JPJC 9758 2023 Prelim P1

Jurong Pioneer Junior College H2 Mathematics JC2-2023 Prelim Paper 1 Question Paper 1 A sequence 1u , 2u , 3u , … is such that 2 12 15 60nnu u n nA−= − ++ , where A is a constant, and 2.n≥ It is given that 1 2u = and 2 4u = . (i) Find A and 3u . [2] It is known that the nth term of this sequence is given by 12(2 )n nu p qn r += −+ . (ii) Find p, q and r. [3] 2 It is known that the nth term of a sequence is given by ln 1 n nu n =  + , where 1n≥ . (i) Determine if the sequence is convergent, stating your reason clearly. [1] (ii) By using the method of differences or otherwise, find 1 N n n u = ∑ in terms of N . [3] 3 A curve C has equation 11y x= − , where 0x≠ . The line l is the tangent to C at xa= , where a is positive integer. Show that the equation of l may be expressed in the form 22 2.ay x a a−= − [3] Find the value of a for which l is parallel to the line 91yx−= . Hence find the equation of l in this case. [3] 4 The curve C has equation 2 3 2 xxy x ++= − . (i) Without using a calculator, find the set of values that y can take. [4] (ii) Sketch the graph of C , stating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts. [3] (iii) By drawing a suitable graph on the same diagram in (ii), find the range of values of k, where 0k > , such that the equation ( ) ( ) 2 22 22 2( 1) ( 2) 3 2x x x x kx+ − + ++ = − has at least one positive real root, giving your answer correct to 3 decimal places. [3]

2 5 The function f, with domain the set of all real values, is given by ( ) 2 for 0 2,4f for 2 4,24 xxx xx <≤ −= <≤− and that ( ) ( )f f4xx= + . (i) Find ( )f 45 . [1] (ii) Sketch the graph of ( )fyx= for 47 x−≤≤ and state the range of f. [3] The function g is defined by ( ) 2 g: 1 2xx −+ for , 0 1.xx∈ << (iii) Explain why the composite function gf does not exist. [1] (iv) Find an expression for ( )fg x and hence, or otherwise, find ( ) 1 1fg ( ) 2 − . [4] 6 The plane p has equation 2 2 12xyz−+= . With reference to the origin O , the point A has position vector 2 +−i j 3k . (i) Find the coordinates of the foot of perpendicular, N, from A to p. Hence determine the coordinates of the reflection of A in p. [5] (ii) Find the cartesian equations of the planes such that the perpendicular distance from each plane to p is 15. [5] 7 (a) Find 3 4 2 d1 x xx+∫ . [2] (b) Find cos 4 sin10 dx xx∫ . [2] (c) Show, by means of the substitution tanx θ= , that ( ) 31 34 300 2 d cos sin d 1 x x x π θ θθ= +∫∫ . Hence find the exact value of ( ) 31 30 2 d 1 x x x +∫ .