2021 H2 Math RI Prelim Paper 2 Solns

2021 Y6 H2 Math Preliminary Paper 2: Solutions with comments Page 1 of 28 2021 Year 6 H2 Math Preliminary Paper 2: Solutions with Comments Section A: Pure Mathematics [40 marks] 1 Functions f and g are defined by 2( 1)f: e , , 1g : , , 1 2.2 xxx x xx x − ∈ ∈ ≤<−   (i) Sketch the graph of f( )yx= . [1] (ii) If the domain of f is restricted to xk≥ , state with a reason the least value of k for which the function 1f− exists. [2] In the rest of the question, the domain of f is xk≥ , using the value of k found in part (ii). (iii) Find 1g () x− and show that the composite function 11gf−− exists. [4] (iv) Find the range of 11gf−− . [1] Solutions Comments (i) [1] The graph and its properties can be easily obtained from the GC. However, a good number of students did not label either or both the y- intercept and minimum point. Students should also note that the graph is symmetrical about the line 1x= . (ii) [2] For 1f − to exist, f must be a one-one function. Least value of 1k = . Many students elaborated the horizontal line test instead of stating that the condition is for f to be a 1-1 function. In this case, students have to take note of the precise phrasing “every horizontal line [ ), 1,y kk= ∈∞ , cuts the graph of f at one and only one point.” 2( 1)e xy −=

Raffles Institution H2 Mathematics 2021 Year 6 __________________________________________________________________________________________ 2021 Y6 H2 Math Preliminary Paper 2: Solutions with comments Page 2 of 28 (iii) [4] 1Let 2 12 12 y x x y x y = − −= = − 1 1g () 2x x − = − For 11gf−− to exist, 11fgRD−− ⊆ . [ )1 ffR D 1,− = = ∞ [ )1 ggD R 1,− = = ∞ Since 11fgR = D−− , ∴ 11gf−− exists. Most students managed to find 1g() x− correctly, with a very small number leaving their answers as 1 1g () 2x y − = − , which is incorrect. Most students could remember the condition to check for 11gf−− to exist. Often mistakes were made in either finding 1gD − or keeping fD =  (unrestricted). (iv) [1] 11 gf[1, ) [1, ) [1, 2) −− ∞ → ∞ → [ )11gfR 1, 2−− = Note: [ )1 ffD R 1,− = = ∞ , [ )1 ffR D 1,− = = ∞ OR Since 11fgRD−− = , 11 1 ggf gR R D [1, 2)−− − = = = Most students who used the arrow diagram successfully found 11gfR −− . Students have to note the order of the functions involved and the horizontal asymptote in the graph of 1g− . A number of students also managed to recognize the relationship in alternative method, thus were able to state the answer correctly. 0 0

Raffles Institution H2 Mathematics 2021 Year 6 __________________________________________________________________________________________ 2021 Y6 H2 Math Preliminary Paper 2: Solutions with comments Page 3 of 28 2 (a) Three consecutive terms of a decreasing geometric progression has a product of 58