MI H2 Pre-U 1 Promo 2023 (Qn)

2023 MI H2 Maths Pre-U 1 Exam Paper Duration: 2 hr 15 mins Marks: 75 Attempt all questions. 1 The sum, nS , of the first n terms of the sequence 123, , , . . .uuu is given by 2 4nSn n . (i) Find nu in terms of n. [2] (ii) Hence show that the sequence is an arithmetic progression. [2] 2 A coffee vendor sells three types of coffee be verages: Latte, Cappuccino and Mocha. The cups of coffee come in three different sizes : Short, Tall and Grande. The number of cups for each type of coffee sold in a particular day is given in the following table. Short Tall Grande Latte 3 12 8 Cappuccino 4 8 7 Mocha 2 5 4 The price of a cup of coffee fo r each size is the same regardle ss of the type of coffee. The total amount collected on that day from th e sale of Latte, Cappuccino and Mocha is $147.90, $120.10 and $70 respectively. (i) Find the price of a short, tall and grande cup of coffee respectively. [3] (ii) The vendor offers a buy-one-get-one-free prom otion if a customer buys a grande cup of Latte and gives a 10% discount if a cust omer buys a short or tall cup of Latte. Andrew wishes to have 1 short cup, 4 tall cups, and 4 grande cups of Latte. How much does Andrew need to pay in total? [2] 3 (i) Sketch the curve C with equation  22 11xy , indicating clearly the coordinates of any points of intersection with the axes and the equations of any asymptotes. [3] (ii) The curve C undergoes a sequence of three transformations as follows: I: A scaling parallel to x-axis with a scale factor of 4, II: A scaling parallel to y-axis with a scale factor of 1 2 , III: A translation of 2 units in the positive y-direction. Find the equation of the resulting curve. [3]

2 4 The diagram below shows the graph of f( )yx . The asymptotes of the graph are 2x and 2y  . The graph crosses the x-axis at 1, 0 and the y-axis at 0, 1 , and has a turning point at 4, 1 . On separate diagrams, sketch the following graphs, indicating clearly the coordinates of any points of intersections with both axes and a ny turning point(s), and the equations of any asymptotes where possible. (i) f1yx , [3] (ii)  1 fy x . [3] 5 With reference to the origin O, the position vectors of points A, B and C are expressed as OA  , OB  and OC   respectively such that 1 OA OB OC        , where  is a non-zero constant. (i) Show that A, B, C are collinear. [2] It is given that 1 6  , 0 2 0 OA    and 1 1 2 OB    . (ii) Find the position vector of the point D that lies on the line segment AC such that AD : DC = 1 : 4. [3] It is given that the point E has coordinates 1, 1, 2 . (iii) Find the exact area of the triangle ABE. [3] (iv) Find the length of projection of OE   onto AB  . [2] x = 2 y = 2 (0,1) (1,0) O y x (4,1)

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