VJC 2024 H2 JC2 Math Prelim P1 Questions_Student

2024 VJC Prelim 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]

2024 VJC Prelim Paper 1 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]

2024 VJC Prelim Paper 1 3 The functions f and g are defined by 6f: 3 axx x − −  for x∈  , 3x≠ , g: e xx − f or x∈  , ln 3x≥ . 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]

2024 VJC Prelim Paper 1 (b) State the exact value of 6 f( π) . [1] (c) Find the exact range of fg. [3]

2024 VJC Prelim Paper 1 4 (a) Given that a, b and c are non-zero vectors such that ( ) ( )ab ac bc+×+= × , and bc≠ , find the relationship between a, b and c. [4] (b) It is given instead that a, b and c satisfy the equation abc 0++= with 2a = , 3b = and 4c = . Find the value of ab bc ca⋅+⋅+⋅ . [3]

2024 VJC Prelim Paper 1 5 It is given that ( ) 1f 5n nn −= where n is a positive integer. (a) By considering ( ) ( )f f1rr−+ , 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]

2024 VJC Prelim Paper 1 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]

2024 VJC Prelim Paper 1 (c) Find 22 1 d23 xx kx k−+ + ⌠⌡ , where k is a positive constant. [4]

2024 VJC Prelim Paper 1 7 (a) The curve C has equation f( )yx= where 2 f( ) ax bx cx xd ++= + , and , , and abc 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]

2024 VJC Prelim Paper 1 (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]