ACJC 2024 Prelim H1 Final

Section A: Pure Mathematics [40 marks] 1 A company sells three types of cars: gas-powered, electric and hybrid. The total selling price of 4 electric cars is $172 800 more than the total selling price of 3 gas -powered cars. The selling price of a hybrid car is 1.2 times that of a gas -powered car. The company sold 3 hybrid cars, 2 gas-powered cars and 2 electric cars for $716 880. Find the selling price of an electric car. [3] 2 Find the set of values of k for which 2 2( 36) ( 6) 0.5 0    k x k x for all real values of x. [4] Hence find the set of values of k for which the expression 2 2ln[( 36) ( 6) 0.5)]k x k x    is defined for all real values of x. [1] 3 (a) Find   425 4 d x x x   . [2] (b) Express 4 3 in the form , where and are integers.4 4 x B A A Bx x    Hence find 4 3 d ,4 x xx   where 4x . [3] 4 The curve C has equation 5ln( 1) 2 2y x x     . (i) Show that curve C has no stationary points. [3] (ii) Sketch the graph of C, stating the x-coordinates of any points of intersection with the x-axes and the equation(s) of any asymptotes. [3] (iii) Find the numerical value of the area bounded by C, the x-axis and the lines x = 3 and x = 5. [1]

2 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 5 The diagram shows the curve C with equation 2 2 ,y k x  where k is a positive constant. The x- coordinate of A is positive and the y-coordinate is 23 4 k . (i) Find the equation of the tangent to C at the point A. Give your answer in the form y mx c  , where m and c are in terms of k. [4] (ii) Find the area bounded by the curve C, the tangent to the curve at point A and the x-axis, giving your answer in the form 3,p kq where p and q are integers. [4] 6 Mr Lim has just set up a company selling handphones. Over a period of 9 months , his total profit P thousand dollars per month can be modelled by 0.2 2(e 4) 9,tP   where t is the time in months after he set up his company. (i) Use differentiation to find the value of t which gives a stationary point on the graph of P against t. Justify whether this point is a maximum or minimum point. [4] (ii) Sketch the graph of P against t for 0 9 t  . [1] (iii) Using integration , find 9 0.2 2 0 (e 4) 9 d ,t t   correct to the nearest integer , showing your workings clearly. Explain, in the context of the question, what this value represent. [4] O C x y A

3 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 (iv) Mr Lim has a budget for advertising his handphones for sale. At any time, t months, measured from the start of his advertising campaign, the remaining budget, A dollars, is modelled by 200 375( 20 75 ).3 2 5A t t     Using differentiation, find d when 3,d A tt  giving your answer to the nearest integer. Explain, in the context of this question, what this value represent. [3] Section B: Probability and Statistics [60 marks] 7 It is known that on average p% of the elderly patients in ABC hospital suffer from hypertension. A research study on 30 randomly selected e lderly patients in the hospital was conducted. The number of elderly patients in the hospital with hypertension is denoted by the random variable X. Assume that X has a binomial distribution. (a) If X has mean 18, calculate P( 20).X  [3] It is also known that on average k% of the elderly patients in ABC hospital suffer from diabetes. The hospital conducted a research study on another 40 randomly selected elderly patients. The number of elderly patients in the hospital with diabetes also follows a binomial distribution. (b) The probability of at least nine but no more than twenty elderly patients suffering from diabetes is 0.25. Given that 50,k find the value of k. [3] 8 Events A and B are such that 7 1 3P( | ) , P( ) and P( ) .10 3 5A B B A B    (a) Find P( ).A B Hence show that 1P( ) . 2A  [4] (b) Find the probability that exactly one of A and B occurs. [2] (c) Determine whether the events A and B are independent. [1]

4 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 9 There are two bags of balls. Bag A contains 2 red balls and 1 black ball. Bag B contains 1 red ball and 1 black ball.  A bag is selected at random, and the first ball is drawn at random from it.  If the first ball is black, it is returned to the selected bag.  If the first ball is red, it is placed in the other bag.  Regardless of whether the first ball is red or black, the second ball is drawn at random from the other bag. (a) Draw a tree diagram to represent this information. [2] (b) Find the probability that both balls are of the same colour. [2] (c) Given that both balls drawn are red, find the probability that the first bag chosen was A. [3] 10 A horticulturist carries out research on the effects of varying amounts of fertilizer on the average crop yield of potatoes. Eight potato plants of the same variety were selected at random. The results are shown in the table below. Plant A B C D E F G H Amount of fertiliser (x grams per square metre) 3.2 1.5 0.8 0 2.5 8.0 4.0 4.1 Average crop yield (y kilograms) 2.6 2.2 2.3 2.0 2.3 3.6 2.7 2.8 (i) Give a sketch of the scatter diagram of the data. [2] (ii) Find the product moment correlation coefficient, r, and comment on its value in the context of the data. [2] (iii) Find the equation of the regression line of y on x in the form ,y a bx  giving the values of a and b correct to 3 significant figures. [1] (iv) Use the equation of your regression line to estimate the amount of fertiliser used on a potato plant of the same variety when the average crop yield of potatoes is 2.5 kilograms. Comment on the reliability of the estimate. [2] (v) Without any further calculations, explain briefly what will happen to the value of r found in (ii) if y was recorded in grams instead. [1]

5 ANGLO-CHINESE JUNIOR COLLEGE 2024 H1 MATHEMATICS 8865/01 11 A family of seven, consisting of a father, mother, two sisters, and three brothers is queuing in a line. (i) Find the number of different possible arrangements (a) if the sisters stand next to each other and the brothers are all next to each other. [2] (b) if either the father is first in the queue or mother is last in the queue (or both). [3] (ii) The family is queuing in a line in a random order. Find the probability that no two brothers stand next to each other. [3] 12 In this question you should state the parameters of any normal distributions you use. In a certain college, the heights of boys and girls have independent normal distributions. The heights of boys are normally distributed with mean 172 cm and standard deviation  cm. The probability that the boys are taller than 156 cm is 0.948. (a) Show that 9.84  , correct to 3 significant figures. [2] Use 9.84  for the rest of this question. (b) A sample of 100 randomly chosen boys are taken. X denote the mean height of the boys in the sample. 80 such samples are taken. (i) Find the expected numbe