RI C8A Graphing Techniques Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 1 of 6 Additional Practice Questions for Chapter 8A: Graphing Techniques 1 PJC Prelim 9758/2018/01/Q1 It is given that 32f( ) 5 7 2xxx k x  , where k is a constant. (i) If the graph of f( )yx is strictly increasing, find the range of values of k. [3] (ii) If k = 0, state the range of values of x where the graph of f( )yx is concave downward. [1] 2 NJC Promo 9758/2020/Q4 A curve has equation ln e xyy . (i) Show that  22 22 2 dd edd xyy yyyxx    . [2] (ii) Explain why the curve is concave upwards for all real values of x. [2] 3 Show that 2d 4 sin 8sin 2cos 4cos 3d2 2 2 xx xxxx     . (Hint: cos 2 = 2cos2 1) Hence, by completing the square, show that 4 sin 8sin 2xxx increases as x increases for all real values of x. Deduce that 4 sin 8sin 2xxx > 0 for x  0. [5] 4 RJC JC1 CT 9740/2006/01/Q8 Sketch a continuous curve f( )yx with the following properties, labelling the coordinates where possible. x y Behaviour x < 2 Strictly decreasing, Concave upwards 2 1 Stationary point 2 < x < 4 Strictl y increasing, Concave upwards 4 4 Point of inflexion 4 < x < 6 Strictl y increasing, Concave downwards 6 7 Stationar y point x > 6 Strictl y decreasing, Concave downwards [3]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 2 of 6 Rational Functions 5 JPJC Promo 9758/2020/Q4 The curve C has equation 2x ay x b  , for non-zero constants a and b. The graph of C passes through the point with coordinates  0,1 . The line 6x is an asymptote to the graph of C. (i) Determine the values of a and b. [2] (ii) Sketch the graph of C, clearly stating the equations of the asymptotes and the coordinates of the axial intercepts. [3] (iii) By adding a suitable graph to your sketch in (ii), deduce the number of real roots for 3220xa xx b   . [3] 6 RVHS JC1 Promo 9758/2019/Q8 The curve C has equation 2 6 2 xa xy x  , where a is a constant. It is given that the line 5yx is an asymptote of C. (i) Show that a = 3. [2] (ii) Prove algebraically that y cannot lie between 1 and 15. [2] (iii) Sketch the curve C, stating clearly the coordinates of any points of intersection with the axes, the coordinates of any stationary points and the equations of asymptotes. [3] (iv) Hence, solve the inequality 2 36 32xx xx   . [2] 7 VJC JC1 Promo 9758/2019/Q10 A curve C has equation 2 1,2xxy x   where  is a real, non-zero constant. Show that if C has 2 stationary points, then 0  or k  , where k is a constant to be determined. [4] Sketch the curve C for 1  , giving the equations of asymptotes , the coordinates of stationary points and points of intersection with the axes. [4] By considering C and an appropriate line, find the range of values of m such that the equation  22 12 3 6xx m x m x      has exactly 1 positive root and 1 negative root. [3]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 3 of 6 8 SRJC Prelims 9740/2009/01/Q7 The curve C has equation 2x by x a  , where 0a and 0b . (i) State the coordinates of the intersection(s) of the curve C with the axes in terms of a and b . [1] (ii) Find the equation(s) of the asymptote(s). [2] (iii) Draw a sketch of the curve C, labeling the equation(s) of its asymptote(s) and coordinates of any intersection with the axes. [2] (iv) Hence find the range of values of k , where k is a positive constant, for which the equation   2x bx a k x a   has no real root. [2] 9 NYJC Promo 9740/2006/01/Q11 The curve C has equation given by 2ax bx cy x d  . It has a vertical asymptote with equation x = 2 and its oblique asymptote cuts the x-axis at x = 2. (i) State the value of d and show that b = 4a. [3] (ii) Find the condition for a and c such that the gradient of C is always positive. [3] (iii) If C passes through the origin, find the value of c. [1] (iv) Given that a = 2 and using the values of b, c and d found in (i) and (iii), sketch the graph of C. [3] (v) Find the equation of the normal line to C at the origin, and determine the coordinates of the point where this normal line cuts C again. [4] Conic Sections 10 VJC JC1 Promo 9758/2019/Q4 The curve C has equation   22 23 1.94yx  (i) Sketch ,C giving the equations of its asymptotes and the coordinates of any turning points. [4] The curve D has equation 2212 48 48 6 3 0y y ax ax a   , where a is a positive constant. (ii) Find the set of values of a for which C and D do not intersect. [2]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 4 of 6 11 TMJC Promo 9758/2020/Q4 The curve C has equation 22 2y x k  where 2.k  It is given that C has only one stationary point. (i) Find d .dyx Hence find the coordinates of th e stationary point in terms of .k [4] (ii) Sketch the graph of ,C stating the equations of any asymptotes, coordinates of any points of intersection with th e axes and the coordinates of the stationary point. [3] (iii) By adding a suitable graph on the same diagram in part (ii), find the range of values of ,h in terms of ,k where h is a positive real constant and hk such that   2 22 22 2khx kh xk  has at least one real root. [3] 12 VJC Prelim 9740/2013/ 02/Q2(modified) The curve C has equation 2 1 ax bx cy x  , where a, b and c are non-zero constants. (a) It is given that C passes through the point 233, 2 and has a minimum point at 2, 10 . (i) Find the values of a, b and c. [3] (ii) Sketch C, giving the coordinates of any turn ing points, points of intersection with the axes and the equations of any asymptotes. [3] (iii) Find the range of values of m and k, such that the curve with equation 22 22 1xykm does not intersect C. [2] (b) In the case when 1a and 1b , find the set of values of c, where 2c , such that C has no stationary point. [3]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 5 of 6 Parametric Equations 13 Sketch, on separate diagrams, the curves de fined by the following pair of parametric equations: (a) 33cos , sin for 0xa ya      where a is a positive constant. (b) 24 3 4, 3 8x ty t t   for t; (c) 2 111, f o r , 0xy tt t tt     Find the equation of the curves in cartesian form