2024 Mid-Year Holiday Assignment 1

Page 1 of 3 RAFFLES INSTITUTION RAFFLES PROGRAMME 2024 YEAR 3 MATHEMATICS MID-YEAR HOLIDAY ASSIGNMENT 1 Total mark: 40 marks Duration: 1 hour Name: ( ) Class: 3( ) Date: Topic 1: Surds 1 Given that ( )42 5 2 3 2 2 3 2 2 ab − + − = + − , find the value of a and of b. [3] [Ans: a = 36, b = −5] 2 Solve 3 3 2 4 6 1x x x− + − = + . [3] [Ans: x = 4] Topic 2: Graphical Solutions of Equations 3 The diagram shows part of the graph of 21 1016 4yx x= − − for 1 82 x (on P.2) . (a) Find, from the graph, (i) the solutions of the equation 21 1014 04 x x− − = , [2] (ii) the range of values of x for which 2 4064 16 2xx x− −  − . [3] (b) By drawing a tangent, find the gradient of the graph at the point where x = 5. [2] (c) By adding a suitable straight line to your graph, solve the equation 32 8 52 40 0x x x+ − + = . [3] [Ans: (a)(i) x = 0.7 or 7 (ii) 0.8 7.6x (b) −2.11 (c) x= 0.9 or 3.5]

Page 2 of 3 Topic 3: Geometrical Properties of Circles 4 In the diagram, AC is a diameter of the circle centre O, ADC = 36 and BAC = 25. Calculate (i) BOC, (ii) CAE, (iii) EBC. [3] [Ans: (i) 50 (ii) 29 (iii) 29] D C 25° O 36° B A E 

Page 3 of 3 5 In the diagram, AE is a tangent to the two circles at B and E. Similarly, AD is a tangent to the two circles at C and D. Given that = 30BAˆC , EBX = 38 , = 34ˆYCD , and CY = DY. Find the values of (i) ABC , [1] (ii) CYX , [1] (iii) EFD , [1] (iv) EFX . [2] [Ans: (i) 75 (ii) 113 (iii) 75 (iv) 30 ] Topic 4: Quadratic Functions 6 (a) By completing the square, find the maximum value of 254y x x= − − and state the value of x when this occurs. [3] (b) Sketch the graph of 254y x x= − − , showing clearly the maximum point and intercepts with the axes. [3] [Ans: (a) max y = 9 when x = −2] 7 (a) Find the least value of k for which x x k k( )+ − is never negative for all real values of x. [3] (b) Prove that the line y x m= +3 will meet the curve y mx x= + +2 2 for all real values of m. [3] [Ans: (a) –4] Topic 6: Binomial Theorem 8 (i) Write down and simplify, in ascending powers of x, the first three terms in the expansion of 9 2 4 x − . [2] (ii) Hence, find the coefficient of 2x in the expansion of ( ) 9 1 (2 ) 2 . 4 xxx − + −  [2] [Ans: (i) 2512 576 288xx−+ (ii) 640] 30 38 34 B F D E A X C Y