EJC H2 Promo 2023 (Qn)

Duration: 3 hrs Marks: 100 2023 EJC H2 Maths Promo Paper Attempt all questions. 1 A function f is defined by  32f xxa xb x c  , where a , b and c are constants. The graph of fyx passes through the points  2, 1 and 2, 3 . The point 2,1 lies on the graph of f1yx . Find the values of a, b and c. [4] 2 The diagram below shows the graph of fyx . The curve passes through the x-axis at  2, 0 and  1, 0 , and has a maximum point with coordinates 2, 2 . The lines 1x and 1y are asymptotes to the graph. Stating the e quations of any asymptotes and the coordinates of any points of intersection with the axes and stationary points, where possible, sketch the graphs of (a) 3fyx a , where a is a positive constant such that 12 a , [3] (b) fyx  . [3 ] 3 (a) Sketch, on the same diagram, the curves with equations 1 2 xy x   and 1ln 1 22 xy   , stating the equations of any asymptotes and the coordinates of an y points of intersection with the axes. Label the two curves clearly. [5] (b) Hence solve t he inequality 11 ln 122 2 xx x     . [1] y x O ×

4 Referred to the origin O, let P, Q and R be distinct points with position vectors p, q and r respectively. (a) Show that       rp rq qqrr pp . [2] (b) Give the geometrical meaning of 1 2 qqrrpp . [2] (c) Given that qqrrp0p , 3PR QR , and that PQ PR , express r in terms of p and q. [3] 5 (a) Verify that   2321 3 1 1! ! ( 1 ) ! 1! rr rr r r    . [1] (b) Hence find  2 1 31 .1! n r rr r   [3] (c) Use your answer to part (b) to find 2 3 3 ! n r rr r    . [3] 6 (a) The first n terms of a series are given by 21log 3 log 27 log 243 ... log 3 n aa a a    , where a is a positive constant. (i) Show that the series is an arithmetic series. [2] (ii) Given that sum of the first 30 terms of the series is 300, find the value of a. [2] (b) A geometric series has first term c and common ratio r, where c and r are non-zero. An arithmetic series has first term b and common difference d, where b and d are non-zero. It is given that the 5th, 8th and 10th terms of the arithmetic series are equal to the 2nd, 3rd and 4th of the geometric series respectively. Show that r satisfies the equation 235 2 0rr  and hence find the sum to infinity in terms of c. [4] 7 A function g is defined by 2 2 if 0,g( ) 1 0i f 0 . xx x x      (a) Give a reason why g does not have an inverse. [1] y x y = g(x) O 2

(b) The function 1g exists if the domain of g is restricted to xk . State the greatest possible value of k. [1] In the rest of the question, the domain of g is xk , where k takes the value determined in part (b). (c) Find 1g( ) x and state the domain of 1g . [3] (d) Sketch, on the axes given below, the graphs of g( )yx and 1g( )yx  . Label the two graphs clearly. Write down the equation of the line in which the graph of g( )y x must be reflected i