SEAB 9758 Sample Paper Solutions / RI Mock Paper 2 2024

RAFFLES INSTITUTION H2 Mathematics 9758 2024 Year 6 T erm 3 Revision Mock Paper 2 Section A: Pure Mathematics (40 marks) 1 () The function fi s de fi ncd as follows: f:x3cosx - 2 sinx, xER,-* <r<. Write f(*) as Rcos(r + a), where R and a are constants to be found. Hence, or otherwise, fi nd the range of f and sketch the curve. (4] (ii) The function g is de fi ned as follows: 8:xH3cosx - 2 sinx, xER,-a<x<b. Given that the function g cxists, write down the largest value of B. Find g'(). The fi rst four terms of a sequence of numbers are 3, 1, 1 and 3. S, is the sum of the fi rst n terms of this sequencc. ) Explain why S, cannot be a quadratic polynomial in n. It is given that S, is a cubic polynomial. () Find S, in tens of n. (üü) Find an cxpression in tems of n for the nth term of the scquence. The angle between the vectors 3i -2j and 6i + dj - 7k is cos- ((a) Show that 2d2– 117d +333 = 0. (2] (4) (3) (3) Given that y = (an(e2r - 1), show that of = ke? (1 +y), where k is to be found. Hence fi nd the values4 ) andwhen x =0. [6] [0) (i) Write down the fi rst three non-zero terms in the Maclaurin series for tan(e2r 1). (iüi) The fi rst two non-zero terms in the Maclaurin series for tan(ezr - 1) are equal to the fi rst two non- zero terms in the series expansion of e In(l + nx). By using appropriate expansions from the List of Fomulae (MF26), fi nd the constants a and n. Hence fi nd the third non-zero term of the series (4]expansion of e In(l + nx) for these valucs of a and n. Section B: Probability and Statistics (60 marks] This question is about six couples. Each couple consists of a husband and a wife.5 The 12 people visit a theatre, and sit in a row of 12 seats. (i) In how many different ways can the 12 people sit so that each husband and wife in a couple sit next to cach other? (2) In how many different wvays can the 12 people sit so that the 6 wives all sit next to cach other, and nonc (3] (it) of the wives sits ncxt to her own husband? The group decides to form a committee to arrange futurce outings. The committee will consist of 3 of the 12 people. At least I of the wives will be on the committce but no husband and wife couple will be included. (ii) In how many ways can the committee be formed? (3] Giant purmpkins are often iregular in shape. In order to account for the different shapes of pumpkins, growers of giant pumpkins measure the size of a pumpkin by a combination of three measurements, called the 'over the top' length. Pumpkin growers keep records so that they can cstimate the mass of giant pumpkins whilc they are still growing. The over the top lengths (d m) and thc masses (m kg) of a random sample of 7 giant pumpkins are as follows. 6 d 2.31 11 2.9 14 4.05 47 5.5 104 6.7 170 7.929.17 282 449 Drawa scatter diagram of these data, and explain how you know from your diagram that the relationship [2 (1) between m and d should not be modelled by an equation of the form y = ax+ b. (ii) Which of the formulac m = ed' +fand m = gd + h, where e, f, g and h are constants, is the better model for the relationship between m and d? Explain fully how you decided, and fi nd the constants for the better fonnula. (ii) Use the formula you chose from part (ii) to estinate the mass of a giant pumpkin with (a) over the top length 6m, (b) over the top length 12 m. Explain which of your two estimates is more rcliable. (S] 3 [3 (b) o- With referenće to origin 0, the points 4, B, C and D are such that 04 - a, OB = b, 4C= Sa and BD = 3b. The lines AD and BC cross at E (see diagram). [6Find OE in terms of a and b. The point Fdivides the Jine CD in the ratio 5:3. Show that O, E andFare collinear, and fi nd OE: 0F . () () (4) Y6 H2 Math Te m 3 Revision: Mock Paper 2 Page I of 4 (3) Y6 H2 Math T erm 3 Revision: Mock Paper 2 Page 2 of 4

