RI 2024 Year 5 H2 Math Timed Practice (Modified) (Qn)

© RI 2024 [Turn Over RAFFLES INSTITUTION 2024 YEAR 5 TIMED PRACTICE (MODIFIED) Higher 2 MATHEMATICS 9758 2 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF26) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 4 printed pages. RAFFLES INSTITUTION Mathematics Department

2 H2 MA 9758/2024 RI Year 5 Timed Practice 1 A smoothie shop sells three different types of smoothie blends, Berrylicious, Tropicality and Pumptein. The smoothie blends are dispensed from machines and each serving is prepared by blending protein powder, mixed fruits and milk. Relative to a serving of Berrylicious, a serving of Tropicality uses 50% the amount of protein powder, 30% more the amount of mixed fruits and 75% the amount of milk. Relative to a serving of Berrylicious, a serving of Pumptein uses 50% more the amount of protein powder, 60% the amount of mixed fruits and 25% more the amount of milk. A serving of Berrylicious, Tropicality and Pumptein weighs 360 grams, 350 grams and 355 grams respectively. Find the amount of protein powder, mixed fruits and milk required to make a serving of Berrylicious, in grams. [4] 2 A graphic calculator is not to be used in answering this question. Given that 3 4i is a root of the equation 3 25 25 0,z az bz    find the values of the real numbers a and b and the remaining roots of the equation. [4] 3 With reference to the origin O, the points A and C have position vectors a and c respectively, where a and c are non-zero and non -parallel vectors. The point E lies on AC such that :AE AC = 2 : , where 2.    (a) Find the area of triangle OAE in terms of  , a and c. [3] (b) Given that the area of triangle OAE is 3 10 a c , solve for the value of  . [1] (c) Given that a is a unit vector, give the geometrical interpretation of (i) a.c , [1] (ii) .a c [1] (d) If 90OAC   , explain why a .c is positive. [1]

3 H2 MA 9758/2024 RI Year 5 Timed Practice [Turn over 4 The line 1l contains the points A and B with coordinates ( 1, 3, 7) and ( , 0, 2)a  respectively, where a is a constant. The line 2l has equation 7 10 , 10.2 3 x y z    It is given that 1l and 2l cross at the point C. (a) Find the value of a and the coordinates of C. [4] (b) The point P has coordinates ( 2,9,8). Find the point on 2l which is closest to P and hence find the exact perpendicular distance between P and 2l . [4] 5 On the same axes, sketch the curves with equations 2 5y x x  and 16 3y x  , stating the exact coordinates of any points of intersection with the axes. Hence solve exactly the inequality 2 5 16 3 .x x x   [8] 6 The curve C has equation 2 1 e tan 0,xx a y   for 0,x where a is a positive constant. (a) Show that    2 2de 2 1 .d x ya x x yx    [2] It is given that C has one stationary point. (b) Find the x-coordinate of this stationary point and d etermine its nature, showing your working clearly. [3] 7 A curve C has parametric equations     2 2 1 , 1 .x t y t    The point A is on C such that the tangent to C at A has gradient 2. (a) Find the equation of this tangent and hence show algebraically that this tangent never meets C again. [5] (b) The tangent and the normal to C at A meet the y-axis at points P and Q respectively. Find the area of the triangle APQ. [3]

4 H2 MA 9758/2024 RI Year 5 Timed Practice 8 The function g is defined by 2g : ln ( 3) 1 , for , 3.x x x x      (a) Explain why 1g exists. [2] (b) Find 1g ( )x and write down the domain and range of 1g . [4] (c) The function f is defined by   2 f : 2 3 , for , 0x a x x x     for some real constant a. Show that the composite function 1fg exists. F ind 1fg ( )x and its range in terms of a. [4] 9 An architect is designing the structure of a slide for an indoor children’s playground. As shown in the diagram below, part of the structure has a rectangular base ABCD and a rectangular slide surface BCEF and two sides ABF and DCE which are perpendicular to the base ABCD. It is given that the base ABCD is part of the plane with equation 2 6 7 7,x y z    and the line BF has equation 1 1 18 8 , . 5 1                       r  (a) Find the position vector of B. [3] (b) Find the angle between the slide surface BCEF and its base ABCD. [3] (c) Show that the cartesian equation of the plane containing side ABF is 62 5 22 138x y z   . [3] It is now given that the other side DCE is part of the plane with equation 62 5 22 200.x y z   (d) The playground’s theme requires the slide surface width to be between 80 cm and 100 cm. Taking each unit in the architect’s design to be a metre, determine if the slide meets the requirement. [3] E F A B C D