DHS 9758 2023 Prelim P2

© DHS 2023 This document consists of 19 printed pages and 5 blank pages. [Turn over Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2023 Year 6 MATHEMATICS 9758/02 Paper 2 19 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, name and class on 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 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 spe cifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using 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. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total Score Max Score 8 9 10 13 5 8 6 7 10 12 12 100

2 DHS 2023 Year 6 H2 Mathematics Preliminary Examination Section A: Pure Mathematics [40 marks] 1 (a) Given that f is a continuous and increasing function, explain with the aid of a sketch, why 1 0 1 f n r r nn − =  ∑ is less than 1 0 f( ) d .xx∫ [3] (b) Find a similar expression that is greater than 1 0 f( ) d .xx∫ [1] (c) It is given that 2f( ) 1xx= + and 10.n= (i) Using the expression in part (a) and your expression in part (b), find the lower and upper bounds of 1 0 f( ) d .xx∫ [2] (ii) Comparing your answers from part (c)(i) with the calculated value of 1 0 f( ) d ,xx∫ comment which limit is a better estimate. [2] 2 (a) A sequence of real numbers 123, , . . .uu u is such that 1 3 0.5 , for 1.nnu un+ = −≥ It is given that 1uc= where c is a constant. (i) Describe how the sequence behaves when (A) 5,c= [1] (B) 2.c= [1] (ii) Find the value of c for which 322 5.uu=− [3] (b) Another sequence is defined by 12 , 2,v pv= = where p is a constant, and 21 2 1, for 1.n nnv vv n++ =+− ≥ (i) If 3 21 12 ,v uu+= − find a relationship between c and p. [2] (ii) Find the value of p for which 5 77.v = [2]

3 DHS 2023 Year 6 H2 Mathematics Preliminary Examination [Turn over 3 Fig. 1 shows the net of a triangular prism cut from a square cardboard of side length 20 cm. The net consists of one rectangle z cm by x cm, two rectangles z cm by y cm and two isosceles triangles each with base x cm and perpendicular height h cm. When the shaded area of the cardboard is removed, the net is folded to form a triangular prism with a rectangular base as shown in Fig. 2. (a) Show that 2 5 20 hy= + and find x in terms of h. [3] (b) Show that the volume of the prism ( ) 43 21 10 100 1000 .10V hh h h= −− + Use differentiation to find the maximum volume of the prism. [7] 4 A tetrahedron has four vertices O, A, B and C where O is the origin. It is given that the coordinates of B are (1, 4, 6 ) and the line AB has equation 4 6 (2 ) aλ= ++ + ++r i j k i jk , where λ is a parameter. It is also g iven that the coordinates of C are (4, 4, 9) and the line AC has equation 4 4 9 ( 2) bµ=++ + ++r i jk ijk , where µ is a parameter. (a) Show that 0ab+= . [3] (b) Using a suitable cross product to find the normal to plane ABC, find the cartesian equation of the plane if its normal is parallel to 33−++ij k . [4] (c) Hence find the coordinates of A and the acute angle between lines AB and AC. [3] (d) By considering the distance of plane ABC from the origin, find the equation of the planes which are 19 units from plane ABC. [3] Fig. 1 20 y y y y y y y y h Fig. 2 z y y y y h z x x

4 DHS 2023 Year 6 H2 Mathematics Preliminary Examination Section B: Probability and Statistics [60 marks] 5 Tim plays a game with 3 outcomes having the score of 2, 5 and 6. The corresponding probabilities are 24, and 77k respectively, where k is a constant. (a) State the value of k. [1] Tim pays $m in total to play the game twice. The outcome of each game is independent. He receives an amount corresponding to twice the absolute difference of the two scores from each game. (b) If Tim is expected make a profit, determine the range of values of m. [3] (c) For m = 2.3, explain why Tim may not profit for every 2 such games that he plays. [1] 6 A student needs to create a 7-character alpha-numeric passcode (that is, at least one letter and one digit must be included). She uses lower case letters from her name “venessa see” and digits 1 to 9. (a) Find the number of different passcodes if all the letters and numbers are distinct such that (i) they begin with a letter and end with a digit, [2] (ii) all the letters and digits are to be separated. [3] (b) Find the number of different passcodes if there are 3 identical letters, and all other letters and digits are distinct. [3] 7 In a supermarket lucky draw, a participant will first spin a wheel, followed by throwing a fair six- sided die. The wheel has 9 sections and an arrow which has an equal chance of coming to rest over any of the 9 sections. The wheel has one section labelled “$100”, three sections labelled “$4” and five sections labelled “$1” (see diagram). The fair six -sided die has two faces labelled “0.02”, three faces labelled “0.5” and one face labelled “2”. The amount of money that the participant wins is calculated by taking the multiplication of the outcomes of the wheel and die. (a) Find the probability that a participant wins $2. [2] (b) If a participant wins $2, find the probability that the wheel came to rest over “$100”. [1] $1 $100 $1 $1 $1 $4 $4 $4 $1

5 DHS 2023 Year 6 H2 Mathematics Preliminary Examination [Turn over (c) A participant plays three consecutive lucky draws. Find the probability that a participant wins $6 in total, such that the die shows a different face value each time. [2] (d) Another participant plays one lucky draw. He is now given an option to either end the game by taking the amount based on the section that the wheel came to rest over, or to proceed to throw the die. Explain, with justification, which option should he take. [1] 8 One round of a game is played by randomly moving the counter from the starting point X to one of the ending points A to E along the grid lines as shown in the diagram below. • The counter can only either move up or to the right along the grid lines. • At every movement after X , the player randomly moves the counter to the right with probability p or up with probability q, 1pq+= . The moves taken at each junction are independent. • For the player to win a round, the counter must pass through Y and end up in point B. (a) Show that the probability of winning a round is 5330 .pq [3] It is given that 4 .5p= Wang plays one game which comprises 15 rounds. Each round is independent of one another. Let the random variable W be the number of rounds that Wang wins. (b) Find the probability that Wang wins at least one-third of the 15 rounds. [1] (c) Wang plays 40 such games independently. Estimate the probability that the mean