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 the turning

ௗమ௬ ௗ௫మ ൌ ଶ ௫య ൅ ଵଶ ௫ ర ൌ0 Using GDC, OR solving ଶ ௫య ൅ ଵଶ ௫ ర ൌ0 ሻ ൌ െ ଵ ଽ (b) [Correct shape and asymptotes x = 0 and y = 0] [min. point (-4, -1/8), inflexion (-6, -1/9) ] [x-intercept (-2, 0) ] (c) Reading off from the intersection points and vertical asymptotes ൏െ 1 ൏ 6 point instead of the point of inflexion. Many students did not indicate the minimum point in their answer. Quite a number of students only show one set of solutions but left out the solution

൏ െ1 6. Two boats departs from the same pier at right angles to each other. The faster boat travels at 8km/h and the slower boat travels at 6km/h. (a) Show that the rate at which the distance between the boats changes is independent of time. [3 marks] (b) A cruise ship departs from the same pier according to ൅3 , where the distance units are km and the time units are hours. Find the speed of the cruise ship. [2 marks] 5 (a) Relative distance, ൌ√ ሺ݂ሻଶሻ ൌ ඥሺ8ଶ ൅6 ଶሻݐ ݐkm Rate of change of distance, ௗ௦ ௗ௧݄/݉݇ . Hence rate is independent of time.) (b) Direction vector of the cruise ship ൌ ቀ3 2ቁ Speed of the cruise ship ൌ √3ଶ ൅2 ଶ ൌ √13 Not well attempted. Many students could not formulate the equation Sൌ ඥሺ8 ଶ ൅6 ଶሻݐ 7. The plane ߨhas equation ൌെ 2 8 . P is a point on the plane with coordinates ሺെ4 ,6 ,5ሻ. A line perpendicular to the plane passes through P, find the exact value of the shortest distance from the line to the origin. 5 Method 1. ሬሬሬሬሬԦหൌอ ൭ െ4 6 5 ൱อ ൌ √4 ଶ ൅6 ଶ ൅5 ଶ ൌ √77 Perpendicular distance of plane to origin ൌ |ௗ| | ൌ ଶ଼ √ଷమାଵమାଶమ ൌ ଶ଼ √ଵସ ൌ2 √14 By Pythagoras Theorem, taking ሬሬሬሬሬԦ as the hypotenuse or otherwise, shortest distance = ට√77 ଶ െሺ 2√14ሻଶ ൌ √21 Method 2. Let ሬሬሬሬሬԦ be the vector perpendicular to the plane. ሬሬሬሬሬԦ൭ 3 െ1 െ2 ൱ Since ሬሬሬሬሬԦ lies on the plane, ൭ ݉ ݉ ݉ ൱∙൭ 3 െ1 െ2 ൱ൌെ 2 8 ൌെ 2 8 ൌെ 2 ሬሬሬሬሬԦൌ൭ െ6 2 4 ൱ Shortest distance = ሬሬሬሬሬԦหൌอ ൭ െ6 2 4 ൱െ൭ െ4 6 5 ൱อ ൌ อ൭ െ2 െ4 െ1 ൱อ ൌ √2 ଶ ൅4 ଶ ൅1 ଶ ൌ √21 Fairly well attempted. Recommend that students sketch to visualise the question. Common error to substitute the line equation into the plane. Ie. ൭ ݉ ݉ ݉ ൱∙ ൭ 3 െ1 െ2 ൱ൌെ 2 8 Note that the above will give ൌ0 since the intersection is just P(-4, 6, 5). O P(-4,6,5) π

Method 3. Equation of line passing through P, perpendicular to the plane ൌ ൭ െ4 6 5 ൭ 3 െ1 െ2 ൱ൌ൭ ݉ ݉ ݉ ࡾ The foot of the perpendicular, N is when line is perpendicular to the normal of the plane. ൭ ݉ ݉ ݉ ൱∙൭ 3 െ1 െ2 ൱ൌ0 ൌെ 2 ሬ ሬሬሬሬሬԦ ൌ൭ െ4 ൅ 3ሺ2ሻ 6െ2 5െ2 ሺ 2 ሻ ൱ൌ൭ 2 4 1 ൱ Shortest distance = ሬሬሬሬሬሬԦหൌอ ൭ 2 4 1 ൱อ ൌ √2ଶ ൅4 ଶ ൅1 ଶ ൌ √21 Question requires the exact value and the answer 4.58 is penalised. 8. Given that ଵ ൌ2 and √3 are roots of the cubic equation ଶ ൅ ൌ 0 where a, b, c ϵ R, (i) write down the third root, ଷ, of the equation; (ii) find the values of a, b and c. (iii) Explain why ൅ 16 ൌ 0 has three distinct roots, ଵ, ଶ and ଷ. 8 (i) √3 (ii) √3√3൯ቁ ൅4 ൅4 ሻ െ8 (iii) െ8 ሻ √3√3൯ቁ Hence three distinct roots, ଵ, ଶ and ଷ. Well attempted. Most were able to observe the repeated root in 8c). 9. A group of 6 men and 6 women are assigned to sit in three distinct rows of four seats each. (a) Find the number of ways in which the 12 people can be arranged if there is no restriction. [2 marks] (b) Find the number of ways to arrange the 12 people if each row has at least one woman. [4 marks] 6 (a) No. of ways = 12! = 479001600 (b) Method 1. No. of ways that 12 people are seated with no women in at least one of the rows ൌ ቀ6 4ቁ ൈ4 !ൈ ቀ3 1ቁ ൈ8 ! [select a row of all men ቀ6 4ቁ ൈ4 !, arrange the rest] No. of ways ൌ1 2 !െ ቀ6 4ቁ ൈ4 !ൈ ቀ3 1ቁ ൈ8 ! [Complement set] ൌ 435456000 Method 2. No. of ways ൌ1 2 !െ ቀ8 6ቁ ൈ6 !ൈ ቀ3 1ቁ ൌ 435456000 Method 3. 9b) is poorly attempted. Most students were not able to use the complement set to solve the question.

Topi c Section B Questions/ Solutions Marks 10. (a) Find the values of h for which the following system of equations has i. no solution, ii. an infinite number of solutions. ൌ െ 2 ൌ 5 ݄ [4 marks] (b) Given that the system of equations can be solved, find the line of intersection of the planes with direction vector ൭ 2 3 4 ൱ . . [5 marks] (c) Show that the line ௫ିସ ଶൌെ 5 lies on the plane ଵ. [5 marks] (d) The z-axis meets the plane ଶ at the point P. i. Find the coordinates of P. [2 marks] ii. Hence find the coordinates of F, the foot of the perpendicular from P to ଵ. [5 marks] 21 (a) ൭ 12 െ 2 െ 12െ 1 0െ 43 െ2 5 ݄ ൱ ோଶାோଷ ሱۛۛۛሮ ൭ 12െ 2 04െ 3 0െ 4 3 െ2 3 ݄ ൱ ோଷାோଶ ሱۛۛۛሮ ൭ 12െ 2 04െ 3 00 0 െ2 3 ൅3 ൱ i. There is no solution whe