VJC 2023 promo prac paper (A)

2023 PROMO PRACTICE PAPER A 1 VICTORIA JUNIOR COLLEGE 2023 PROMO PRACTICE PAPER A (Modified from 2017 VJC H2 Math PROMO) 1 TJC Promo 9758/2020/Q1 The sum, Sn, of the first n terms of a sequence 123,,,uuu  is given by ( )ln 1nSn= + . (i) Find nu in terms of n. [2] (ii) Describe the behaviour of the sequence. [1] 2 By using an algebraic approach, solve 10 3 xx ≥− . [4] 3 2017 A-level Paper 1 Q5 (Modified) When the polynomial 32x ax bx c+ ++ is divided by ( ) 1x− , ( ) 2x− and ( ) 3x− , the remainders are 8, 12 and 25 respectively. (i) Find the values of a, b and c. [3] A curve has has equation ( )fyx= , where ( ) 32f x x ax bx c=+ ++ , with values of a, b and c found in part (i). (ii) Show that the gradient of the curve is always positive. Hence, explain why the equation ( )f0 x = has only one real root. [2] 4 Water is poured at a rate of 9 m 3 per minute into an open container in the form of a frustum of a right circular cone as shown in the diagram above. The container has bottom radius of 2 m, top radius of 3 m and height of 4 m. After t minutes, the radius of the water surface is r m and the depth of water is h m, where 04 h≤< . (i) Show that 24 hr = + . [2] (ii) Find, in exact form, the rate of increase of the depth of water when h=1. [4] [The volume of a frustum of a right circular cone of bottom radius 1r , top radius 2r and height h is given by ( ) 22 1 2 12 1 π3V rrr r h= ++ .] mh 3 m 2 m 4 m mr • Exam conditions (one sitting, 3 hours) • Manage your time well • Check against solutions and learn from your mistakes before the next practice

2023 PROMO PRACTICE PAPER A 2 5 (i) Find the expansion of 12 4 x x − − in ascending powers of x, up to and including the term in 2x . [3] (ii) By substituting 3 4x= into the expansion in part (i), find an approximation for 13 in the form a b , where a and b are integers to be determined. [2] 6 The diagram below shows the cross section of a cooling tower of height 125 m. The radii of its circular base and its circular top are 50 m and 37.5 m respectively. The narrowest part of the tower occurs at a height of 80 m above the base of the tower. Take the origin to be at the center of the base, and 1 m to be 1 unit on both axes. The walls of the tower is part of the curve C with equation ( ) 22 22 1ykx ab −−= . (i) Find the values of a2, b2 and k. [3] For the rest of the question, use 80, 900 and 3600 as the values of k, a2 and b2 respectively. (ii) Sketch C, indicating the equations of any asymptotes and coordinates of the points where C crosses the axes, where appropriate. [3] (iii) Given that r is a positive constant and C intersects the curve with equation ( ) 222 80xy r+− = at exactly two distinct points, state