2025 EJC Promo (Qn)

1 2025 EJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. 1 Four housewives bought three different kinds of organic vegetables at a Goodprice supermarket. They cannot remember the individual prices per kilogram, but three of them can remember the total amount that they each paid. The weights of vegetables purchased and the total amounts paid are shown in the following table. Amy Betty Carol Denise Pumpkins (kg) 1.20 1.50 1.30 2.30 Carrots (kg) 2.50 2.70 2.60 3.10 Broccoli (kg) 1.25 1.35 1.10 2.10 Total amount paid ($) 21.86 25.20 21.90 ? Given that, for each variety of vegetable, the price per kilogram paid by each of the housewives is the same, find the amount that Denise paid for her purchase. [4] 2 The diagram shows the sketch of the curves 2 5yx= − and 2yx= + . Solve the inequality 2 52xx−>+ exactly. [4] 3 (a) Given that ,×=ab 0 what can be deduced about the vectors a and b? [2] (b) Find a unit vector n such that ( )23.× −+ =n ij k 0 [1] (c) Find the cosine of the acute angle between 23−+ij k and the y-axis. [2] 5 2 -2

2 4 The diagram below shows the curve f( )yx= . The curve crosses the x-axis at point ( ,0)Aa , where 1a<− and has a maximum point 1 10,29B−− . The curve also has a horizontal asymptote 1y=− and vertical asymptotes 1x=− and 0x= . On separate diagrams, sketch the graphs of (a) 3f ( 1)yx= − , [2] (b) 1 f( )y x= , [3] showing clearly in each case, the equations of any asymptotes, the coordinates of any stationary points and axial intercepts in terms of a. The curve f( )yx= undergoes the following transformations in succession: P: Reflection in the y-axis Q: Scaling parallel to the x-axis by a scale factor of k (c) Determine the coordinates of the x-intercept of the transformed curve. [2] y x 0

3 5 The sequence 123, , , ...xxx satisfies the recurrence relation ( ) 2 1 2 for 1nn nxx x n+ = −≥ (a) Prove algebraically that, if the sequence converges, then it converges to either 1 or 0. [2] (b) Determine the behaviour of the sequence for each of the following cases. (i) 1 0.5x = [1] (ii) 1 1.5x = [1] (iii) 1 1x =− [1] (c) By considering the graph of 232y xxx= −− , show that if 01 nx<< , then 1 0nnxx+ −< . State briefly what this result tells us about the sequence in part (b)(i). [2] 6 A curve C is defined parametrically by 26,x ty t= = . (a) Show that the equation of the normal at the point ( ) 26,P pp is ( ) 3 18 3 0p yp x+ − −= . [3] (b) The normal to the curve at the point P meets the curve again at the point ( )54, 81Q − . Find the possible values of p. [2] (c) Sketch the graph of C, indicating clearly the coordinates of Q and the possible points of P. Sketch also the normals at each of these points P. [3]

4 7 [It is given that a sphere of radius r has surface area 24 rπ and volume 34 3 rπ .] A model of a toy is made up of three parts. • The top of the toy is modelled by a circu