2021 J2 RVHS H2 Math CT QP

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 RVHS 2021 JC2 H2 MATHS COMMON TEST Section A: Pure Mathematics [60 marks] Sequence & Series not tested in 2023 H2 MA CT 1 A M C a P O b N B As shown in the above parallelogram OACB, OA = a and OB = b. It is given that M is the midpoint of AC and 21::ON NB = .The point P is the intersection of the line OM and AN. (i) By letting OP OM= and AP AN= , find the value of  and  and hence the position vector of P in terms of a and b. [3] (ii) Given that the point Q lies on AN produced and is such that 21::AN NQ = , find the position vector of Q and hence determine if Q, B and C are collinear. [3] 2 Solve the following simultaneous equations for complex numbers z and w. * * 2i 1 4 zw zw −= += [6] 3 Let 1 3iz=+ ( )arg 4w = and 22zw = . (i) Find w in the form ixy+ where ,xy  . [2] (ii) Find the least positive integer value of n such that ( ) n zw is purely imaginary. [3] 4 It is given that 3f1 2!() ()r r=− + , where r is a non-negative integer. Show that 1f f 1 2! ()( ) ( ) () arrr r +− − = + for some constant a to be determined. [1] (i) Using the above result, find 1 1 2!() n r r r= + + in terms of n . [3] (ii) By using the result in part (i), deduce the value for 3 1 ! n r r r= −  , simplifying your answer. [2]

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 Integration not tested in 2023 H2 MA CT 5 The functions f and g are defined by 2 1f 1 for , 1 1x x x x→ −  −−: , 2 4 5 for , 1g x x x x x→ + +  : . (i) Find 1f ()x− and state the domain of 1f− . [3] (ii) State whether the composite functions fg and gf exist, justifying your answer. Hence find the range of the composite function(s) that exist(s). [4] 6 A curve C has parametric equations etx= , sinyt= where 0 t  . (i) Find e sin dt tt . [4] (ii) Sketch the graph of C. [1] (iii) Find the area of the region bounded by C, the x-axis, the lines 1x= and eax= , where a is a positive real constant, in terms of a. [2] 7 The plane  passes through the points with coordinates ( 1, 0, 1),− (2, 1, 1)− and (1, 3, 2)− . The line l passes through the point P (16, 20, 16) and is parallel to the vector 1 2 1    . (i) Find the cartesian equation of  . [3] (ii) Find the coordinates of the point of intersection of l and  . [2] (iii) Find the acute angle between l and  . [2] (iv) Determine a vector equation of the line of reflection of l in  . [4] 8 The R0 value of a virus refers to the average number of new infections resulting from each infected person. (a) A new virus is discovered. Extensive testing reveals that by the nth day, a total of 268 people were infected. In the following day, 686 people were newly infected. In the day following that, 2401 people were newly infected. Virologists propose to use a geometric progression to model the number of infections due to the virus. They propose the model: 1 0 n nu aR −= where un is the number of new infections on the nth day after the virus emerged, and a is the number of people infected initially. (i) Using this model, estimate the value of R0. [2]

3 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 (ii) Contact tracers identify 16 people who were likely originally infected by the virus. Estimate how long the virus had been spreading before it was identified. [3] (iii) The government introduces a series of measures that reduces the value of R0 to less than 1. Explain the trend in the number of new infections in the long run. [1] (b) Administrators of a hospital in a region suffering from an outbreak observe a pattern in their data. As new patients are admitted overnight, the number of patients each morning is 20% higher than the night before. During the course of the day they are able to discharge 10 patients. The hospital has a total of 500 beds available. The hospital has 200 patients on the morning of the first day of the observation. Given that nv denotes the number of patients in hospital on the morning of the nth day, show that 1140 1.2 n nvk −=  + , where k is a constant to be determined. Hence, find the least value of n when the hospital first runs out of capacity. [6]

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 Section B: Statistics [40 marks] 9 An ice-cream café sells a mega sundae in which the cu stomer will order 5 scoops of ice- cream. The staf f will arrange the 5 scoops on a plate in either a circular or row manner. The café offers exactly 10 distinct flavors of ice-cream daily. (a) Find the number of ways the mega sundae can be arranged in a circular manner if 5 distinct flavors of ice-cream are to be chosen. [1] (b) A customer decides to order the vanilla, chocolate and banana flavour among the flavours for all 5 scoops of ice-cream but does not specify the number of scoops for each flavor, i.e. some flavor of ice-cream can have 2 or more scoops. Find the number of ways the staff can arrange the mega sundae in a row manner.[2] 10 Mr Tan, who is the school Admin Manager, believes that JC students in his school take 22 minutes on average to buy and consume their lunch in the school canteen. In the data collection process, Mr Tan uses the school’s computer programme to assign every JC students in his school a number based on the alphabetical order of their statutory names and select 50 students using a random number generator. The selected stu dents do a survey on the duration they spent for lunch on a particular day in the school canteen. The random variable X , which denotes the time spent for lunch in the school canteen, in minutes gives the following results: 21117 25061, xx== . (i) Justify whether the group of 50 JC students chosen by Mr Tan is a random sample. [1] (ii) Calculate the unbiased estimates for the population mean and variance for the variable X . [2] (iii) Carry out a test at the 5% level of significance to determine if Mr Tan’s belief is valid. [3] 11 A game is played in which a tetrahedral die is thrown. The sides of the die are numbered 1, 2, 3, 4. Whatever number on the face that the die land on, that many number of fair coins are tossed. A player’s final score X, is the number of coins showing heads. (i) Determine the probability distribution table of X. [3] (ii) Find E X() and Var X() . [3]

5 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 12 (a) A particular disease is known to infect approximately 1 in every 10 000 people. A test for this disease is 99.9% accurate, meaning the outcome of the test matches the true result 99.9% of the time. Find the probability that a person has the disease given that the person is tested positive. [2] (b) Researchers looking into whether wearing masks helps reduce the spread of a particular disease. A representative sample of 50 people is taken. In that sample, 7 people who wore masks got sick, and 6 people who did not w