TJC H2 MATHS P2

Section A: Pure Mathematics [40 marks] 1 Given that    sin 1 sin 1 2cos sinnx nx n x x    , show that  11 22 1 1 2 sin sincos 2sin n r nx xrx x  . [4] Hence express  22 2 2 31 1cos cos cos ... cos22 2 x xxx               in the form sin sin bxad cx   , where ,,abc and d are real numbers. [3] 2(a) The diagram above shows two curves 1C and 2C which are reflections of each other about the line y = x. State with justification, whether the following statement is true: “If 1C is the graph of fy x , then 2C is the graph of   1fy x .” [1] (b) The functions f and g are defined as follows 2 1f: , 6x x x  x , x < 0, 1g: t a n 2 xx     , x . (i) Sketch the graph of  fy x . Determine whether 2f e x i s t s . [ 3 ] (ii) Find  1f x . [2] (iii) Given that gf(a) = ,4  find the exact value of a. [2] y = x C1 y (1, 2) (2, 1) C2 x 0

3 Given that ee s i ny x x  . Show that 22 22 2 dd2e 4e sin 0 dd yy yy xxx   . [2] (i) Find the values of y, d d y x and 2 2 d d y x when 0x . Hence, find in terms of e, the Maclaurin’s series for  ln e sinx x , up to and including the term in 2x . [4] (ii) By using appropriate standard series expansions from the List of Formulae (MF26), verify the correctness of the first three terms in the series expansion for  ln e sinx x found in part (i). [3] (iii) Use your answer to part (i) to give an approximation for  1e 20 2e 4 d,el ne s i n x xxx    giving your answer in terms of e. [3] 4 With reference to origin O, the points A, B, C and D are such that OA a  , OB b  , OC a   and 2OD b   . The lines AB and DC meet at E. Find OE  in terms of a and b. [4] Hence show that 3BE AB  . [1] It is given that A and E have coordinates (1, 4, 3) and (3, 15, 5) respectively. (i) Show that the lines AC and BD are perpendicular. [4] (ii) Find the equation of the plane p that contains E and is perpendicular to the line BD. [2] (iii) Find the distance between the line AC and p. [2] B C D E O A

Section B: Statistics [60 marks] 5 Four classes CG40, CG41, CG42 and CG43 are task ed to organise a College event. Each class sends 3 representatives for a meeting. (i) In how many different ways can the 12 re presentatives sit in a circle so that representatives from CG40 are not seated next to each other and representatives from other classes are seated with their respective classes? [3] The 12 representatives are to be split up in to 3 groups for bonding activities. Each group must consist of a representative from each class. (ii) In how many ways can the groups be formed? [2] 6 In a game at the carnival, a player rolls discs onto a board containing squares, each of which bears one of the numbers 1, 2, 5 or 10. If a di sc does not land within a square, the player receives nothing. The probability that the di sc does not land within the square is 3 4 . If a disc lands within a square, the player receives the same amount (in dollars) as the number in the square. Given that a disc falls within a square, the probabilities of landing within a square with the numbers 1, 2, 5 and 10 are 0.5, 0.3, 0.12 and 0.08 resp ectively. It is assumed that the rolls of the discs are independent. (i) A player pays $5 to play the game and is given n discs. Find n if the game is fair. [4] (ii) If a player is allowed to roll 3 discs for $2, find the probability that the player will have a profit of $10. [4] 7 A factory manufactures large number of pen refills. From past records, 3% of the refills are defective. A stationery store manager wishes to purchas e pen refills from the factory. To decide whether to accept or reject a batch of refills, the manager designs a sampling process. He takes a random sample of 25 refills. The batch is accepted if there is no defective refill and rejected if there are more than 2 defective refills. Otherwis e, a second random sample of 25 refills is taken. The batch is then accepted if the total number of defective refills in the two samples is fewer than 4 and rejected otherwise. (i) Find the probability of accepting a batch. [4] (ii) If a batch is accepted, find the probability that there are 2 defective refills found in the sampling process. [3] The stationery store manager purchases 50 boxes of 25 refills each. (iii) Find the probability that the mean number of defective refills in a box is less

than 1. [2] 8 A study is done to find out the relationship be tween the age of women and the steroid levels in the blood plasma. Sample data collect ed from 10 females with ages ranging from 8 years old to 35 years old is as shown below. Age (years) x 8 11 14 17 20 23 26 29 32 35 Steroid Level (mmol/litre) L 4.2 11.1 16.3 19.0 25.5 26.2 24.1 33.5 20.8 17.4 (i) Give a sketch of the scatter diagram for the data. Identify the outlier and suggest a reason, in the context of the question, why this data pair is an outlier. [3] For the remaining part of the question, the outlier is to be removed from the calculation. (ii) Comment on the suitability of each of the following models. Hence determine the best model for predicting the steroid le vel of a female based on her age. Model A: lnLa bx Model B:  2 25Lcd x  Model C:  4 25Lef x  where a, b, c, d, e and f are constants. [3] (iii) Using the best model in (ii), estimate the steroid level of a woman at age 40. Comment on the reliability of your estimate. [3] (iv) It is known that body muscle mass and ster oid level has a linear correlation. The muscle mass percentage m % of the 9 females were measured. An additional female, Jane, participated in the study. Jane has her muscle mass percentage and steroid level measured. The mean muscle mass percentage of the 10 females is now found to be 26.28 %. The equation of the least squares regression line of m on L for the 10 pairs of data is 2.22 1.25mL  . Calculate Jane’s steroid level. [3]

9 A flange beam is a steel beam with a “H”-sh aped cross section, and is used as a supporting structure in construction and civil engineering. A factory manufactures both Grade X and Grade Y flange beams. The load that can be supported by a Grade X flange beam follows a normal distribution with mean 524 3 1 0 k N.  and standard deviation 44.5 10 kN . The load that can be supported by a Grade Y flange beam is 1.5 times of the load that can be supported by a Grade X flange beam. (i) Find the probability that the combined load that can be supported by two randomly chosen Grade Y flange beams is within 411 0k N of the combined load that can be supported by three randomly chosen Grade X flange beams. [4] (ii) A construction company wants to buy 100 sets of three Grade X flange beams. Find the probability that fewer than 95 of these sets can support more than 561 0k N . [3] The company decides to place an order with the factory for a custom-made flange beam such that the probability of being ab le to support a load of at least 561 0k N must be at least 0.999 . It is assumed that the load that can be supported by the cu stom-made flange beam also follows a normal distribution. (iii) By taking the standard deviation of a custom-made flange beam to be 431 0k N , find the smallest possible mean load in kN, giving your answer correct to the nearest thousand, for the factory to meet the company’s requirements for the custom-made flange beam. [5]

10(a) College students intending to further their studies overseas have to sit for a mandatory Overseas Universities Test (OUT). Researcher Mr Anand wishes to find out if male college students tend to score higher for OUT compared to female co