HCI 8865 2022 Prelim

HCI_H1Mathematics_Prelim Paper 1 Determine the set of values of k such that the equation 2 30x kx k− + + = has real roots. [4] 2 (a) Find ( ) 33 d xx x x −  . [2] (b) (i) Differentiate 3 26ln 3 4x + with respect to x, simplifying your answer. [3] (ii) Hence find 2 d .34 x xx + [2] 3 A window at a grand church has the shape of a semicircle AEB joined to a regular trapezium ABCD of height x m, and a rectangle DCGF as shown in the diagram. It is given that 2 mAB x= , 4DC x= m and the angle ADC and the angle BCD are both equal to o45 . The total perimeter AEBCGFD of the window is 10 m. (i) Show that 5 2 22DF x = − + + m. [2] (ii) Hence use a non-calculator method to find the exact value of x such that the area of the window is maximum. [5] G F C D B A E 4x m 2x m

2 © HCI 2022 [Turn over 4 The curve C has equation 33e xy . (i) Sketch the graph of C, stating the exact coordinates of the points of intersections with the axes and the equation of the asymptote. [3] (ii) Find the equation of the tangent to C at the point where the curve C meets the x-axis, giving your answer in the form ln 3y ax b=+ , where a and b are integers to be determined. [3] (iii) Hence find the exact value of the area enclosed by the curve C, the y-axis and the tangent at the point where the curve meets the x-axis [4] O x y

3 © HCI 2022 [Turn over 5 A company sells three types of toys: boats, cars and planes. • The selling price of a toy boat is twice as much as that of a toy car. • The total selling price of 9 toy boats, 5 toy cars and 3 toy planes is $335. • The total selling price of 3 toy boats and 4 toy cars is $38 more than the total selling price of a toy boat and 2 toy planes. (i) Find the selling price of a toy plane. [3] The company decides to trial the production of a new type of toy: trains. Based on market research, the manager estimates that when the toy trains are priced at p dollars apiece, the number sold each month, D thousands, can be modelled by 0.083e pDa −=+ where a is a constant. When the toy trains were given out for free to spark public interest, the demand for the toy was 20 thousand units. (ii) Find the exact value of a. [1] (iii) Sketch the graph of D against p, stating the coordinates of any point(s) of intersection with the axes and the equation(s) of any asymptote(s). [2] (iv) Find d d D p . [1] The manager estimates that t months from its production, the selling price of a toy train, $p, will be ( ) 2 0.5 9 312pt= − + , 0 30t . (v) Use differentiation to find the rate of change of monthly demand for toy trains at 20 months from its production. [4] (vi) Explain in context what the answer in (v) suggests about the demand for toy trains at 20 months from its production. [1]

4 © HCI 2022 [Turn over Section B: Probability and Statistics [60 marks] 6 A company has 8 vacancies to fill from 17 applicants. The applicants consist of 8 women and 9 men. One of the 8 women is the wife of one of the 9 men. (i) How many different ways can these vacancies be filled if there are no restrictions? [1] On the day of the interview, all 17 applicants are seated in a row in the waiting room. (ii) How many different ways can the 17 applicants be arranged such that the men and women alternate and the husband and wife are seated together? [3] The company then decides that four of the eight vacancies shall be filled by women and four by men. (iii) How many different ways can the eight vacancies be filled under this new decision? [1] (iv) Of all the possible selections of four women and four men, one selection is picked at random. Find the probability that this selection includes either the wife or her husband, but not both. [2] 7 A group of customers in a restaurant are asked which fruits they like from a choice of mangoes, strawberries and apples. The results are as follows. 65 like mangoes 40 like strawberries 45 like apples 21 like mangoes and apples 14 like strawberries and apples 26 like mangoes and strawberries x like all three fruits 3 like none of these three fruits. One of the customers is chosen at random. M is the event that the customer likes mangoes. S is the event that the customer likes strawberries. A is the event that the customer likes apples. (i) Represent this information on a Venn diagram. [3] (ii) Given that events M and S are independent, show that 8x= . [2] (iii) Explain, in the context of this question, what is meant by P( | )MA , and find its value. [2]

5 © HCI 2022 [Turn over Four customers are chosen at random, without replacement. (iv) Find the probability that three customers like exactly two of these three fruits and one customer likes only one of these fruits. [3] 8 Alice sold homemade Nutella brownies when she was jobless during the Covid -19 outbreak to earn a living. The numbers of brownies, x, sold in the first eight months of the year 2022, together with the profits, y dollars, on the sale of the Nutella brownies are given in the following table. Jan Feb Mar Apr May Jun Jul Aug x 400 560 290 360 620 190 520 600 y 800 1100 560 k 1200 390 970 1100 (i) It is known that the regression line of y on x is given by 1.834352 40.8yx=+ . Show that the value of k is 700, correct to the nearest integer. [3] Take 700k = for the rest of the question. (ii) Find the product moment correlation coefficient and comment on its value in the context of the data. [2] (iii) On the same diagram, give a sketch of the regression line of y on x given in part (i) and the scatter diagram of the data as shown in your calculator. [2] (iv) Use the equation of regression line in part (i) to calculate an estimate of the profit when 480 brownies were sold. Explain whether you would expect this estimate to be reliable. [2] (v) Alice realised that she had recorded her profits wrongly. There is a shortfall of 80 dollars for each of the monthly profit in the table above. Comment on whether this recording error will affect the value of the product moment correlation coefficient found in (ii). [1]

6 © HCI 2022 [Turn over 9 In epidemiology, particularly in the discussion of infectious disease dynamic s, the duration of infection period is the time interval during which a host (individual or patient) is infectious. Researchers claim that the mean duration of a particular viral infection among children is 4.3 days. In a random sample of 120 children in fected with the particular virus, the average duration of infection is 4.07 days and the standard deviation is 1.215 days. (i) Test at the 5% significance level whether the researchers’ claim is supported by the data. [4] (ii) State, giving a reason, whether it is necessary to assume that the durations of the infection for the particular virus among children are normally distributed for this test to be valid. [1