SAJC 9758 2023 Prelim P2

Question 1 2 3 4 5 6 7 8 9 10 11 TOTAL Marks 7 9 9 6 9 8 8 10 10 12 12 100 This document consists of 26 printed pages and 2 blank pages including this page. ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS HIGHER 2 9758/02 Wednesday 13 September 2023 3 hr Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) NAME:_________________________________________( _____ ) C.G.: __________ TUTOR’S NAME: _________________________________________ SCIENTIFIC / GRAPHIC CALCULATOR MODEL: _______________________ NUMBER OF ADDITIONAL PIECES OF WRITING PAPER :________________ READ THESE INSTRUCTIONS FIRST Write your name, civics group, index number and calculator models on the cover page. 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. Total marks : 100 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. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calc ulator are allowed unless a question specifically 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.

2 [Turn Over Section A: Pure Mathematics [40 marks] 1 Do not use a calculator in answering this question. The complex number z satisfies the equation 23 (5 i) 0,zz α−+ += where α is a real number. It is given that one root is of the form ikk+ , where k is real and non-zero. Find α and k, and the other root of the equation. [7] 2 In triangle PQR, 2 3PRQ π∠= , 6RPQ π θ∠= + , and PR a= units, where a is a positive real constant. (i) Show that 3 cos 3 sin aPQ θθ− = . [4] (ii) Given that θ is sufficiently small, show that ( ) 231 cPQ a b θθ≈+ + , where b and c are real constants. [3] (iii) Given that 0θ > , find the range of values of θ such that the percentage error of the approximation in (ii) is less than 5%. [2] 3 (a) Use the substitution πtan , where 0 , 2xt t= << to find 22 1 d 1 x xx +∫ . [4] (b) (i) Write down ( ) 3d cosd xx . [1] (ii) Hence, find 53sin .x x dx∫ [4] 4 The point P travels along the curve C with equation 2 90xy x y+− = . (i) Without expressing y in terms of x, find an expression for the gradient of the tangent at P in terms of x and y. [2] The x-coordinate of P is increasing at the rate of 0.02 units per second when 3x= . (ii) Determine the rate at which the gradient is changing at this instant. [4]

3 [Turn Over 5 (i) Show that 11 ! ( 1) ! ( 1) ! r rr r−= ++ . [1] (ii) Using the result in (i), evaluate the sum 123 ...2! 3! 4! ( 1)! n nS n= ++ ++ + in terms of n. [3] (iii) Explain why the series converges as n→∞ and state its sum to infinity S∞ . [2] (iv) The Comparison Test for Convergence states that if 1 r r a ∞ = ∑ and 1 r r b ∞ = ∑ are two infinite series with 0 and 0rrab≥≥ and rrab for all r +∈  , then 1 r r b ∞ = ∑ converges implies that 1 converges.r r a ∞ = ∑ Using the Comparison Test for Convergence and result from part(iii), explain why the series 2 1 ( 1) ( 2)!r r r ∞ = − +∑ converges. [3] Section B: Probability and Statistics [60 marks] 6 A bag contains 5 red balls and 3 blue balls. In a game, Anne removes balls at random from the bag, one at a time, without replacement, until she has taken out 2 red balls. The total number of balls Anne removes from the bag is denoted by X. (i) Find P( )Xx= for all possible values of x. [3] (ii) Find E( )X and Var( )X . [3] Anne pays $1 every time she removes a ball and she will receive $y once she gets 2 red balls. (iii) Calculate the value of y if the game is fair. [2] 7 Ben has a collection of baby toys of various shapes and colours. There are four different shapes: Circle, Square, Triangle and Star. Each shape has four different colours: Yellow, Green, Red and Purple. Ben’s collection consists of 40 toys and the numbers of each shape and colour are shown in the table. Yellow Green Red Purple Circle 1 2 2 0 Square 3 1 5 2 Triangle 4 4 2 3 Star 2 5 3 1

4 [Turn Over (i) Ben puts all the toys in a box and chooses one toy at random. Given that this toy is not Yellow, find the probability that this toy is either a Triangle or a Star. [2] (ii) Ben now puts the toy back into the box and chooses two toys at random. (a) Find the probability that the two toys chosen are purple in colour and of different shapes. [3] (b) Ben has two favorites among the 16 possible colour-shape combinations. The probability of choosing these two at random from the 40 toys is 1 39 . Write down all his possible favourite colour-shape combinations. [3] 8 Professor A claims that the mean score of the students in his Calculus class is 52. His student Barbie decides to test this claim. She randomly selects a sample of 30 students from the Calculus class and finds that their mean score is 46 with a standard deviation of 15. (i) Test, at the 5% level of significance, whether Professor A’s claim is valid. You should state your hypotheses clearly and define any symbols that you use. [5] Barbie also attends the Statistics class conducted by Professor B, who claims that the mean score of the students in his Statistics class is 48. It is known that the standard deviation of the Statistics scores is 13. Barbie also decides to test whether Professor B has understated the mean Statistics score. She randomly selects a sample of 30 students from the Statistics class and finds that their mean score is k. Testing at the 5% level of significance, she concludes that the professor has indeed understated the mean score. (ii) Find the range of values of k. [4] (iii) Explain if it is necessary for Barbie to select a sample of at least 30 students in both (i) and (ii). [1] 9 On average, 1 in 15 patients who visit polyclinics have diabetes mellitus. On a certain day, 50 patients are randomly selected from a polyclinic. The number of these patients having diabetes mellitus is the random variable X. (i) State, in the context of this question, two assumptions needed to model X by a binomial distribution. [2] Assume now that X has a binomial distribution. (ii) Find the probability that the 50th patient is the 4th patient with diabetes mellitus. [1]

5 [Turn Over The test for diabetes mellitus consists of two stages. In the first stage, a fasting blood glucose test is conducted. The percentage of patients with the respective fasting blood glucose levels is given in the table below. Fasting blood glucose level Percentage less than 100 mg/dL 84 % 100 – 125 mg