RVHS 9758 2024 Promo

RIVER VALLEY HIGH SCHOOL 2024 JC1 Promotional Examination Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Additional Materials: List of Formulae (MF27) Cover Page Answer Papers 9758/01 26 September 2024 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number in the space at the top of this page. Write your name and class on the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for poor presentation in your answers. At the end of the examination, place the cover page on top of your answer paper and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 9758 / 01 / 2024 1 A curve C has equation 2 abyc xx= + + , where a, b and c are constants. It is given that C passes through the point with coordinates 112, 2 −− and has a turning point at 1x= . The gradient of C is 3 2− at the point where 2x= . Find the values of a, b and c and state the equation of C. [5] 2 The diagram below shows the graph of ( )fyx= . The curve crosses the x-axis at the origin and at the point ( )6, 0 . The curve also has a minimum point at ( )2, 8− . The equations of the asymptotes of the curve are 1x=− and 2y= . On separate diagrams, sketch the following graphs, indicating clearly the coordinates of any points of intersections with both axes and any turning point(s), and the equations of any asymptotes where possible. (i) f (3 2)yx=+ [2] (ii) 1 f ( )y x= [3] 3 (i) Solve the inequality 2 9 15x −− exactly. [3] (ii) Hence solve 2 9 1e5x −− exactly. [2] y x O 1x=− 2y= (6,0) (2, 8)−

3 9758 / 01 / 2024 4 It is given that ( ) 232 1 1 14 n r r n n = =+ . (a) Find ( ) 2 3 1 n rn r = + where 1n . Express your answer in the form ( ) ( ) 22 21 1 2 1n a n na + + − where a is a constant to be determined. [4] [4] (b) Find 3 3 3 3 31 3 5 ... (2 3) (2 1)kk+ + + + − + − where k is a positive integer. [2] 5 The curve C has equation 22 20x axy xb ++= + , where a and b are constants. The equations of the asymptotes of C are 4x= and 23yx=− . (i) Find the values of a and b. [3] (ii) Sketch C, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points. [3] (iii) Hence deduce the number of real roots of the equation ( ) 322 20x ax x b+ + =− + , where xb− . [2] 6 (a) Find ( ) 4 P d1 x xx+ where (i) ( ) 3P xx= , [2] (ii) ( )P xx= . [2] (b) Find 12tan dx x x−  . [4] 7 (a) The function f is defined by 2f( ) 6 2, 1x x x x= − − − . Define 1f− in a similar form, stating its domain. [4] (b) The function g is defined by g( ) , , axx x axa= − where a is a non -zero real number. (i) Find 2g ( )x , giving your answer in simplified form. [2] (ii) Without finding an expression for 1g ( )x− , show that 1g( ) g ( )xx −= . [1] (iii) Find the range of values of a such that 1fg− exists. [2]

4 9758 / 01 / 2024 8 The curve C has cartesian equation 21 1yx x=− . (i) Sketch the graph of C for 1x . Label the coordinates of the axial intercept and equation of the asymptote. [2] (ii) Find the equation of the normal to C at the point 12, 2 P  , leaving your answer in exact form. [3] (iii) Using the substitution secx = , find the exact value of 2 2 1 1 1 dxxx − . [3] (iv) Hence, find the exact value of the area of the region bounded by C, the normal to C at P, and the x-axis. [2] 9 A town developed a waste management system in the year 2014 to treat waste generated. The waste generated can either be sent to the landfill, or incinerated at the incineration plant. (a) In the landfill, 1250 tonnes of waste is deposited in the year 2014. Due to the increase in activit ies in the town, experts predict that the amount of waste deposited in the landfill will increase by 12 tonnes in each subsequent year. The landfill is projected to run out of space at the end of the year 2090. Determine the theoretical maximum amount of waste that the landfill can contain. [2] (b) In the incineration plant, 1500 tonnes of waste is incinerated in the year 2014. As the town aims to reduce the amount of carbon emissions from incinerating waste, the amount of waste incinerated will decrease by 2% in each subsequent year. (i) Determine the least number of years from the start of the year 2014 in which the total amount of waste incinerated exceeds 50 000 tonnes. [3] (ii) In its lifetime, the incineration plant can handle 80 000 tonnes of waste. Explain, with working, whether the incineration plant can cope with the total amount of waste to be incinerated in the future. [2] (c) Determine the number of years from the start of the year 2014 in which the amount of waste deposited in the landfill first exceeds the amount of waste incinerated in the incineration plant in that particular year. [3]

5 9758 / 01 / 2024 10 (a) It is given that ( )ln 2 sin 2yx=+ . (i) Show that 22 2 dde 4sin 2 0dd y yy xxx  + + =   . [2] (ii) By further differentiation of the result in part (a)(i), find the Maclaurin series for ( )ln 2 sin 2yx=+ , up to and including the term in 3x . [3] (iii) Hence find the Maclaurin series for 2cos2 2 sin 2 x x+ , up to and including the term in 2x . [2] (b) Given further that x is small such that 3x and higher powers of x can be neglected, use appropriate standard series from the List of Formulae (MF2 7) to verify the correctness of the series of 2cos2 2 sin 2 x x+ obtained in part (a)(iii). [3] 11 The curve C has parametric equations