•Bings' are sweets that are sold in packets of 6. Each packet is made up of randomly chosen colourcd sweets. On average 10% of Bings are yellovw. 7 ) Explain why a binomial distribution is appropriate for modelling the number of yellow sweets in a packct. Find the probability that a randomly chosen packet of Bings contains no more than one yellow swcct. (3) (G) Kev buys 90 randomly chosen packets of Bings. Find the probability that at least 80 of these packels (2] contain no more than one yellow swect. On average the proportion of Bings that are red is p. It is known that the modal number of red swects in a packet is 2. (üi) Use this information to fi nd exactly the range of values that p can take. (4) A bag contains 3 blue counters, I red counter and y ycllow counters. Darvina chooses 3 counters at random from the bag, without replacement. The random variable S is the sum of the number of bluc counters chosen and twice the number of red counters choscn. 6(3y + 1) (Ö) Show that P(S = 3)=(v+ 4)o+3)(+2) (G) Given that P(S = 3) = calculate y. Hence fi nd the probability istribution of S. 121 [6 9 A type of metal bolt is manufactured with a nominal radius of 0.8 cm. In fact, the radii of the bolts, measured in cm, have the distribution N(0.8, 0.012). 1) Find the percentage of bolts that have a radius between 0.79 cm and 0.82 cm. Metal washers are manufactured to fi t on the bolts. The inside radii of the washers, measurcd in cm, have the distribution N(0.81, 0.0122). (ii) Write down the distribution of the inside circumference of the washers, in cm, and fi nd the circumference that is excccdcd by 5% of the washers. (4) A bolt and a washer are a 'good fi t' if the insidc radius of the washer is greater than the radius of thc bolt and the inside radius of the washer is not more than 0.04 cm grcater than the radius of the bolt. (ii) A washer is chosen at random, and a bolt is chosen at random. Find the probability that the washer and bolt are a good fi t. (3) The outside radii of the washers, ncasured in cm, have the distribution N(4, o²). It is known that 1 5% of the washers have an outside radius greater than 1.25 cm and 25% have an outside radius of lcss than 1.1S cm. (iv) Find the values ofu and o. [4 The average time requircd for the manufacture of a certain type of clctronic control pancl is 17 hours. An alternative manufacturing process is triallcd, and the time taken, ! hous, for the manufacture of each of 50 randomly chosen control panels using the altemative process is recorded. The results are summarised as follows. 10 n= 50 E=835.7 P= 14067.17 The Production Manager wishes to test whether the average time taken for the manufacture of a control pancl is different using the altemative process, by carrying out a hypothesis test. ) Explain whether the Production Manager should use a 1-tail test or a 2-tail test. [0) (ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything [21about the distribution of the times taken to manufacture the control panels. (iii) Find unbiased estimates of the population mean and variance and carry out the test at the 10% level of [6 signi fi cance for the Production Manager. (iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average timc than thc original process. The Finance Manager wishes to test whether the average time taken for thc manufacture of a control panel is shortcr using the altermative proccss. The Finance Manager fi nds that the average time taken for the manufacture of each of 40 randomly chosen control pancls, using the altermative process, is 16.7 hours. He carrics out a hypothesis test at the 10% level of signi fi cance. Explain, with justi fi cation, how the population variance of the times will affect the conclusion madc by the Finance Managcr. (v) [3) Y6 H2 Math T erm 3 Revision: Mock Paper 2 Page 3 of 4 Y6 H2 Math Te n 3 Revision: Mock Paper 2 Page 4 of 4

Page 1 of 15   Raffles Institution H2 Mathematics (9758) Solution for 2024 Mock Paper 2 Solution 1(i) f 3cos 2sin cos ( ) cos cos sin sinxx x R x R x R x      cos 3R   sin 2R     22 cos sin 1