2023 NYJC Promo (Qn)

2023 NYJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. 1 (a) Sketch, on the same axes, the graphs of 15 1y x   and 2 1 . (5 ) y x   [2] (b) Solve the inequality 2 115 1 (5 )x x    . [3] 2 (a) Find  f' x where    2fl n1x xx  , giving your answer in its simplest form. [3] (b) Find  g' x where  1 2 12gt a n xx x       , giving your answer in its simplest form. [3] 3 With reference to the origin O, the points A and B have position vectors a and b respectively. (a) A point C has position vector a + b. State the geometric shape formed by O, A, B and C. [1] Point M lies on AC, between A and C, such that AM : MC = 1 : 2. (b) Show that the area of triangle OAM can be written as  ab , where  is a constant to be found.[2] It is given that | b| = 5 and the shortest distance from A to OB is 3 . (c) Using your result from part (b), find the exact value of the area of triangle OAM. [2] (d) Point N lies on OM, between O and M, such that ON : NM = 3 : 1. Show that A, N and B are collinear and hence find the ratio AN : NB. [3] 4 (a) Describe fully a sequence of transfor mations which transforms the curve 2yx  onto the curve 2 14 y x . [2] (b) A curve has equation y = f(x), where  21 f o r 0 1 ,f 0 otherwise. xxx    (i) Sketch the curve of y = f(x) for 22 x , stating the coordinates of the end points and the coordinates of any points where the curve crosses the axes. [3] (ii) Sketch the curve with equation f1yx for 22 x , stating the coordinates of the end points and the coordinates of any points where the curve crosses the axes. [3]

2 5 The function g is defined by  21 2g: e , , 1 .xx xx a     (a) State the greatest value of a for which the function g −1 exists. [1] For the rest of the question, let a = 0. (b) Sketch the graphs of y = g(x), y = g −1(x) and y = g −1g(x) on the same diagram, showing clearly the relationship between them. You should also st ate the coordinates of the end points of the graphs. [3] The function h is defined by  h: 1 ln , , 0.xx x x    (c) (i) Find hg(x) and state its domain. [2] (ii) Hence, or otherwise, find the exact value of   1 3 2hg .  [2]   6 Let f( ) l n ( )rr , where r is a positive integer and r  2. (a) Show that 2 1f( 1 ) 2 f( ) f( 1 ) l n 1rr r r      . [1] (b) Show that 2 2 11ln 1 ln 2 N r N Nr     . [3] (c) Explain why the series 2 2 1ln 1 r r     converges and state the exact value of the sum to infinity of this series. [2] (d) Find an expression for the series    22211 1 111ln ln ln ... 21 22 23kk k             , giving your answer as a single logarithmic function in terms of k. [3] 7  The parametric equations of a curve are 1 e2 e2 ttx  and e2 e .tty  (a) Using calculus, find the exact gradient of the normal to the curve at t