RVHS 9758 2023 Prelim P2

1 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 Section A: Pure Mathematics [40 marks] 1 The complex numbers 12 and zz are such that i 3 1 4ze π = and i3 2 2ze π− = . (i) On an Argand diagram with origin O, mark out the points which represent 12 and zz respectively. Deduce the exact value of 12 zz− . [3] (ii) Write down a complex number a such that 12z az= . Hence or otherwise, describe geometrically the relation between 1z and 2z . [2] 2 A sequence 0u , 1u , 2u , … is such that 1 41 2 33 3 nn nnuu Q+  −= − +    , where Q is a constant and 0n≥ . (i) Given that 0 3u = , 1 4 3u = , find Q and 2u . [2] (ii) Show algebraically that for any real positive constant ε , there exists an integer 0n such that 1nnuu ε+ −≤ for 0nn≥ . Find an integer 0n when 0.001ε = . [4] 3 (a) A curve has equation ( )fyx= , where ( ) 2 for 0 1, 13f for 1 3,22 0 otherwise. xx xx x  ≤≤ =− + <≤   (i) Sketch the curve for 14 x−≤ ≤ . [3] (ii) On a separate diagram, sketch the curve with equation 1f1 2yx = − , for 14 x−≤ ≤ . Write down the coordinates of the end points of this curve. [2] (b) State a sequence of transformations that will transform the curve with equ ation lnyx= onto the curve with equation ( ) 3 ln 2yx= − . [2] 4 The line 1l has equation ( )3 λ= ++r k ij where λ is a parameter. Referred to the origin O , the points A and P have position vectors 4+ik and 345−+ +i jk respectively. (i) Find the cartesian equation of the plane 1π containing the line 1l and the point A. [2] (ii) The point Q is the foot of perpendicular of P on 1l . Find the position vector of Q . [3] (iii) The line 2l passes through P and is parallel to the vector 3 m++ij k where m is a constant. Find the value of m given that 2l is parallel to 1π . h denotes the distance between 2l and 1π . Find the value of h . [3]

2 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 Section B: Statistics [60 marks] (iv) The plane 2π is parallel to 1π . The 3l passing through P and Q , intersects 2π at the point R . The points P , Q and R are such that : 2:5PQ PR = . Without finding the equation of 2π , find the possible value(s) of the distance between 1π and 2π in terms of h . [2] 5 (i) Using the substitution 2 1ux= + , evaluate 1 1 1d a xx x − − +∫ , giving your answer in terms of a. [5] (ii) The curve C has equation 1y xx= + . The region R is bounded by C, the lines 2y= and 1x=− . Find the exact volume of the solid obtained when R is rotated through 2π radians about the line 2y= . [7] 6 Two fair six-sided dice are thrown and the highest score X is recorded. (i) State the probability distribution for X. [1] (ii) Find the expected value and variance for X. [3] 7 A group of 8 people consisting of 2 children, 4 women and 2 men goes for lunch, photo - shoot and a game. (i) At lunch, they are seated at a round table such that each child is seated between 2 women and no men are seated together. Find the number of ways they can sit. [3] (ii) After lunch, they go for a photo-shoot. They stand in a row such that all the women are in descending order of height, from their left to right. Find the number of ways they can s tand, assuming that all the women are of distinct height. [2] (iii) After the photo-shoot, they are split into 2 teams to play a board game. Each team consists of a child, 2 women and a man. Find the number of ways the 2 teams can be formed. [2]

3 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 8 A blind box series named MathBots includes 4 bots and 4 number sets as accessories for the bots. Bots are named α , β and γ with ω as the special character bot. The number sets are  ,  ,  and  . Each box of the series contains one of the bots and one of the number sets. The data in the table below shows the probabilities of getting boxes with the specified contents.     α 18 18 0 120 β 0 18 18 120 γ 18 0 18 120 ω 0 0 0 110 (For example, there is a 110 probability that a randomly chosen box contains ωand  .) A collector randomly chooses 4 boxes for purchase. (i) Show that the probability of the collector getting all the bots and number sets is 3 320 . [2] (ii) Find the probability that the collector gets all the bots given he manages to get all the numbers sets. [3] (iii) Determine if the event that he gets all the bots and the event that he get all the number sets are independent. [2] 9 Professor Aok conducts a course with grading at the end of each run of the course for certification. Each run of the course takes in exactly 10 students. It is found that on average a student has %p chance of achieving a distinction. Let X be the number of students who achieve a distinction during a run of the course. (i) State an assumption for X to be well modelled by a binomial distribution. [1] Assume that X is well modelled by a binomial distribution. (ii) Given that the probability 7 students scoring distinction is greater than the probability of any other number of students scoring distinction, find the possible range of values of p. [2] (iii) There is a probability of 0.00228786 that there are only 2 stu dents who achieved distinction in a particular run of the course, find the value of p. [2] (iv) A check revealed that of the 8 recently concluded courses, at least 5 had more than 6 students achieving distinction. Find the probability that out of those re cently concluded courses, there are at most 7 runs with more than 6 students achieving distinction. [3]

4 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 10 (a) With the aid of diagrams, explain the difference between the least square regression lines of y on x and that of x on y. [3] (b) To investigate the effect of fertilizers on plant growth, 10 randomly selected plants of the same species are given varying amount of fertilizers and the increase in height of the plants is measured after a week. Amount of fertilizers, x (ml) 7 5 7 8 6 10 4 9 6 9 Increase in height of plant, y (cm) 9 7 8 10 9 10 3 8 8 10 (i) Using a scatter diagram, explain if a linear relationship between the amount of fertilizers used and increase in height of plants is appropriate. [2] (ii) Another relationship lny ab x= + is considered more appropriate. Use this to estimate the amount of fertilizers needed to achieve an increase of 12 cm in the height of the plant. [2] (iii) Justify if this prediction is reliable. [1] (iv) Suppose that the unit of y is in millimetres and that of x remains unchanged. State how the value of the product -moment correlation coefficient will change and find the corresponding equation of the regression line of y′ on ln x , where y′ is the increase in height in millimetres. [2] 11 Rectangular blocks of wood undergo crafting and lacquering to become ornaments. The masses of the rectangular blocks, X g, are known to have normal distribution with mean 300 g and standard deviation 20 g. Crafting the company logo on a block reduces its mass by 10%. After that, a layer of lacquer is applied to the crafted wooden block that increases its mass by 5%. (i) Show that the probability that a piece of completed ornament has mass between 290 g and 350 g is approximately 0.365. [3] (ii) During an inspection, completed ornaments are selected randomly and their masses are recorded. Find the probability that the 10 th ornament selected is the 3 rd piece that has mass between 290 g and 350 g.