RI 9758/01 Prelim 2023

 RI2023   [Turn over                 CANDIDATE NAME   CLASS 23     MATHEMATICS 9758/01 Paper 1 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. Give non-exact numerical answers correct to 3 signifi cant 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 questi on specifically states otherwise. Where unsupported answers from a graphing calculat or are not allowed in a question, you are required to present the mathematical steps usi ng mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. FOR EXAMINER’S USE Q1  Q2  Q3  Q4  Q5  Q6  Q7 4  4  4  5  8  9  10 Q8  Q9  Q10  Q11  Q12  Total 10  10  11  12  13                               100    This document consists of 23 6 printed pages and 1 blank page.   RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2   H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 1  1 Without using a calculator, solve the inequality 2 95 .11 x xx  [4]   2 Griffles popcorn is available in bags of 3 sizes: small, medium and large. At the Griffles physical store, Amanda bought 3 small, 7 medium and 1 large bag of Griffles popcorn and paid a total of $85.50 after enjoying a 5% discount off the usual selling price. Beatrice found out that at the Griffles onlin e store, for every 2 bags of the same size purchased at the usual selling price, a customer will receive an additional bag of the same size free. Beatrice paid $18.50 less than Amanda at the Griffles online store and received 3 small bags, 7 medium bags and 1 large bag of Griffles popcorn, inclusive of the free bags. Given that the usual selling price of a large ba g of Griffles popcorn is 2.4 times that of a small bag of Griffles popcorn, write down and solve equations to find the usual selling price of a bag of Griffles popcorn for each of the sizes. [4]   3 The volume of a sphere with radius r is given by 34 3 r and the surface area of a sphere with radius r is given by 24 r . (a) A meteorite enters the Earth’s atmosphere and starts burning up in such a way that its surface area is decreasing at a constant rate of 100 cm 2 per second. Assuming the meteorite is always spherical, find the rate at which the radius is changing when the radius is 5 cm. [2] (b) Another meteorite enters the Earth’s atmosphe re and burns up at a rate such that its volume is decreasing at a rate proportional to its surface area. Assuming the meteorite is always spherical, show that the radius decreases at a constant rate. [2]   4 (a) Describe a sequence of transformations that will map the graph of  fyx onto the graph of  12 fyx . [3] (b) A point P ,ab is said to be R-invariant if it lies on both the graphs of f( )yx and 1 .f( )y x Sketch the graph of a function f ( )yx , which is defined for all real x, for which f0 x  for all real x and there are no R-invariant points. [2]

3   [Turn over  H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 1  5 The function f is defined by 32f: , 1 xx x   for x  , 1x . (a) Find 1f( ) x and state its domain. [3] (b) Sketch on the same diagram the graphs of f( )yx and 1f( ) ,yx  giving the equations of any asymptot es. Hence find the exact solution of the equation 1f( ) f ( ) .x x [5]   6 Alice bakes a large pie and removes it from th e oven at 1 pm. The temperature reading of the pie using a food thermometer is 200 C at that instant. She left the pie on the kitchen countertop to c ool down. The temperature of the kitchen is kept at a constant 32 C. The temperature of the pie t minutes after it is removed from the oven is  C. This temperature decreases at a rate proportional to the difference between the temperature of the pie and the temperature of the kitchen. At 1.15 pm, Alice checks the temperature of the pie again and it is 180 C. (a) Find an exact expression for  in terms of t, simplifying your answer. [5] (b) Sketch a graph of   against t. [2] (c) Food safety experts caution that bacteria grow fastest in food within the temperature range of 5 C to 60 C, known as the temperature danger zone. Find the latest time to the nearest minute to safely stor e the pie to avoid this danger zone. [2]   7 (a) Find  1 1 21 21 n r rr  . (There is no need to express your answer as a single algebraic fraction.) [4] (b) Explain why  1 1 21 21 n r rr  is a convergent series and state the sum to infinity. [2] (c)  Deduce  4 1 23 25 n r rr  .   [4]

4   H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 1  8 (a) Use the substitution 1e , xu or otherwise, to find  2 3 1 e d 1e x x x    in exact form. (You do not need to simplify your answer.) [4] (b) Find 1 1 0 tan dx x  in exact form. [3] Function f is defined by   1 3 tan , for 1 1, ef , for 1 2, 1e x x xx x x       and   ff 3xx for all real values of x. (c) Find  2 2 fd x x  , leaving your answer in exact form. [3]   9 The curve C has equation 2 ,ax xy x b   where a and b are positive constants such that 1a b . (a) Given that C has no stationary points, use diffe rentiation to find the relationship between a and b. [4] It is now given that 2a and 1.b (b) Sketch C, stating the equations of any asymptot es, coordinates of any turning points and of the points where C crosses the axes. [4] (c) Hence find the volume of the solid fo rmed when the finite region bounded by C and the x-axis is rotated completely about the x-axis. [2]   10 Do not use a calculator in answering this question. (a) Solve the equation 32 46 4 0zzz   . [4] (b) Hence solve the equation 32i46 i 4 0ww w  , giving the roots in the form ie,r  where 0r  and 02   . [4] (c) For the value of w found in part (b) with the largest  , find the smallest positive integer n such that nw is a positive real number, s howing your working clearly. [3]

5   [Turn over  H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 1   11 One geometric optimization problem concerns finding the largest area axis-parallel rectangle inside an object. The largest area rectangle problem h