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