ACJC 2025 JC1 H2 Math Promo QP

ANGLO-CHINESE JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION Higher 2 MATHEMATICS 9758/01 Paper 1 3 October 2025 QUESTION PAPER 3 hours 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 . 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. _________________________________________________________________________________ This document consists of 8 printed pages and 2 blank page. [Turn over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 1 Given that cos sin 2xxyx = , find the exact value of d d y x at x = . [5] 2 It is given that 11 tan 2yx −=+ where 11 22 x−   . Show that 22 2 2 2 d d 8 0d d (1 4 ) y y xy x x x + + = + . Hence find the Maclaurin series for y, up to and including the term in 3x . [5] 3 Given that ( )1 1 11 n r n r r n= =++ . (a) Show that ( )2 11 11 n r r r n= =−− . [2] (b) Find ( )2 11 21r r rr  =  + −  . [3] 4 (a) The diagram shows the graphs of y a bx=− and 6y mx=− where a, b and m are positive constants. The graph of y a bx=− meets the x-axis at the point 9 ,04   . The solution set of the inequality 6mx a bx−  − is given as { : , 1 or 3}x x x x   . Find the values a, b and m. [4] (b) Given that is a constant andk 1k  , solve the inequality 2 2 1 1,( 1) kx x k x k + −+ + + giving your answer in terms of k. [4] y x y a bx=− 6y mx=−

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 5 A curve 1C has parametric equation 23sec , 2 tan , 0 . 2xy   = =   (a) Sketch the curve 1C , stating the coordinates of the endpoint(s). [1] (b) Show that d1 d 3tan y x = . [2] 1L and 2L are tangents to curve 1C . (c) Find (i) the equation of 1L which is parallel to the y-axis. [2] (ii) the gradient of 2L which is tangent to the curve 1C at the point ( )3.75,1 . [2] (d)  is the acute angle between 1L and 2L . State the value of tan  . [1] (e) Describe a transformation that will map 1C onto the curve 2C with equation 23sec , 2 tan( ), 0 . 2xy    = = −   [1] [Turn over

4 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 6 (a) The figure below shows the sketch of the graph f ( )yx= . The graph has asymptotes at 1x= , 3y= and the x-axis. The graph has stationary point at ( 1, 3)−− and cuts the axes at (0,0) and (2,0) . Sketch, on separate diagrams, the following graphs, indicating clearly any asymptotes, axial intercepts and stationary points, where possible. (i) ( )2f 1yx=− + [3] (ii) 1 f ( )y x= [3] (iii) ( )f1yx=+ [3] (b) Describe fully a sequence of transformations that transform the graph of 2yx= onto the graph of 24 4 1y x x= − + . [2] y x

5 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 7 Referred to the origin O, points A, B and C have position vectors a, b and c respectively. It is given that +−a b c and −+a b c are parallel. (a) Using vector product, show that OA → and BC → are parallel. [4] It is now given that a and b are unit vectors, 2BC OA →→ = and angle 60OBC= . (b) Using scalar product, show that OB → and OC → are perpendicular. [3] (c) The point H has position vector h. Given that h b c= and 3c = , find the exact volume of the pyramid HOBC. [3] [The volume of a pyramid is 1 base area height3 .] 8 The function f is defined by f : , for , ,axx x a x axa  −  − where a is a positive constant. (a) Sketch the curve f ( )yx= , stating the equation of the asymptote and the coordinates of the endpoint. Explain, stating the reason clearly, why f has an inverse. [3] (b) Find 1f ( )x− and state its domain. [3] (c) Find the range of values of x in terms of a, for which 1f ( ) f ( )xx −= . [1] (d) For positive integers k, find 21 1 2f k a+ − in terms of a. [2] (e) Given that fg( ) for , x x a x x a= +  − , find g( )x stating its domain. [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 [Turn over 9(a) In the diagram, point Q is 400 m vertically above point A. Initially, a balloon is at point Q and moves horizontally in a straight line at a constant speed of 10 m per second away from an observer stationed at point A. At time t seconds, after the start, the balloon is at point P where QP is x m and the angle of elevation of the balloon from the observer at A is  radians. Find exactly the rate of change of  at the instant when x is 100 m. [4] 9(b) A gardener designs a flower bed ABC in the shape of an isosceles triangle inscribed in a circular plot of land with radius 6 meters, as shown in the diagram. The vertices of the triangle lie on the circumference of the circle and AB = AC. The angle BAC is  , where  is acute. Express the area of the flower bed ABC in terms of  . [2] As  varies, use differentiation to find the exact maximum possible area of the flower bed , and show that it is a maximum. [4] P A Q x 400 m A B C 6 m

7 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 10 The plane 1p has equation 0x y z− + = and the plane 2p has equation 2x y z+ + = . (a) Find the acute angle between planes 1p and 2p . [2] (b) Find the vector equation of the line of intersection of planes 1p and 2p . [1] With reference to the origin O, the points A and B are such that 2OA → =− + −i j k and OB → =+ij . The line l has equation ( )2=− + − + −r i j k i k , where  is a parameter. (c) Find the shortest distance between line l and the line with cartesian equation 3 2, 1x z y− =− + = . [2] (d) The point C lies on l such that angle 90ABC= . Find the position vector of point C. [3] (e) Find the position vector of the foot of perpendicular from point A to 2p . [3] (f) A plane 3p is equidistant to both point A and plane 2p . Find the cartesian equation of plane 3p . [2] [Turn over

8 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 11 Caleb is training for a vertical marathon, by running up the stairs of a building with 62 floors. He decides to train three days a week starting with 10 floors on his first day of training. On each subsequent training day, he runs 2 floors more than the previous training day. (a) Find the number of floors he completes on his 10th training day. [2] Once he can complete 62 floors in a training day, he is considered as ready for the vertical marathon. Then for subsequent training days, he will continue to complete 62 floors each day. (b) He is ready for the vertical marathon at the end of k weeks of his training. Find k. [2] (c) Find the total number of floors he complete s at the end of 20 weeks of his training. [3] After training, Caleb’s heart rate decre