RVHS 9758 2025 Prelim P1

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 RIVER VALLEY HIGH SCHOOL 2025 JC2 Preliminary Examination Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Additional Materials: Printed Answer Booklet List of Formulae (MF27) 9758/01 17 Sep 2025 3 hours READ THESE INSTRUCTIONS FIRST This document consists of 6 printed pages and 2 blank pages. Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100.

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 1 The function f is defined by ( ) 32f x ax bx cx d= + + + where a, b, c and d are real numbers. Given that 4i+ and 1− are roots of ( )f0 x = , find b, c and d in terms of a. [4] 2 Without using a calculator, solve the inequality 2 95 .11 x xx +−+ [3] Hence solve 2 9 e 5 .1 e e 1 x xx +−+ [3] 3 Do not use a calculator in answering this question. Find the roots of the equation ( ) 2 1 2i 1 7i 0zz− + + + = , giving your answers in the cartesian form iab+ . [6] 4 It is given that 1 11 1( 1) 1 n r r r n= =−++ . (a) Find ( )( )1 1 12 N r rr= ++ in terms of N. [3] (b) It is given that 11 118 ( 1) ( 1) k r k r r r r r  = + = =++ . Find the value of k. [3] 5 (a) Find 4 d 16 x x x− . [2] (b) Find 2 1sin d24 xx x−   . [5]

3 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 6 For any vectors m and n, explain why ( ) 0.  =m m n [1] With respect to the origin O, the points A, B and C have position vectors a, b and c respectively. O, A, B and C are non-coplanar. The point M is the mid-point of AC and p denotes the plane OAB. (a) The point R is such that MR is perpendicular to p. Show that R lies on a line with equation ( ) ( ),,k = + +  r a c a b where k is a constant to be determined. [2] (b) Given that a and b are unit vectors perpendicular to each other, 2 =−ac and 4=bc , find in terms of a and b, the position vector of the point of intersection between and p. [3] 7 A curve C has equation 22 4x y y xy+ − = , where 0x . (a) Show that ( ) d2 4 2 d yy x y x x− − = − . [2] The diagram below shows the curve C. M is a point on C with coordinates ( ),xy and N is a fixed point ( )3,0 . The area of triangle OMN is denoted by A. (b) Find A in terms of y. [1] (c) Show that d 3 2 d 2 2 4 A y x x y x  −=  −− . [2] (d) Hence find the exact value of x for which A is a maximum. (You do not need to show that A is a maximum for the value of x found.) [2] y x O

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 8 It is given that ( )2 d1 9 3 d yxy x+= , and the curve ( )fyx= passes through the y-axis at ( ) 20,e . (a) Show that ( ) ( ) 2 2 2 dd1 9 18 3 0dd yyxx xx+ + − = . [2] (b) Find the Maclaurin series for y , up to and including the term in 3x , giving the exact coefficients for each term. [4] (c) Given that 1ln 2 tan 3yx −=+ , find the Maclaurin series for 1tan 3e x− , up to and including the term in 3x . [2] 9 A curve C has equation 2 2 bay ax b x −= + + + , where a and b are real constants such that 0a  , 1 2ab and 2x− . (a) Given that C has stationary points, use differentiation to find the relationship between a and b. [3] It is now given that 1a = and 3b= . (b) Prove algebraically that y cannot lie between –1 and 3. [3] (c) Sketch C, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points. [2]

5 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 10 The function f is defined by ( ) 2 1f : , , 2. 2 x x x x  − + The domain of f is further restricted to be x > a, where a is an integer. (a) State the least value of a such that the function 1f− exists. [1] (b) Hence find ( ) 1f x− and state its domain. [3] (c) Sketch the graphs of ( ) ( ) 1f and fy x y x −== on the same diagram, giving the equations of any asymptotes and the coordinates of the points where the curves meet the axes. [2] The function g is defined on the domain  )1,− . The graph of ( )gyx= has minimum point at (2, a) and the equation of its asymptote is y = b as shown below. (d) Determine if gf exist. If it exists, find its range. [3] 11 The curve C has cartesian equation 242yx=− . (a) Sketch the curve C, labelling the exact coordinates of any axial intercepts. [2] (b) Find the equation of the tangent to C at the point ( )1, 2P , leaving your answer in exact form. [2] (c) Using the substitution 2 sinx = , find the exact value of 2 2 1 4 2 dxx− . [4] (d) Hence, find the exact value of the area of the region bounded by C, the tangent to C at P, and the x-axis. [2] (−1, 2) (2, a) y = b ( )gyx=

6 ©RIVER VALLEY HIGH SCHOOL 9758/01/2025 END OF PAPER 12 The curve C is defined by the parametric equations 65xt=− and 221yt=+ , where 5 6t  . (a) Sketch the curve C, labelling the exact coordinates of any axial intercept(s). [1] (b) Find the equation of the normal to C at the point P where 1t = . [2] (c) Find the acute angle between the normal to C at the point P where 1t = and the tangent to C at the point M where 3t = . [2] (d) The curve C is translated 5 units in the positive x-direction and translated 3 units in the negative y-direction, to form the curve D. Find the equation of D in parametric form. [2] The curve E is defined by the parametric equations 7xu= and 9y u= , where 0u  . (e) Show that at the point of intersection of the curves C and E, 326 5 3 34 0t t t− + − = . Deduce that there is only one point of intersection and find the coordinates of this point. [4] 13 A cafe roasts its own coffee beans and packages them to be sold. The amount of roasted coffee beans remaining in the cafe at time t days is denoted by x kg. The cafe produces roasted coffee beans at a fixed rate of 10 kg/day, and sells the roasted coffee beans at a rate