ACJC 9758 2024 Promo Qns

ANGLO-CHINESE JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION Higher 2 MATHEMATICS 9758/01 Paper 1 27 September 2024 QUESTION PAPER 3 hr Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Fo llow the instructions on the front cover of the answer booklet. Give non-exact numerical answers corr ect 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. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. [Turn over

2 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 1 Solve exactly the inequality 32 4 12 x xx   . [3] Hence solve 32 4 12 x xx   . [2] 2 It is given that 1t a nyx . (i) Show that  22d21 1d yyy x   and  22 2 2 dd d 21dd d yy yyy yxx x     . [3] (ii) Hence find the first three terms in the Maclaurin expansion of y. [2] 3 The diagram below shows the graph of the curve fyx , with turning point at ,22 k  and asymptotes with equations xk and 0y . The curve undergoes the following sequ ence of transformations in succession A: Translate k units in the negative x-direction B: Reflection about the y-axis C: Translate k units in the positive x-direction (i) Sketch the curve after the transformations, indicating clearly the equations of the asymptotes and the coordinates of the turning point. [2] (ii) Find the equation of the curve after the transformations in the form fya x b where a and b are constants to be determined, in terms of k where appropriate. [1] (iii) Another curve gyx is such that   gg 4xx for all x. State the equation of a line of symmetry of this curve. [1] y x O

3 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 [Turn over 4 The diagram shows the graph of fyx  , which passes through the origin and has turning points at 2, 5 and 6, 0 . The graph of fyx  has a vertical asymptote 1x and a horizontal asymptote 5y . (i) Sketch the graph of fyx  , stating the equation of asymptote(s) and the coordinates of any intersection(s) with the axes where possible. [3] (ii) Write down the x-coordinate(s) of the stationary point(s) of fyx and determine their nature. [2] (iii) If the tangent to the graph of fyx at 2x  passes through the origin, find the equation of the normal to the graph of fyx at 2x  . [2] 5 The functions f and g are defined as follows ln 3 2 2f: , 3 xxx x   2g: 3 1 2 1 3 ,x xxx   (i) Sketch the graph of fyx , indicating the equations of asymptotes, coordinates of intersections with the axes and turning points, if any. [3] (ii) Show that fg exists and find its range. [2] The function h is defined as h: g ,xx x k  such that 1h exists. (iii) State the largest value of k. For this value of k, find 1h in a similar form. [4] y x O

4 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 6 Relative to the origin O, points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel vectors. The points M is the midpoint of OA and the point N lies on OB such that :ON NB is 1:2 . The point C with position vector c is the point of intersection of lines AN and BM. (i) Show that 21 55ca b . [3] (ii) Give the geometrical meaning of ac , where c is the unit vector in the direction of c. [1] The points O, A and C lies on a circle where OA is the diameter of the circle. It is further given that 21 2ab a . (iii) Find the ratio of :ac . [3] (iv) Hence find the area of triangle OAC, giving your answer in terms of a . [2] 7 The parametric equations of a curve C are π2c o s , s i n ,6xa t y at    where 02 πt and a is a positive constant. A sketch of curve C is shown below, with C intersecting the positive x-axis at P and the negative y-axis at Q. (i) Show that the normal to a point on the curve C with parameter  has gradient 13 t a n  . [3] (ii) Show that 4π 3t  at the point Q. [1] (iii) Hence, find the equation of the normal to the curve C at Q. [2] The normal to the curve C at Q intersects the x-axis at T. (iv) Show that the ratio OT OP is independent of a and find its value exactly. [3] y x O P Q

5 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 [Turn over 8 (a) It is given that 2 1 1 41 2 1 n r n rn  . (i) Explain why the series 2 1 1 41r r    converges and write down the value to which it converges. [2] (ii) Find 6 1 21 ( 23 ) N r rr  in terms of N, express your answer in a single fraction. [3] (b) The sequence 123,,,uuu  is defined by 1 2u  , 1 11, 1n n un u    . Find the value of 2u , 3u and 4u . Hence find the value of 50 1 r r u   . [4] 9 As part of a game, a person stands x m away from a wall and tries to throw an object through a 0.40 m wide hole in the wall. The point B denotes the bottom of the hole, which is located 3.05 m above the ground. The object is thrown from point P at a height of 1.75 m above the ground. The angle APB is denoted by  in radians. (i) Show that 11 1.7 1.3tan tan xx      . [2] (ii) Using differentiation, find the value of x which gives the stationary value of  , giving your answer to four significant figures. [3] (iii) Show that the value of x in (ii) corresponds to a maximum value of  , and suggest why knowing this value of x may be useful in this situation. [2] (iv) The person walks away from the wall at a cons tant rate of 0.1 m/s. Find the rate of change of  at the instant when the person is 1 m away from the wall. [2] 0.40 m 3.05 m x m 1.75 m A B P 

6 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 10 The curve C has equation 2 22 2 ax ax ay x    , ,0aa . (i) Find d d y x . [2] (ii) Find the range of values of a for which C has no stationary points. [3] For 1a  , (iii) sketch the curve C, indicating the equations of asymptotes, coordinates of intersections with the axes and turning points, if any. [3] (iv) By sketching a suitable curve on the diagram in (iii), state the number of positive and negative roots of the equation   32 22xx x k x  where 0k  . [2] 11 The line 1l has the vector equation 0 11 0 a b        r ,   , where a and b are real constants. The plane 1p has cartesian equation 244xyz . The line 1l is perpendicular to the line 2l with cartesian equation 1 32 y z  , 5x . The line 1l intersects the plane 1p at the point with coordinates  ,1 ,0a . (i) By first finding the vector equation of line 2l , show that 2ab . [3] (ii) Verify that the point A with coordinates  2, 2, 2 lies on 1l . [1] (iii) Find the coordinates of the foot of perpendicular, F, from point A to the plane 1p . Hence find the shortest distance from point A to the plane 1p . [5] (iv) The plane 2p contains the line 1l and the point F. Find the vector equation of the plane 2p , giving your answer in the form drn , where n is a vector and d is a real constant to be determined. [2]

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