2024 VJC H2 JC2 Math Prelim P1 Qn (updated)

2024 Victoria Junior College Preliminary Examination H2 Mathematics Paper 1 1 (a) Express 2 33 8 22 15 x xx − ++− as a single algebraic fraction. Hence, without using a calculator, solve exactly the inequality 2 33 8 22 15 x xx − −+− . [4] (b) Using your answer to part (a), find the set of values of x for which 2 42 33 8e 2e 2e 15 x xx − −+− . [2] 2 The sum of the first n terms of a sequence, ru , is given by 1 1 ( 1)! n r r nu n= =− + . (a) Find nu in terms of n, for 2n , expressing your answer as a single algebraic fraction. [2] (b) Show that 5 1 30 n r r u =  for all 5n . [2] (c) Explain why 1 r r u  =  is a convergent series. [1] 3 The functions f and g are defined by 6f: 3 axx x − − for x , 3x , g : e xx − for x , ln3x . The function f is such that 1 f( ) f ( )xx − = for all x in the domain of f. (a) Find the value of a. [3] (b) State the exact value of 6 f( π) . [1] (c) Find the exact range of fg. [3] 4 (a) Given that a, b and c are non-zero vectors such that ( ) ( )a b a c b c+  + =  , and bc , find the relationship between a, b and c. [4] (b) It is given instead that a, b and c satisfy the equation a b c 0+ + = with 2a = , 3b = and 4c = . Find the value of a b b c c a +  +  . [3] 5 It is given that ( ) 1f 5n nn −= where n is a positive integer. (a) By considering ( ) ( )f f 1rr−+ , find an expression for 2 41 5 n r r r = −  . [3] (b) Hence find an expression for 1 1 46 5 n r r r + = +  . [3]

6 (a) Find the exact value of 3 12 21 2 2sin d 1 x x x − −   . [3] (b) Find the exact value of π 3 0 cos 2 dxx . [3] (c) Find 22 1 d23 xx kx k− + +  , where k is a positive constant. [4] 7 (a) The curve C has equation f ( )yx= where 2 f ( ) ax bx cx xd ++= + , and , , and a b c d are constants, and 0a . Given that C has asymptote 1yx=+ , find the value of a and show that 1bd=+ . [2] If f is an increasing function for all x , xd− , show that cd . [3] (b) It is further given that 1c= and 2d = . (i) Sketch C. [3] (ii) By sketching a suitable graph in the same diagram in part (b)(i), find the number of real roots to the equation 22 231 4 162 xx xx  ++ += + . [2] 8 (a) The diagram shows the graph with equation f (2 )yx= . The graph passes through the points ( )4,0A − , ( )0,0B and ( )3,6C , and has asymptotes 2x=− and 1y= . On separate clearly labelled diagrams, deduce the graphs of (i) f (2 2)yx=− , [2] (ii) f ( )yx= . [2]

(b) The curve 1C undergoes the transformations in the order given below: 1. A translation of 2 units in the negative x direction. 2. A stretch parallel to the x axis, factor 2, y axis invariant. 3. A translation of 1 unit in the positive y direction. The resulting curve 2C has equation 2 9 22 4 xxy x ++= + , x , 4x− . Find, in the simplest form, the equation for 1C . [4] 9 Find the area of the region bounded by the graphs of 223yx=+ and 27yx=− + . [3] State the area of the region bounded by the 2 graphs if both graphs are translated 7 units in the negative y-direction. [1] The region R is bounded by 223yx=+ , 27yx=− + , the x-axis and the y-axis. Find the exact volume of the solid generated when R is rotated 2π about the y-axis. [5] 10 Given that 2iz=− is a root of the equation 4 3 24 12 17 0z z z pz q− + + + = , where p and q are real, find p and q . [4] Using the values of p and q found, find the other roots of the equation 4 3 24 12 17 0z z z pz q− + + + = in exact form. [4] 11 Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass various activities related to the identification, classification, biology, and ecology of trees. A group of dendrologists are studying the growth of 2 species of trees, codenamed Tree Vee and Tree Jay. In the 1st year, the height of Tree Vee and Tree Jay are both H cm. In the 2nd year, Tree Vee’s height increases by s cm and subsequently, the increase in height every year is 10% less than the previous year’s increase. Show that the height of Tree Vee in the 4th year is given by ( )2.71Hs+ cm. [1] Show that the height of Tree Vee in the nth year is given by ( ) 110 1 0.9 nHs − +− cm. [3] Hence, write down in terms of H and s, the theoretical maximum height (in cm) of Tree Vee. [1] In the 2nd year, Tree Jay’s height increases by t cm and subsequently, the increase in height every year is 0.5 cm less than the previous year’s increase. Show that the height of Tree Jay in the 10 th year is given by ( )18 9Ht−+ cm. [2] It is now given that 20t= . After the 10 th year, Tree Jay’s height increases at a constant rate of 7 cm per year. Express Tree Jay’s height (in cm) in the nth year (where 11n ) in terms of H and n. [2] It is further given that 30s= , and the 1st year is the year 2024 . Find the years in which the heights of Tree Vee and Tree Jay are within 7 cm of each other, after 2034. [3]

12 Game developers closely monitor the number of people playing their game. Understanding player numbers and behaviour can not only help in optimising in -game purchases, advertisement placements, and other revenue -generating aspects, it can also help the company manage server loads and ensure the game runs smoothly without performance issues. Two game developers are interested in the number of players playing the mobile game “Mobile Saga”. They attempt to model the number of players x, in hundred thousands, at time t months after the launch of the game using a differential equation. On the day of the launch, there were 55 000 players. (a) One game developer suggests that x and t are related by the differential equation 2d3 d5 x x ktt =− , where k is a positive constant. (i) By substituting 3 5e t xu= , show that the differential equation can be written a s 3 2 5d ed tu ktt − =− . [2] (ii) Hence show that 3 2 55 50 250 11 250 e3 9 27 20 27 tk k k kx t t = + + + −  . [4] (iii) Company A intends to place an advertisement in the game only if there are more than 76 000 players playing the game. Given that 1 10k = , find the length of time for which Company A will place an advertisement in “Mobile Saga”, giving your answer correct to the nearest month. [2] (b) The o ther game developer suggests that x and t are related by the differential equation ( ) 2 23 d 10 d 1 x t t =− + . Given further that there were 180 000 players playing “Mobile Saga” after 1 month, find x in terms of t. [4]