2024 Y6 Timed Prac Rev Paper 3 (Qns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6   2024 Year 6 Timed Practice Revision Practice Paper 3 (Source: 2021 Year 6 Term 3 Common Test) The solution will be released in Ivy on 10 June (Monday) Total Marks: 100 Duration: 3h   **** Note that Qn 4 (Complex Numbers) will NOT be examined in the coming Timed Practice. So, you should complete this paper within 2 hrs 47 mins. 1 An arithmetic series has first term a and common difference d, where a and d are non-zero. If the first, fifth and eleventh term s of the arithmetic seri es are equal to the first, second and third terms of a geometric series respectively, find the exact sum of the first ten terms of the geometric series, in the form ka , where k is a simplified rational number. [4] 2 (i) By writing   2 21 23rr in partial fractions, find an expression for  2 2 21 23 n r rr  in terms of n. [3] (ii) Hence find the exact value of 21 21 21 19 21 21 23 23 25  . [2] 3 For a curve with equation 23 38xx y y  , find (i) the x-coordinate of the point at which the tangent is parallel to the x-axis, [4] (ii) the equation(s) of the normal(s) at the point(s) where 2.y [3]   4 The polar form of a complex number z is given by iezr  , where 0r and 0 π , and the complex number 13 i.33wz  (i) Find w in exact polar form in terms of r and . [3] (ii) Given that 5 * z w is real and positive, find the possible value(s) of  in exact form, leaving your answer(s) in terms of . [4]

5 The function f is defined by 2f : 2 5 3, , .x xx x x a   (i) State the greatest value of a such that the function 1f exists. [1]   For the rest of the question, use the value of a found in part (i). (ii) Find 1f in a similar form. [3]   (iii) Find the exact solution of 1f. x x  [3] (iv) The function g is defined for specific integer values of x as follows. x 5 4 1 0 2 g(x) 6 4 23 0 3 Find the value of b, where 1fg ( ) 3 .b  [1]   6 The curve C has equation 221 0 1 7 ,, 3 .3 xxyx x x     (i) Without using a calculator, find the set of values of y that C can take. [4]   (ii) Sketch C, labelling the relevant features in exact form. [5]

7 To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to the origin O, every point in the virtual world is represented as coordinates (x, y, z). Perspective projection is a techni que used to project points in the virtual world onto a virtual screen while resp ecting the scaling prope rties of objects at various distances. Getting this right in VR he lps in the perception of depth and scale. In addition, only points that li e within a zone in front of the eye positioned at O, known as the viewing frustum (see Fig. 1), will be projected onto the virtual screen. Fig. 1 In Fig. 1 , a triangular object ABC   in the viewing frustum with 36,36,12A   and 24,48,72B is projected onto the virtual screen to give an image of ABC . It is given that the coordinates of A are 3, 3,1 and the line that passes through and AB is parallel to the vector 2+ 2 .ij k (i) Show that the coordinates of B'  are  1, 2, 3 . [4] (ii) Given that the coordinates of C are 1, 4, 2 , find the equation of the virtual screen in scalar product form. [2] (iii) Find the exact area of triangle ABC . [2] (iv) If the length of projection of BC  onto the normal of the virtual screen is 57 29 units, find the possible position vector(s) of ,C leaving your answer(s) in exact form. [4]   B C Object ABC in viewing frustum is projected onto the virtual screen as A'B'C' Viewing frustum O A' B' C' A

Section B: Statistics [48 marks] 8 The continuous random variable Y has the distribution  2N, . It is known that  PP 5 0 . 1 .Ya Ya    Express and   in the forms and ka ma respectively, where k and m are constants to be determined. [4] 9 A card game called “Happy Family” is played with 28 cards, consisting of 7 sets of 4 cards. Each set consists of a father, a mother, a son and a daughter from the same family. The family names are Painter, Postman, Plumber, Butcher, Carpenter, Singer and Teacher. So for example, the complete set of Teacher fam ily cards consists of father Teacher, mother Teacher, son Teacher and daughter Teacher. The objective of the game is to collect as many complete sets of family cards as possible. At the end of the game, all the cards are collect ed. Each player will either be left with no cards or complete sets of family cards. The winner is the one with the most number of complete sets of family cards. A and B play a game of “Happy Family”. (a) At the end of the game, A has exactly 3 complete sets of family cards and one of the sets is the Teacher family cards. How many possible combinations of complete sets of family cards can B have? [1] (b) If A is the winner at the end of the ga me, how many possible combinations of complete sets of family cards can A have? [2]   (c) Given that A has exactly 3 particular complete sets of family cards at the end of the game, and he arranges these 12 cards in a circle. How many different ways can these cards be arranged so that no two mothers are next to each other? [3] 10 For events A and B, it is given that P( ) 0.38A  , P( | ) 0.5AB  and P( ) 0.52AB . (i) Find P( )B . [3] For a third event C, it is given that P( )C = 0.6 and   P |A' B C' = 0.85. (ii) Find P( ).ABC [3] (iii) State, with a reason, whether the events A and C are mutually exclusive. [2]

11 In a game, a player tosses a fair die, whose faces are numbered from 1 to 6. If the player obtains a 6, he tosses the die a second time, a nd in this case, his score is the absolute difference of 6 and the second number. Otherwise, his score is the number obtained in the first toss. Let the player’s score be denoted by X. (i) Show that  7P1 36X  and tabulate the probability distribution of X. [3] (ii) Find the exact value of E(X). [1] Mr Lim plays this game 4 times. (iii) Find the probability that he obtained a score of 5 no more than 2 times. [2] (iv) Find the probability that his total score is less than 2 in the 4 games that he played.[2]   12 A chocolate manufacturing company produces chocolate bars. Th e packaging of the chocolate bar states that the mass of each bar is 200 g. The production manager has been receiving complaints that the mass is overstated and he wants to carry out a hypothesis test. (i) Explain whether the manager should carry ou t a 1-tail test or a 2-tail test. State appropriate hypotheses for the test, defining any symbols you use. [2] The masses, x grams, of a random sample of 30 chocolate bars are summarised as follows.   2 30 200 54 200 550nx x     (ii) Calculate unbiased estimates of the popul ation mean and variance of the mass of chocolat