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1. Maximum mark: 31] This question asks you to explore the behaviour and some key features of the function ()-xa-x, where aR and Z. In parts (a) and (b), only consider the case where a-2. Consider ()-(2-3). () Sketch the graph of y-C), stating the values of any axes intercepts and the coordinates of any local maximum or minimum points Consider (0)-2-x), where 2.8>1. (b) Use your graphic display calculator to explore the graph of y-fr) for the odd values -3 and a-3; . the even values -2 and Hence, copy and complete the following table Number of local maximum points Number of local minimum points -3 and 3 -2 and and Now consider /0)-(-x) where aeR and acZ.*>1. (c) Show that 00-m (a-2xia-x (d) State the three solutions to the equation 0-0. (e) Show that the point horizontal axis (This question continues on the following page) (Question 1 continued) Hence, or otherwise, show that >0. for = 2 (a) By using the result from part (f) and considering the sign of -1). show that the point (0, 0) on the graph of y-() is a local minimum point for even values of , where >1 and R (1) (4) a point of intexion with zero gradient for odd values of a, where Consider the graph of y-xa-a-k, where acZ, ac R and ke R >1 and ack. P (h) State the conditions on a and à such that the equiton aa-x-& has four solutions for . (5) 16) Number of points of Inflexion with zero gradient 153 121 131 2221-2013 (2) on the graph of y(x) is always above the EE
2 Ma mark 24 This question ask you to investigats and prove a geometric property involving the roots of the equation where :C for integers , where 22. where ze Ci The roots of b quation - root can be represented by a point whese Each ctively on an Argand diagram For example, the roots of the equation are 1 and Onan And represented dagan the point P Coneider The noche Bagan poet P. P, and (0) Show that (-1011-1 Hence deduce that 1-0 (This question continues on the following page) Question continued) Line segments and are added to the Augand c on the following Argand diagram 40-52 nghof (Pwd the gh (+* Show that PP-PP-3 -1 where point P, and the root can be represented by and on the flowing Argand dus I unt with cenie O.0) B& 2301-110 and are shown PI
2221-7113 (Question 2 continued) On the following Argand diagram, the points P. P. P. and P. lie on a circle of radius 1 unit with centre 0(0, 0). [P.P.). [P.P.] and [Pare fine segments. Im Re (d) Show that PP, x P.P.x P.P. - 4. [4] For the case where =5, the equation == I where : C has roots I, e, er, co' and es. It can be shown that PP, PP, x P.P.P.P.-5. Now consider the case for integer values of #, where # 22. The roots of the equation=1 where e Care 1,0, these roots can be represented by the points P. P.P. [P.P.]. [P.P.]. with centre 0(0,0). e. On an Argand diagram, respectively where P [P.P.] are tine segments. The roots lie on a circle of radius tunit (e) Suggest a value for P.P. x P.P. x_xP_P__ PP, can be expressed as 11-el. (1) (1) Write down expressions for P,P, and PP, in terms of e (i) Hence, write down an expression for P.P., in terms of " and e. Consider :-1-(2-1) +++:+1) where : e C. (9) (1) Express: +2 ++ +:+1 as a product of linear factors over the set C. Hence, using the part (g)) and part (f) results, or otherwise, prove your suggested result to part (e). (1) References: ਭਾਈ (3) [4]