2024 Y6 Timed Prac Rev Paper 1 (Qns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6 2024 Year 6 Timed Practice Revision Practice Paper 1 (Source: 2019 Year 6 Term 3 Common Test) The solution will be released in Ivy on 27 May (Monday) Total Marks: 100 Duration: 3h **** Note that Qn 2 (Complex Numbers) will NOT be examined in the coming Timed Practice. So, you should complete this paper within 2 hrs 47 mins. Section A: Pure Mathematics [59 marks] 1 The curve 1C with equation 22 22 14 xy aa is transformed to curve 2C by a translation of a units in the positive x-direction, followed by a stretch with scale factor 2 parallel to the x-axis, and followed by a reflection in the y-axis, where a is a positive constant. (i) Find the equation of 2.C [3] (ii) Describe the shape of 2C geometrically. [2] 2 Do not use a calculator in answering this question. The equation 2 23 4 0zz  has two complex roots 1z and 2 ,z where 10a r g ( ) 2z  . (i) Find 1z and 2z in polar form. [3] (ii) Show that 4 1 2 2 4z z  . [1] (iii) Find the set of possible values of n, ,n for which 1 13 i nz  is a real number. [3]

2 2022 Year 6 Common Test Revision Practice Paper 3 3 The function f is defined as follows. 2f: 3 ,x xx for x a . (i) State the largest value of a for which the function 1f exists. Hence find  1f x and state the domain of 1f for this value of a. [3] In the rest of the question, use 1a . The function g is defined on an interval A as follows. g: l n 3 2 ,xx  for .x A (ii) A student suggests 2 possible intervals for A as follows. (a) 21, 33  (b) 1 , 02  Determine which of the above intervals will result in the existence of the composite function fg, justifying your answ er. In the case(s) that fg exists, find the range of fg. [4] 4 (a) A geometric series has first term 22 s i n and second term . (i) Show that the series is convergent for 2    . [2] (ii) It is given that 3 4   and nS denotes the sum of the first n terms of the series. Find nS and hence determine the exact value of .S [3] (b) (i) Show that 1 1 1 nn nn    , where n   . [1] (ii) Hence find the least possible value of N such that 4 1 100. 1 N n nn   [2] sin 2

3 2022 Year 6 Common Test Revision Practice Paper 3 5 The parametric equations of a curve are 2xt , 21y t t  , where t  , 1 .2t  (i) Sketch the curve, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] (ii) The tangent to the curve at the point 2, 21 pp p    intersects the x-axis at point A and the y-axis at point B. Find, in terms of p, an expression for the area of the triangle OAB. [5] 6 (a) Given a and b are two non-zero and non-parallel vectors and show that the length of projection of c onto a and the length of projection of c onto b have the same magnitude. [3] (b) The equations of line 1,l planes and are 2 1 1 31:, 22 :2 3 5 , :3 2 , xylz xy z ax y z b         respectively. ( i ) If lies on , find the values of a and b. [2] For the rest of the question, does not lie on and intersects at point F. ( i i ) Find the coordinates of F. [3] ( i i i ) Find, in terms of a, a direction vector of the line of intersection between and . [2] (iv) Find the relationship between a and b if F also lies on . State, in terms of a, a vector equation of . [2] ,cb aa b 1 2 1l 2 1l 2 1l 1 2l 1 2 2 2l

2022 Year 6 Common Test Revision Practice Paper 3 7 The S-I model is used to study the spread of infectious diseases across different scenarios. In this question, it is assumed that we are studying a homogeneous population in a closed community of constant size N throughout the period of cons ideration. The population is divided into two groups – one group of infected individual s who have the diseases and another group of healthy and susceptible in dividuals who becomes infected when they come into some form of contact with the other group of infected individuals. Using x and y to represent the number of infected indivi duals and number of healthy and susceptible individuals at time t respectively, we can model the situation using the following differential equation where k is a positive constant. (i) By expressing y in terms of N and x, show that equation (I) can be rewritten in the form of a first order differential equation in terms of x and t. Given that x  when 0,t solve this differential equation a nd show that the solution can be expressed in the form  ,11 e Nkt Ax B   where A and B are constants expressed in terms of N and . [8] (ii) This model was tested using some data from the 2003 SARS epidemic in Singapore where k was estimated to be and time was measured in days. Write down the equation of the corresponding solution curve and sketch the part of the curve which is releva nt to this context. (Your sketch should be suitably labelled on the axes.) State what happens to x for large values of t. [4] Section B: Probability and Statistics [41 marks] 8 (i) A code consists of 10 digits which ar e either zeros or ones, for example, 1011011010. Calculate the number of such codes if there is no restriction. [1] Given further that the 10 digits consists of 4 zeros and 6 ones, calculate the number of such codes if (ii) there is no other restriction, [1] (iii) all the zeros must be separated and the first and last digits must be different, [2] (iv) no more than 4 ones are together. [2] 9 In an online shop, the time taken, in hours, to sell a watch is a normally distributed continuous random variable X. The standard deviation of X is 0.68 hours and the expected value of X is 1.75 hours. After an aggressive advertising campaign, the total time taken to sell 8 watches is found to be 11 hours. Test, at 5% level of significance, whether there is evidence that the mean time taken to sell a watch has decreased. State an assumption that you have used in your calculation. [6] d , ( )d x kxyt  I 1, 206,N  48.1835 10 

5 2022 Year 6 Common Test Revision Practice Paper 3 10 In this question you should state the parameters of any distributions that you use. Crispy Cream Donut Shop sells 2 types of donuts: Ring Donuts and Filled Donuts. The masses in grams of Ring Donuts and Filled Donuts have normal distributions and respectively. (i) Find the probability that the total mass of 3 randomly chosen Ring Donuts is more than twice the mass of a randomly chosen Filled Donut. [3] 12 donuts are packed into a paper box. The mass in grams of an empty paper box has a normal distribution . (ii) The probability that the total mass of a box containing 6 Ring Donuts and 6 Filled Donuts is more than m grams is 0.95. Find m. [4] (iii) State an assumption that you have us ed in your calculations in parts (i) and (ii). [1] 11 (a) Two digits X and Y are chosen independently at random from the set of 10 digits {0, 1, 2, . . . , 9}. Events A and B are defined as follows: A : 1,XY B : X and Y are both less than 6. Find (i) P( ),A [1] (ii) P( ),B [1] (iii) P( ).A B [2] (b) On a particular afternoon in June, 5 girls and 4 boys were in the Shaw Library and 6 girls and 9 boys were in the Hullett Li brary. A teacher selects at random 2 students from each library to distribute 4 free concert tickets. (i) Calculate the probability that 2