ACJC EJC NJC RVHS FM 2024 Prelim P2

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) Section A: Pure Mathematics [50 marks] P2 Q1 Prove by induction that ( ) ( )2 7 3 5 5nn +− is divisible by 24 for all positive integers n. [5] P2 Q2 Show that the substitution sinxt= , where ππ 22 t−   , reduces the differential equation ( ) 2 2 2 dd1 4 0 dd yyx x y xx− − + = to 2 2 d 4 0.d y yt += [4] Hence find the general solution for y in terms of x, giving your answer in non-trigonometric form. [4] P2 Q3 (a) Determine the eigenvalues and corresponding eigenvectors of the matrix 11 31 = − M . [4] (b) By expressing M in the diagonalised form 1−QDQ , find nM for positive integers n. [4] (c) Hence, or otherwise, show that nM is a diagonal matrix for all even values of n. [1]

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) P2 Q4 A Gregorian telescope consists of a parabolic mirror 1C and an elliptical mirror 2C , both of which are concave. The major axis of the ellipse coincides with the axis of the parabola . In the diagram below, this common axis is the x-axis. The vertex of the parabola is at the origin O. A light ray travels along the line PQ parallel to the x-axis, and reflects off 1C . It then travels along the line QR and reflects off 2C . Finally, it travels along the line RS, going through a small hole near the vertex of 1C to reach a detector located at S on the x-axis. (a) Explain why, if the ellipse and parabola have a common focus, incoming light rays which are parallel to the x-axis will all pass through a common point, S after reflecting off both mirrors. [2] The equation of 1C is 2 37.6yx= . The vertex of 2C is at ( )9.6,0 , and S is at ( )0.2,0− . (b) State the coordinates of F, the common focus of 1C and 2C . [1] (c) Find the equation of 2C . [4] (d) Given that the range of y-coordinates of 1C is 4.2 4.2y−   , find the smallest possible range of y-coordinates of 2C so that all the light reflected off 1C will reach 2C . [5] x y O P Q R S

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) P2 Q5 For 2n , the complex numbers 12, , ..., nz z z are the n roots of the equation 10nz −= with ( ) ( )0 arg arg 2 πijzz   for 1 i j n   so that 1nz = . (a) (i) Express kz in the form ier  where 0r  and 02 π , leaving your answer in terms of k ( )1, 2, ..., kn= . [1] (ii) In the case where n is odd, evaluate the series 1 2 3 1 2 2 2 2 2 ...1 1 1 1 1 nnz z z z z − + + + + ++ + + + + , leaving your answer in terms of n. Full working must be shown. [4] (b) Let 7n= . (i) Sketch the roots of 7 10z −= on an Argand diagram, showing clearly the relationships between their moduli and arguments. [2] (ii) For 1,2,3,4,5,6m= , the locus mL is the set of points z satisfying ( ) ( )arg 1 arg 1 mzz− = − and 1z  . Sketch, on the same Argand diagram in part (b)(i), the six loci 1 2 6, , , L L L . [2] (iii) Show that ( )( )( )( )( )( ) 6 1 2 3 4 5 6 0 r r z z z z z z z z z z z z z = = − − − − − − . Hence, find the product of the lengths of the six loci from part (b)(ii). [4] (iv) Let ,,abcz z z , with 17 abc    , be three of the roots of 7 10z −= . Given that ,,abcz z z satisfy cos isinab cb zz zz − =+− , state all possible values that  can take, where 02 π . For each value of  that you have stated, write down the condition satisfied by a, b and c. [3]

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) Section B: Probability and Statistics [50 marks] P2 Q6 A psychologist is conducting an experiment to determine if there is a difference in the reaction times of participants under two different conditions: before taking a stimulant (Condition A) and after taking a stimulant (Condition B). The reaction times are recorded in seconds as shown below. Participant 1 2 3 4 5 6 7 8 9 10 Condition A 21 24 20 22 25 24 23 27 28 32 Condition B 18 22 19 24 23 24 18 25 27 31 (a) Carry out a sign test at the 2% significance level for the above investigation. [5] (b) Explain if the conclusion of the test would change if the investigation is about whether the reaction times of participants are shorter after taking the stimulant. [2] (c) Give a reason why it is more appropriate to conduct a sign test than Wilcoxon matched pair signed rank test in this context. [1] P2 Q7 The webpage of a company receives an average of 5 views per hour, and the views are independent of each other. (a) Find the probability that in a 3-hour duration, the webpage receives more than 20 views. [2] A quality view is one where the viewer spends more than 3 minutes on the webpage. On average, 100p% of views are quality views. Let Y represent the number of views of the webpage, up to and including the first quality view. (b) State, in the context of the question, two assumptions needed to model Y by a geometric distribution. [2] Assume that the assumptions in part (b) hold and that the mean of Y is 4. (c) Find the probability that the first quality view occurs after the 6th view. [3] (d) Find the probability that the 10 th view is the second quality view given that the 3 rd view is the first quality view. [2]

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) P2 Q8 (a) Show that 1 20 1 d πx xx = − . [2] (b) The continuous random variable X has probability density function 2 , oth , 0 1 0 e ise,, rw f ( ) k x xxx   −  = where k is a constant. (i) State the value of k. [1] (ii) Without the use of a calculator, find the exact value of E(X). [3] (iii) Find the lower quartile of X. [3] P2 Q9 Two brands of a health supplement, Brand A and Brand B, are known to enhance short-term cognitive focus in users after consumption. A pharmaceutical laboratory would like to investigate the effectiveness of the two brands of supplement and conducted clinical trials where cognitive focus of participants from a random sample, as measured by a standardised cognitive focus scale, before and after consuming a dosage of the supplement was assessed. The increase in cognitive focus scores, x and y, of participants after consuming Brand A and Brand B respectively are recorded as two sets of data for analysis, with 6 readings belonging to participants for the clinical trials of Brand A and 6 readings belonging to participants for the clinical trials of Brand B. x 7.8 7.3 7.2 7.4 7.5 7.0 y 6.4 7.9 7.1 6.8 6.6 7.2 (a) Carry out an appropriate test, at the 2% level of significance, to determine whether there is a difference between the effects of the two brands of supplement. State any other assumptions made in applying the test. [6] The statistician later found out that the two sets of data above were collected from 6 participants, with each column of data coming from a particular participant. Each participant had gone through two rounds of clinical trials, one for Brand A and the other for Brand B, with sufficient wait time between the trials for the effects of the supplement to wear off. (b) Construct a 98% confidence interval for the mean difference in the increase of cognitive focus after consuming the two brands of supplement. [2] (c) Hence, explain whether there is evidence, at the 2% significance level, that there is a significant difference between the effects of the two brands of supplement. [2] (d) Discuss whether the test in part (a) or part (c) is more suitable for the purpose of the investigation. [2]

2024 H2 FM 9649 Prelim Paper 2 (ACJC_EJC_NJC_RVHS) P2 Q10 An analyst suggests that the number of emergency calls received at a particular fire station in a day follows a Poisson distribution. The number of emergency calls received in each day over a span of 60 days has the following frequency distribution. No. of emergency calls in a day 0 1 2 3 4 5 6 Frequency 7 14 16 12 7 4 0 (a) Carry out an appropriate test to determine whether the analyst’s claim is vali