2026 RVHS H2 J2 Revision Package (Probability,Vectors, Complex Numbers) - Questions

1 Probability 1 NYJC Prelim 9758/2023/02/Q6 In a funfair, a vendor sells lucky draw tickets at $1 per ticket. He accepts payment using $1 or $2 notes only. In the queue for tickets, there are m people each paying with a single $1 note and n people each paying with a single $2 note. Each person in the queue wants to buy a single ticket and each arrangement of people in the queue is equally likely to occur. Initially, the vendor has no money but a large supply of tickets. The vendor will stop selling tickets if he cannot give the required change. (a) In the case of 1m and 1n= , find the probability that the vendor can sell one ticket to each person in the queue. [2] (b) By considering the first three people in the queue in the case of 2m and 2n= , (i) show that the probability that the vendor is able to sell one ticket to each person in the queue is 1 1 m m − + , [3] (ii) find the probability that the vendor is unable to sell any ticket to the third person, given that the vendor is unable to sell tickets to the first three people in the queue. [3] 2 PJC Prelim 9758/2017/02/Q10 For events A and B, it is given that 11P( ) 20A = and 1P( ) 2B = . (i) Find the greatest and least possible values of P( )AB . [2] It is given in addition that 7P( | ') 9BA = . (ii) Find P( )AB . [2] (iii) Determine if A and B are independent events. Justify your answer. [2] (iv) Given another event C such that 2P( ) 5C = , 19P( ) 20A B C  = , 1P( ) 10A B C  = and P( ) 2P( )A C B C =  , find P( )AC . [3] River Valley High School 2026 JC2 H2 Mathematics Revision Package for Lecture Test 1

2 3 MI Prelim 9758/2020/02/Q8 A group of n people each takes turn to pick one card, with replacement, from a pack of cards. (a) Assume that there are 12 unique cards in a pack. The probability that all the people in the group picked different cards is denoted by P. (i) For the case where 3n= , show that 55 72P= . [2] (ii) Find P for the case where 4n= and deduce the smallest value of n such that 1 2P . [2] (iii) State, with a reason, the least possible value of n such that 0P= . [2] (b) Assume now that there are 120 unique cards in the pack. It is given that, for the case where 12n= , the probability that all the people in the group picked different cards is 0.56640, correct to 5 decimal places. Find the smallest value of n such that the probability that at least two people picked the same card exceeds 1 2 . [4] 4 CJC MYE 9758/2021/Q10 A group of 11 friends, including Alan, Ben and Chris, forms a team to play football friendlies matches. In the team, there has to be 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. Everyone in the team can play in any position. (i) Find the number of different teams that the 11 friends can form. [2] (ii) Find the probability that Alan and Ben play in different positions. [3] After the match, the team goes to a restaurant for a meal where they all sit at a round table. (iii) Find the probability that Alan and Ben are seated next to each other. [2] (iv) Find the probability that Chris sits with either Alan or Ben, given that Alan and Ben are not seated next to each other. [3] 5 SAJC MYE 9758/2023/01/Q5 For events A and B, it is given that P( ) 0.45B = , P( | ) 0.4BA = and P( ) 0.75AB = . (i) Find P( )A and P( )AB . [4] (ii) Explain why A and B are not independent. [1] For a third event C, it is given that P( ) 0.4C = , and that B and C are mutually exclusive. (iii) Find the greatest and least possible values of P( )A C . [3]

3 Vectors 1 CJC Promo 9758/2021/Q4 The points ( )1,0, 2A − , ( )3, 1, 2B −− and ( )3,7,0C − lie on plane 1p . Another plane 2p has equation 3 2 3x y z− + = . (i) Find a vector equation of plane 1p in the form d=rn . [3] (ii) Find the acute angle between 1p and 2p . [2] The equation of plane 3p is given to be 9 3 6 7xyz− + − = . (iii) Find the shortest distance between 2p and 3p . [3] 2 CJC Promo 9758/2025/Q11 The plane 1 contains the points ( )1, 4, 2A , ( )1,0,5B and ( )0,8, 1C − . (a) Find a cartesian equation of 1 . [3] (b) Find the shortest distance between the point P (2,1,2) and the plane 1 . [2] A second plane 2 contains the point ( )2, 2,3D and is perpendicular to the vector 22++i j k . The point ( ),0,pq lies in both planes. (c) Find p and q. [3] (d) Hence find an equation of the line of intersection of the two planes in the form =+r a b , where  is a real constant. [2] (e) Find the acute angle between the line AC and the plane 2 . [2] 3 RI Promo 9758/2024/Q9 The planes p and q have equations ( ) ( ) ( )2 2 3 3 2r i j k      = + − + − + + − + and 11 a b  −= r. respectively, where a and b are constants and  and  are parameters. The line l passes through the point ( )5, 4,0 and is parallel to the vector 2 2 .− − +i j k The planes p and q meet in the line l. (a) Show that 1a= and 1 .2b= [2] (b) Find the exact acute angle between the planes p and .q [3] (c) Find the distance from the point ( )2,0,3A to the plane .q Hence deduce the shortest distance from A to l. [4] The plane q is reflected in the plane p to obtain the plane .q (d) Find a Cartesian equation of the plane .q [3]

4 4 HCI JC2 Prelim 9758/2019/01/Q12 At an airport, an air traffic control room T is located in a vertical air traffic control tower, 70 m above ground level. Let (0,0,0)O be the foot of the air traffic control tower and all points ( , , )x y z are defined relative to O where the units are in kilometres. Two observation posts at the points (0.8,0.6,0)M and (0.4, 0.9,0)N − are located within the perimeters of the airport as shown. An air traffic controller on duty at T spots an errant drone in the vicinity of the airport. The two observation posts at M and N are alerted immediately. A laser rangefinder at M directs a laser beam in the direction 2 7 1   − at the errant drone to determine D, the position of the errant drone. The position D is confirmed using another laser beam from N, which passes through the point (0.8,0.75, 0.3) , directed at the errant drone. (i) Show that D has coordinates (0.56, 0.24,0.12)− . [4] A Drone Catcher, an anti-drone drone which uses a net to trap and capture errant drones, is deployed instantly from O and flies in a straight line directly to D to intercept the errant drone. (ii) Find the acute angle between the flight path of the Drone Catcher and the horizontal ground. [2] At the same time, a Jammer Gun, which emits a signal to jam the control signals of the errant drone, is fired at the errant drone. The Jammer Gun is located at a point G on the plane p containing the points T, M and N. (iii) Show that the equation of p is 10.5 2.8 6.72 96 −  =−− r . [3] It is also known that the Jammer Gun is at the foot of the perpendicular from the errant drone to plane p. (iv) Find the coordinates of G. [3] (v) Hence, or otherwise, find the distance GD in metres. [2]

5 5 ACJC JC2 Prelim 9758/2024/01/Q11 At a ski resort, engineers are installing cables for a new cable car system to transport skiers to ski slopes. The system involves installing cables running between support towers. Cables are laid in straight lines and the widths of cables can be neglected. The cable AB is used to transport skiers up the slope and another parallel cable CD is used to transport skiers down the slope. Straight