2023 EJC Promo (Qn)

2023 EJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. [You may skip Q10 for now, as they are on Integration. Remaining total marks is 90, Duration is 2 hr 42 mins.] 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 equations 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 the 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(