 50 5 10 yx xy xx y xx xx = = −− = = −− − + −= = − += − −= y O x

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 3 3. (a) 108 4 2718 333 +− ( ) 33 6 43918 333 63 4 3318 333 2 18 4 3 16 3 ××= +− ×=+− =+− = (b) (i) Since they are on the same straight line, ( )( ) ( ) ( )( ) ( )( ) ( ) ( ) OR 15 3 1 1535 5 35 5 5 35 15 2 52 5 15 3 5 35 5 25 2 5 25 2 35 3 5 5 35 3 5 5 2 10 PQ PR PQ QRmm mm x xx x xx x x x xx x = = − − −−= = −− −− − = − − −=− − − = − −− + = − −+ = 2 10 5 5 x xx = = = OR Equation of the line, ( ) 115 5 When 5, 5 yx yx −= − = = (ii) Length of PR ( ) ( ) ( ) 2 2 35 5 31 45 4 24 2 6 units = − +− = + = =

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 4 4. ( ) ( ) ( ) 3 23 33 31 323 3 3 5 125 log log 5 5 5 log 5 5 5 x y x y xy xy xy = +=  = ×=  = ( ) ( ) 2 3 25 2 log 5 3 .....(1) 3 ......(2) subs (1) into (2), 243 xy x y xy yy = ∴= = = 3 3 3 9 243 27 3 When 3, 9 y y y y x = = = = =

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 5 5. (a) 52 03 BCm −= − 1=− Equation of the line L1, which passes through C and is perpendicular to BC is ( )51 0 5 yx yx −= − = + (b) ( ) 1 5 10 2 .................(2) 5 10 2 5 25 10 2 3 15 5 , 5..............(1) Subs (2) into (1), 5 yx x xx x x Ly x x −= −= +−= =− =− = + + When x = - 5, y = 0. Hence, the Coordinates of D are (-5, 0). (c) Area of quadrilateral ABCD ( ) ( ) ( ) ( ) 2 0 5 4301 5 0 2252 1 0 10 8 15 25 0 6 02 1 482 24 units = −− − = + +− + −− −−− − = =

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 6 6. (a) (i) 2 2 2 3 11 33 1 43 1 x xx xx x x x +−+ +−= − −= − (ii) 2 2 3 011 43 01 4 30 3 4 x xx x x x x +=−+ − =− −= = (b) ( )33 4 2 26 4 x xx − −+<≤− ( ) ( )33 33 4 2 26 44 8 24 9 3 18 6 4 16 11 33 22 3 xx xx x x xx xx x −− −+−< ≤ − < − − ≤− + <≥ < 1 13 x x ≥ ∴≤ < Integers that satisfy the inequality are 1 and 2.

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 7 7. (a) 2225 4 20 25a ab b−+ − ( ) ( ) ( ) ( ) ( )( ) 22 22 2 25 4 20 25 25 4 20 25 25 2 5 5 25 5 25 52 5 52 5 a ab b a ab b ab ab ab ab ab = −+ − = −−+ = −− = −− +−     =−+ +− (b) ( )( ) 2 2 2 8 5 8 4 54 8 2 32 216 2 22 22 5 8 40 52 20 5 20 o r 20 2 25 x x x x xx xx xx xx xx − = = = ∴ − −= + −= += −= =−=

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 8 8. (a) 2 3 40mx x− −= 2 3 202 3 2 2 xx αβ αβ − −= ∴+= =− (b) 22αβ+ ( ) ( ) 2 2 2 3 222 9 44 25 4 α β αβ= +− = −− = + = (c) Sum of roots Product of roots ( ) 22 22 2 2 22 22 2 4 2 33 5 24 2 4 β α βαα β αβ α β α αβ βα β αβ α β αβ    =+++ =+ +       =+++ =++ + += += + = ( ) 2 529 25 4 84 5 8 95 048xx −= + = − +=

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 9 9. (a) By using Pythagoras’ Theorem, 22AB BC+ ( ) ( ) 22 2 2 11 + 4 3 121 16 3 13 AC = = + = = Hence, angle ABC is a right angle. (b) (i) (ii) ( ) cos cos 180 cos 11 13 DAC BAC BAC ∠ = −∠ = −∠ =− tan .cos 11 43 11 13 11 13 1143 13 43 13 3 12 ACB BAC ∠ ∠ = = × = = (c) Given that AF is the reflection of AB in the line of AE, EF = BE, ( ) ( ) 22 2 2 22 22 43 2 8 3 16 3 4 8 3 52 13 13 3/ 623 EF FC EC xx xx x x x += − += − + += = = OR Using trigonometric ratio cos cos 43 2 13 26 43 ACB ECF x x ∠= ∠ = =

ACS(Independent)/Y3IPCoreMathP1/2023/Final Exam 10 10 (a) Let y = 0, ( ) ( )( ) ( )( ) 2 2 2 2 ( 3) 4 0 Discriminant 0 44 3 0 16 4 12 0 3 40 4 10 14 p x xp pp pp pp pp p − − += > −− − > −+> − −< − +< −< < Given that the curve has a minimum point, 0 30 3 34 a p p p > −> > ∴< < (b) (i) 2 45xx−+ (ii) 2 2 45 02 2 40 xx xx −+ >+− Since, ( ) 22 45 2 1xx x− += − + which is always positive for all real values of x, ( )( ) 2 2 2 2 40 0 20 0 5 40 5 or 4 xx xx xx xx +−> +− > + −> <− > ( ) ( ) ( ) 22 2 2 25 21 x x = − −− + = −+