RVHS H2 Math Curves Sketching Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Curve Sketching 1 Curve Sketching. 1. IJC/II/1 The curve C has equation 2 9 16 2 xxy x ++= + . (i) Find the equations of the asymptotes of C. [2] (ii) Find, leaving your answers in exact form, the range of values of k for which the equation 2 9 16 2 xxk x ++= + has no real roots. [4] (iii) Sketch C. Show, on your diagram, the equations of the asymptotes, the coordinates of the stationary points, and the coordinates of the points of intersection of C with the x- and y- axes. [3] 2. IJC/2015/I/7 The curve 1C has equation 11 2y a x x=+ − , where a is a positive constant. (i) By using differentiation, or otherwise, find the coordinates of the stationary point and determine the nature of the stationary point. [4] (ii) Sketch 1C , indicating clearly the equations of any asymptotes and the coordinates of the stationary point. [2] Another curve 2C has equation ( ) 2 2 22 1xa y ab − += , where b is a positive constant. (iii) By considering the graph of 2C , find an inequality satisfied by b such that 1C and 2C intersect at more than one point. [2] (i) Given that 2a = and 6b = , find the range of values of x such that 2 2 1 1 ( ) 12 xaba x x a   −+  − −   . [2] 3. NYJC/I/7 The curve C has equation 224 3( 1) xy x  += ++ . It is given that C has a vertical asymptote 0x = . (i) Determine the value of . [1] (ii) Find the equation of the other asymptote of C. [2] (iii) Prove, using an algebraic method, that C cannot lie between 44 33 aa y−   , 0a  , where a is to be determined. [3] (iv) When 0 = , the graph of 224 3( 1) xyk x  +=+ ++ cuts the x-axis at 2 distinct points where k is a positive integer. (a) Sketch the graph of 224 3( 1) xyk x  +=+ ++ when 0, = indicating clearly all asymptotes and coordinates of the axial intercepts. [4] (b) Deduce the least possible integer of k. [1]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Curve Sketching 2 4. PJC/I/12 A curve C has the equation 2ax bx cy xd ++= + for some constants a, b, c and d. Given that the asymptotes are 5x = and 7yx=+ , find the values of a, b and d. [3] Given that the curve has a stationary point at 1x =− , find the value of c. [2] Draw a sketch of C, indicating clearly the asymptotes, the coordinate(s) of the stationary points and the point(s) of intersection with the axes. [3] By drawing a sketch of another suitable curve on the same diagram, state, with a reason, the number of roots of the equation ( ) ( ) 21 2 5 1 0xx− + + + = (i) when 1 = , (ii) when 01  . [3] 5. AJC/2011/I/8 The curve C has the equation 2 where axya xa= + is a negative constant. (i) Show that the curve C has two stationary points for all negative values of a. [2] (ii) Sketch the curve C, showing clearly all the asymptotes, axial intercepts and turning points. [3] (iii) Using the sketch in (ii), find the range of values of k for which ( )( ) 242 1x k x x= − − has exactly two roots. [3] 6. EJC Prelim/2020/02/Q2 The curve 1C has equation 23 2 3 1 xxy x −+= − . The curve 2C has equation ( ) ( ) 2 2 2 411 yx b −− − = , where 0b  . (i) Sketch 1C , stating the equations of any asymptotes and the coordinates of any turning points and points where the curve crosses the axes. [4] (ii) Hence, find the range of values of b such that there is no point of intersection between and 2C . Using the maximum value of b found, sketch 2C on the same diagram as part (i). [3] 1C

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Curve Sketching 3 7. NJC/2012/I/10a The curve C has equation , , where a is a positive constant. (i) Find the equations of the asymptotes, leaving your answers in terms of a. [2] (ii) Show that if C has two turning points, then . [3] (iii) For , sketch the curve C, stating the equations of any asymptotes, the coordinates of the turning points and axial intercept(s) if any. [2] (iv) Hence, find the range of values of h such that the graph of , where h is a positive integer, intersects C more than once. [2] 8. MJC/2013/I/Q8 The curve C1 has equation 23 28 xy xx=+ −− . (i) Express y in the form 3 AB x c x d++ ++ and show by differentiation that C1 has no stationary points. [3] (ii) Sketch C1, stating the axial intercepts and the equations of any asymptotes. [3] (iii) Find the exact area bounded by the curve C1, the line 2x = and the axes. [3] (iv) The curve C2 has equation ( ) 2 2 22 5 132 x y− += . Sketch C2 on the same diagram as C1 and find the coordinates of any points of intersection between C1 and C2. 9.VJC/2013/II/2 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 turning points, points of intersection with the axes and the equations of any asymptotes. [3] (iii) Find the set of values of m k , where m and k are positive constants, such that the curve with equation ( ) 22 22 1 1yx km −−= 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] 2 2 2 22 24x a x ay xa −+= − xa 06 a 4a = ( ) ( ) 2 222 6 2 5 1h x y h− + + + =

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Curve Sketching 4 10. PJC/2013/II/5 The curve C has equation 2x ax by cx ++= − . The vertical asymptote of C is 2x =− , and the coordinates of the turning points are ( )4, 2− and ( )0, 6− . (i) Find the values of a, b and c. [3] (ii) Sketch C, stating the equations of the asymptotes. [2] (iii) By drawing an appropriate graph on the sketch of C, find the range of values of k ( )0k  such that the equation ( ) 222 24 x ax bxk cx  +++ + =  − has no real roots. [2] (iv) When C undergoes a translation in the direction of y-axis by p units, C intersects the line 1y =− at one negative value of x. State the set of values for p. [2] 11. PJC/2014/1/10 The curve C has equation 2 px qy xr += + , where p, q and r are constants. It is given that C has an asymptote at 2x = and a turning point at 14, 4   . (i) Find the values of p, q and r. [4] (ii) Sketch C, stating clearly equations of asymptotes, and the coordinates of turning points and intersections with the axes. [3] (iii) Hence, solve the inequality 2 1px q xr + + . [3] 12. SAJC/2014/I/6 The equation of a curve C is y = 2x + 8k 2 x - 2 , where k > 0. (i) Prove, using an algebraic method, that y cannot lie between two values (to be determined in terms of k). [3] (ii) Sketch C for the case where k = 1, stating the equations of any asymptotes and the coordinates of any turning points. [3] (iii) State, with a reason, the range of values of b, where b > 0, su