EJC_H2_2022_Prelim_P1

EJC 2022 JC2 Prelim/9758/01 1 1 On the same axes, sketch the graphs of y x a=− and y x b=− , where a and b are constants such that 0 ab . You should show clearly the axial intercepts of both graphs. Hence solve the inequality x a x b−− . [5] 2 (a) Without using a calculator, solve the inequality 2 117 5 5 14 x xx+− −− . [3] (b) Hence solve the inequality 17 5 11 5 14 x xx − − +− . [3] 3 The non-zero vectors a, b and c, where a, b are non-parallel vectors, satisfy the equation 35 = c b a c . (a) Determine, with clear reasons, the relationship between the vectors c and 53+ab . [4] Referred to the origin O, it is given further that the vectors a, b and c are the position vectors of the points A, B and C respectively. Point D lies on the line segment AB such that it divides AB in the ratio :1 − , where 01  . (b) Write down an expression for d, the position vector of point D in terms of a and b. [1] The point D also lies on the line segment OC. (c) Determine the exact value of  . [3] 4 (a) The function f is defined by ( ) 2 f : 3 1xx +− for , 3xx − . Find ( ) 1f x− , stating the domain of 1f − . [3] (b) The function g is defined by ( ) 2 1 for 0 1,g 1 for 1 2. xxx xx −  = −   and ( ) ( )g g 2xx=+ for all values of x. Sketch the graph of g for 13 x−   . Hence state the range of g. [3] (c) The domain of g is now restricted to 11 x−  . (i) Explain why the composite function 1fg− exists. [1] (ii) Find ( ) 1fg x− in the form ( ) ( ) 1 p for 1 0,f g( ) q for 0 1. xxx xx −  −  =  where ( )p x and ( )q x are expressions in terms of x to be determined. [3]

EJC 2022 JC2 Prelim/9758/01 2 5 The curve C has equation 2 26 23 xy xx −= +− . (a) State the equations of the asymptotes of C. [2] (b) Without using a calculator, find the range of values that y can take. [4] (c) Sketch the graph of C, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s). [4] (d) Describe one transformation that will transform the curve C onto the curve 2 28 4 xy x −= − . [1] 6 A curve C has parametric equations 23 cos 6xt =+  , 2sinyt= , where 0 t  . (a) Find the exact Cartesian equation of l, the normal to C at the point ( )0, 3P . [4] (b) Let Q be a point on C such that the x-coordinate of Q is the minimum x-coordinate of C. The tangent to C at Q intersects l at point R. Find the exact coordinates of R. [3] (c) Without using a graphing calculator, show that C has only one stationary point, and determine the nature of this stationary point. [4] 7 The line 1l has equation ( )11 3 2 = − − + − +r i j k i j k , where  is a parameter. The point A has coordinates ( )2,0, 1− . (a) The plane p contains the line 1l and the point A. Find a cartesian equation of the plane p. [3] (b) Find the position vector of the point A’, the reflection of the point A