RI C8A Graphing Techniques Add Prac (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 1 of 23 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] 1(i) 32f( ) 5 7 2xxx k x  '2f( ) 1 5 1 4xxx k  f is strictly increasing, so we consider 'f( )x > 0 for all x  215 14xx k > 0 for all x  Discriminant < 0 ,   2 14 4(15) 0 k  49 15k  or k <3.27 (ii) 320f ( ) 5 7 2kx x x    For graph of f( )yx to be concave downward, ''f() 3 0 1 4 0 7 or 0.46715 xx xx       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] 2 (i) ln e xyy d d d ed x y y x xy

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 2 of 23  22 2 2 22 222 22 22 22 22 2 dd d dd ed dd d edd d dd e ( s h o w n )dd x x x yyyy xx xy yy yyy yxx x yy yyyxx       (ii)  22 22 2 dd edd xyy yyyx x   Since ln y is defined, 0y , and e0x  for all x, 2 2 2 22 ded d 0d x yyy xxy y   for x. The curve is concave upwards for x. 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] 3 2 2 dLHS 4 sin 8sind2 14 cos 8( ) cos22 14 (2cos 1) 8( ) cos 22 2 2cos 4cos 3 RHS (Shown)22 xxxx xx xx xx          22 2 2 2cos 4cos 3 2(cos 2cos ) 322 22 2(cos 2cos 1 1) 322 2(cos 1) 12 xx xx xx x           22(cos 1) 1 1 02x x    , i.e. the gradient of 4 sin 8sin 2xxx is always positive x , 4 sin 8sin 2xxx  increases as x increases for all real values of x. (Shown)

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 3 of 23 When 0x , 4s i n 8 s i n 0 0 0 02xxx     And 4 sin 8sin 2xxx increases as x increases for all real values of x 4s i n 8 s i n 2xxx  > 0 for x  0. 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 Strictl y decreasing, Concave upwards 2 1 Stationar y point 2 < x < 4 Strictl y increasing, Concave upwards 4 4 Point of inflexion 4 < x < 6 Strictly increasing, Concave downwards 6 7 Stationary point x > 6 Strictly decreasing, Concave downwards [3] 2 (4, 4) (2, 1) (6, 7) 6 y

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 4 of 23 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] (i) 22 2xa a by xb xb  Since 6x is an asymptote, 6b  Graph passes through 0,1 1 6 6 a a    (ii) (iii) 3220xa xx b    2 2xx a x b   2 21xa x bx   Since the graphs of 2xay xb  and 21y x intersect only once, there is 1 real root for 3220xa xx b   .

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 5 of 23 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] (i) 2 2 6 5 , for some 22 3 ( 10) 2 xa x kyx k xx xx k x        2 6xa x = 2 3 ( 10)xx k By comparing coefficients of x, a = 3 (shown) Alternatively, 2 6 2 21 0( 2) (performing long division)2 xa xy x axa x       Hence, 25 3aa  (shown) (ii) 2 2 2 36 2 23 6 (3 ) (2 6) 0 xxy x xy y x x xy x y       For this quadratic equation to have real roots, discriminant ≥ 0 i.e. 2(3 ) 4(2 6) 0yy   2 14 15 0 (1 ) (1 5 ) 0 1 or 15 yy yy yy         y cannot lie between 1 and 15.

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 6 of 23 (iiii) Graph of C: 2 36 1 6 522xxyx xx   Coordinate of (2 ,1 )A  and (6,15)B (iv) 2 36 32xx xx   Hence, sketch additionally the graph of 3yx  : The two graphs intersect at A ( 2x ) and at 0x . Solution to 2 36 32xx xx   is 20 o r 2xx  3yx  x y B (0,–3) A O O x y B –3 A

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8A: Graphing Techniques Page 7 of 23 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     