2024 JC2 H2 Math prelim - SAJC

1 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION HIGHER 2 Candidate Name CLASS 2 3 H2 MATHEMATICS 9758/01 Paper 1 26 AUGUST 2024 (Monday) Candidates answer on the Question Paper. 3h Additional Materials: MF 26 N umber of pieces of additional writing paper : _____________________ (N.A. if none) Q 1 2 3 4 5 6 7 8 9 10 11 TOTAL M 4 4 7 5 8 8 11 12 12 13 16 100 READ THESE INSTRUCTIONS FIRST W rite your name, civics group, index number and calculator models on the cover page. 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. A nswer all the questions. Total marks : 100 W rite your answers in the spaces provided in the question paper. G ive 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. The use of an approved graphing calculator is expected, where appropriate. U nsupported 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. Y ou are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 26 printed pages and 2 blank pages including this page.

2 [Turn Over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 1 Show that 2 24xx++ is always positive for all real values of x. Hence solve the inequality 2 2 24 032 xx xx ++ <+− . [4] 2 A function is defined as f ( ) 3 sin cosθ θθ= + . (i) Show that f ( ) 3 sin cosθ θθ= + can be written in the form sin( )R θα+ where exact values of R and α are to be found. [1] (ii) Hence, state a sequence of transformations that will transform the graph with equation siny θ= on to the graph with equation 3sin cos 5y θθ= ++ . [3] 3 (i) Show that 2121 1 ( 1) ! ! ( 1) ! ( 1) ! rr r rr r −−−+ =− ++ , where r +∈  . [1] (ii) Hence find an expression for 2 2 1 ( 1) ! n r rr r= −− +∑ . [4] (iii) Show that 2 2 1 ( 1) !r rr r ∞ = −− +∑ is convergent. [2] 4 A curve has equation 2335 2x xy y− += . Find d d y x in terms of x and y. Hence, deduce the number of tangent(s) to the curve that is/are parallel to the y-axis. [5] 5 (a) Using the substitution exu= , find d2 eexx x−−∫ . [3] (b) (i) Find cos sin 3 dx xx∫ . [2] (ii) Without the use of