ACSI 2016 Promo Paper 2 Ans

2016 HL Final Examination Paper 2 Section A Questions/ Solutions Comments 1. A polynomial ൅8 has factor ൅2 . Find the values of a and b. [6 marks] 6 Let ൅8 Since െ1 ሻ െ2 ሻ and െ1 ሻ are linear factors. By Factor theorem, ൅8 ൅ 24 ൌ 0 ൅8 ൅ 9 ൌ 0 Solving (using GDC or otherwise), ൌെ 3 and ൌ െ 6 Generally well attempted 2. (a) If ሻ, express ݔin terms of k and ߙ . b) Hence find the value(s) of x, where 0°൏ 360°, when ൌ ଵ ଶ and ൌ 70°. Give your answer(s) to the nearest degree. 8 (a) ሻ ሻ ሻ ୱ୧୬ ௫ ୡ୭ୱ ௫ ൌ ሺଵି௞ሻ ୡ୭ୱ ఈ ሺିଵି௞ሻ ୱ୧୬ ఈ ൌ ሺ௞ିଵሻ ሺ௞ାଵሻ ୲ୟ୬ ఈ (b) ൌ ଵି଴.ହ ሺି଴.ହିଵሻൈ୲ୟ୬ ଻଴° ൌ െ0.1213 [A1] ିଵሺ0.1213ሻ ൌ 173°, 353° (to 3 sf) Well attempted 3. (a) Describe a sequence of two geometric transformations which maps the function ሻ ൌ3 ଶ௫ାଵ to ൌ ଵ ௚ሺ௫ሻ. [4 marks] (b) If ଵ ௚ሺ௫ሻ intersects ݇at only one point, find the value of k corrected to 3 decimal places. [4 marks] 8 (a) 1. Reflect ݂ሻ in the x-axis. 2. Stretch by scale factor 4 parallel to the y-axis. Method 1: ଵ ௚ሺ௫ሻ ൌ3 ିଶ௫ିଵ ൌ3 ሾଶሺି௫ሻାଵሿିଶ ൌ ଵ ଽ ൈ3 ሾଶሺି௫ሻାଵሿ = ଵ ଽሻ 1. Reflect ሻ about the y-axis. 2. Scale by factor ଵ ଽ parallel to the y-axis. Method 2: ଵ ௚ሺ௫ሻ ൌ3 ିଶ௫ିଵ ൌ3 ሾଶሺି௫ሻାଵሿିଶ ൌ ଵ ଽ ൈ3 ሾଶሺି௫ሻାଵሿ = ଵ ଽሻ 1. Scale by factor ଵ ଽ parallel to the y-axis. 2. Reflect about the y-axis. Method 3: ଵ ௚ሺ௫ሻ ൌ3 ିଶ௫ିଵ ൌ3 ሾଶሺି௫ିଵሻାଵሿ ൌ ଵ ଽ ൈ3 ሾଶሺି௫ሻାଵሿ = ଵ ଽሻ Many students did not show any working to show the sequence of transformation .

1. Translate 1 unit parallel to the x-axis. 2. Reflect about the y-axis. Method 4: ଵ ௚ሺ௫ሻ ൌ3 ିଶ௫ିଵ ൌ3 ൣଶ൫ିሺ௫ାଵሻ൯ାଵ൧ ൌ ଵ ଽ ൈ3 ሾଶሺି௫ሻାଵሿ = ଵ ଽሻ 1. Reflect about the y-axis. 2. Translate -1 unit parallel to the x-axis. (b) ݇ ݇[M1: attempt to formulate equation] The minimum point is at x = 0.254 Hence, k = 0.254 Not well attempted. A lot of students do not think of finding the minimum point of the graph y= 3 ଶ 4. Solve the simultaneous equations ൌ 0 and ∗ ൌ3 , giving ݖand ݓ in the form of ܾ݅ where ܽand ܾare real. [6 marks] 6 ∗ ൌ0 Let ܾ݅ , ൌ0 ሻ ൌ0 Comparing real and imaginary parts, ൌ0 and ൌ0 Solving (using GDC or otherwise), ൌെ 2 ݅ ݅ Quite a number of students do not fully understand the concept of w* and mistakenly assume w*= - w or iw or ଵ ௪ 5. A curve has equation ൌ ଵ ௫ ൅ ଶ ௫మ് 0 . (a) Using differentiation or otherwise, find the coordinate of the non-stationary inflexion point of the curve. [3 marks] (b) Sketch the curve, stating the equations of any asymptotes and the coordinates of any turning points and points of intersections with the axes. [3 marks] (c) Hence find the solution set of the inequality െ ଶ ଽ൅ ଶ଴ ଽ ൐ ଵ ௫ ൅ ଶ ௫మ. [2 marks] 8 (a) ൌ ଵ ௫ ൅ ଶ ௫మ ௗ௬ ௗ௫ ൌെ ଵ ௫మ െ ସ ௫య Quite a number of students find