EJC_H2_2021_Prelim_P1

EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2021 General Certificate of Education Advanced Level Higher 2 1 The Jiuzhang Suanshu (Nine Chapters on the Mathematical Art) is a Chinese mathematical manuscript written somewhere in the middle of the 3rd century. It demonstrated that the Chinese had an understanding of negative numbers way before the Europeans did. The following problem is translated and modified from the original text found in the Jiuzhang Suanshu: “Sell 2 cows and 5 sheep to buy 13 pigs: there is a surplus of 1000 coins. Sell 3 cows and 3 pigs to buy 9 sheep: there are exactly enough coins. Sell 6 sheep and 8 pigs, then buy 5 cows: there is a deficit of 600 coins. Tell me: what is the surplus or deficit from buying 3 cows and selling 4 pigs and 3 sheep?” Solve this problem, giving your answer in context. [4] 2 Using the substitution 2cosx k = , find the exact value of 1 2 0 d1 k kx xkx−   in terms of k, where k is a positive constant. [5] . 3 (i) Let k be a positive constant. Sketch the curve with equation 2 1 kxy x += − , stating the equations of the asymptotes. On the same diagram, sketch the line with equation 2yx=− + . [3] (ii) Hence, solve, in terms of k, the inequality 2 21 kx xx + − +− . [2] (iii) Solve, in terms of k, e2 e21e x x x k + − +− . [2]

2 4 (a) The graph of ( )fyx= is given below. It has one vertical asymptote at 1x= and one horizontal asymptote at 0y= . The graph cuts the x-axis at 3x=− and has turning points at ( )2,2− and the origin. Sketch the graph of ( )f'yx= , stating the equations of any asymptotes, and indicating any axial intercepts. [3] (b) It is given that 3g( ) 3x x x p=−+ , where p is an unknown constant. (i) Given that the graph of 1 g( )y x= has a vertical asymptote at 2x= , find the value of p. [1] (ii) Given instead that the graph of 1 g( )y x= has a minimum point at 1 5y= , find the value of p. [3] 5 (a) (i) It is given that 22d .d y xy xyx + = + Using the substitution 22w yx= + , show that the differential equation can be transformed to ( )d fd w wx = , where the function ( )f w is to be found. [2] (ii) Hence, given that 4y= when 3x= , solve the differential equation 22d .d y xy xyx + = + [3] (b) Solve the differential equation 2 2 d1 d y xx = , where 0x . [4] y x O 0y=

3 6 It is given that ln( 1) 1 tanyx+ = − . (i) Show that 2d ( 1)secd y yxx =− + . [1] (ii) By further differentiation of the above result, find the Maclaurin series for y, up to and including the term in 2x . [4] (iii) Verify the correctness of your result in part (ii) by using small angle approximation and standard series from the List of Formulae (MF26). [3] (iv) Use your series from part (ii) to estimate 0 1 tan 0.1 e d , x x− − correct to 5 decimal places. [2] 7 (i) Show that 2 41 3r r + − can be expressed as 2 ( 2) 33rr Ar B r + +− , where A and B are constants to be determined.