SAJC_9758_2022_Prelim_P1

1 [Turn Over 1 The equation of a curve C is given by ( ) ( )224 20x y x y+ + − = . (i) Show that the gradient of C at the point ( ),xy is given by d 5 3 d 3 5 y x y x x y +=− + . [3] (ii) Find the equation(s) of the tangent(s) to the curve C which are perpendicular to the line yx= . [6] 2 The curves 1C and 2C are defined by the equations 2 6 4y x= − and 26yx=− respectively. (i) On the same axes, sketch the graphs of the curves 1C and 2C , stating the equations of any asymptotes , the exact coordinates of the turning point(s) and any points where the curve crosses the x- and y-axes. [5] (ii) Solve the inequality 2 2 6 64 xx −− . [2] (iii) The transformations A and B are given as follows: A : Reflection about the y-axis; B : Translation of 4 units in the negative x-direction. The graphs 1C and 2C undergo in sequence, the transformations A and B. The resulting equations of the transformed graphs of 1C and 2C are ( )fyx= and ( )gyx= respectively. Deduce the solution set of the inequality ( ) ( )fg xx . [2]

2 [Turn Over 3 The function f is defined by 32f: x ax bx cx d+ + + , where x and a, b, c and d are constants. The graph of f intersects the y-axis at 3y=− and passes through the points ( )01,− and ( )2 ,0 . (i) Explain why f does not have an inverse. [1] (ii) Given also that the tangent to the graph of f at 1x= is a horizontal line, find f(x). [3] (iii) Sketch the graph of y = f(x), giving the coordinates of the turning points and the points which the graph intersects the axes. [2] (iv) Given that the function f has an inverse if its domain is restricted to xk , state the smallest possible value of k. [1] For the rest of the question, use the domain given and value of k found in part (iv). (v) Describe the relationship between the graphs of f ( )yx= and 1f ( )yx −= . [1] (vi) Show that the solution of the equation 1f ( ) f ( )xx −= satisfies the equation 33 11 6 0xx− − = . Hence, find the solution of the equation 1f ( ) f ( )xx −= . [3] (vii) It is g iven that g( ) ln( 5)xx=+ , where 5.x− A student attempts to find the composite function gf. The student’s solution is shown below: Comment on the validity of the student’s solution. [1]  ) ( ) gf 2 f 3 g( ) ln( 5) D D , gf ( ) l . ,n 5 , xx k x ax bx cx d x x k =+ ==  = + + +  +

3 4 (i) By sketching the graph of 1 21 xy x += − , find the range of values of x for which 1 021 x x + − . [4] (ii) Hence, without the use of a calculator, show that 0 2 1 3 3d ln 3 ln 52 1 2 4 x xx− + =−− . [4]