ACSI 2023 Y3EXP FYE AMath P2 Solutions

1 FINAL EXAMINATION 2023 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 SOLUTIONS 1 It is given that a and b are the roots of the quadratic equation 2 2 10xx− −= and that ab> . Show that 3 22a b = −− . [5] ( ) ( ) ( )( ) ( ) 2 2 2 41 1 21x −− ± − − −= 28 2 ±= 222 122 ±= = ± 1 2 and 1 2ab∴=+ =− 12 12 1 21 2 a b ++= × −+ 122 2 322 12 1 ++ += = −− 322= −− 2 A circle with centre O passes through the points ( )1, 7P − and ( )0,8Q . (i) State the relationship between the perpendicular bisector of PQ and the point O. [1] (ii) Find the coordinates of O, given that the line 22yx= − passes through the centre of the circle.[5] (iii) Hence find the equation of the circle. [2] (i) The perpendicular bisector of the chord PQ passes through O, the centre of the circle. (ii) ( ) 87 101 PQm −= =−− 1m⊥⇒= − Midpoint of PQ 10 78,22 −+ += 1 15,22 = − Equation of perpendicular bisector of PQ : 15 1 22yx  − = − −−   MARKER’S COMMENTS: Students should infer from the question that exact answers are required. Therefore, they should not be relying on non-exact roots from the calculator. COMMON MISTAKES: 1) The perpendicular bisector is the radius. No, the perpendicular bisector is a continuous line. 2) A line drawn through point O is perpendicular to the perpendicular bisector. No, there are so many ways to draw this line. COMMON MISTAKE: Finding the gradient of the line perpendicular to 22yx= − .

2 7yx∴= −+ ------------(1) 22yx= − -------------(2) Sub (1) into (2): 72 2xx−+= − ( )3 ; 2 3 2 4xy∴= = −= ( )3, 4O∴ (iii) Radius of circle ( ) ( ) 22 3 0 4 8 5 units= − +− = Equation of circle: ( ) ( ) 22 3 4 25xy− +− = 3. The polynomial ( )fx is given by ( ) 329 30 23 4fx x x x=− −− (a) Factorise ( )fx completely. [4] (b) Hence, prove that the equation 9 23 4 30xx x x−= + has only one real root. Find the solution. [3] (a) By Trial and Error Let 4:x= ( ) ( ) ( ) ( ) 32 9 4 30 4 23 4 4 0fx = − − −= ( )4 is a factorx⇒− . 9 30 23 4 4 36 24 4 9 6 10 −−− ( ) ( )( ) 249 6 1fx x x x=− ++ ( )( ) 2 43 1xx= −+ (b) For ( ) 0fx = , 14, 3x= − 9 23 4 30xx x x−= + 329 30 23 4 0xxx⇒ − − −= Replace “ x ” with “ x ” (NA) 14 or 3x⇒= − 16 one real rootx∴= ⇒ MARKER’S COMMENTS: • Students should remember to write all their coordinates in brackets. • Some students used an alternative method by equating the distance from P and Q to the centre O. This is not recommended as it is highly tedious and likely to lead to careless mistakes. CO