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 t