SOTA 2023 Prelim MAA HL Paper 3

Yr6/5/MATAA/HP3/Aug2023 4 pages © School of the Arts, Singapore Year 6 Mathematics: analysis and approaches Higher Level Paper 3 Preliminary Examinations Monday 28 August 2023 1 hour Instructions to candidates • Do not open this examination paper until instructed to do so. • A graphic display calculator is required for this paper. • Answer all questions on the answer sheets provided. Write your name and class on each answer sheet, and attach them to this examination paper. • Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. • A clean copy of the mathematics: analysis and approaches formula booklet is required for this paper. • The maximum mark for this examination paper is [55 marks]. Question Marks 1 2 Name: ____________________________________ Class: ____________________________________ Index: ____________________________________ 55

Yr6/5/MATAA/HP3/Aug2023 - 2 - Answer all questions on the answer sheets provided. Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. 1. [Maximum mark: 27] This question asks you to work with parametric equations. These are types of equations that employs an independent variable called a parameter (here denoted by ) and in which dependent variables,and, are defined as continuous functions of the parameter and are not dependent on another existing variable. Figure 1 shows the section ABCD of a mound at a skateboard park. Figure 1 The section from B to C is part of the curve OBCE with parametric equations. , for and where is a constant. (a) Find in terms of, (i) the length of the straight-line OE. (ii) the maximum height of the mound. [4] (b) (i) Find and . (ii) Hence find in terms of . [4] The straight-line sections AB and CD are inclined at 30° to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the axis. BF is parallel to the axis. (c) (i) Show that at the point B the parameter satisfies the equation (ii) Find the value of that is a solution of this equation in degrees. (iii) Show that . (iv) Hence, find OF in terms of . [5] (This question continues on the following page) qxy ( )sinxa qq=- ( )1c o sya q=- 02qp££0a>aad d x q d d y q d d y x qx-y-( )1sin 1 cos 3 qq=- q1.5BF a=a Diagram not to scale.

Yr6/5/MATAA/HP3/Aug2023 Turn over - 3 - (Question 1 continued) (d) Find BC and AF in terms of . [3] (e) Given that the straight-line distance AD is 20 metres, calculate the value of . [3] Figure 2 Another part of the skateboard park is shown in Figure 2. The cross-section MN is 25 metres, where MP and QN are horizontal and not equal in length. The section from P to Q is part of the curve with parametric equations. , for The length of the dip PTQ can be calculated by the length of an arc formulae. where and are the values at P and Q. (f) Find the length of the dip PTQ. [5] (g) Find the length of the track MPTQN from start to finish. [3] aa <latexit sha1_base64="5RnTi4195+CRcD5HbhkxwgKZs+Y=">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</latexit> 250 P Q T M N ( )4s i nx qq=- ( )41 c o s 2y q=- 3 22 ppq££ 22 dd ddd b PQ a xyL qqq æö æö=+ç÷ ç÷èø èøò abq Diagram not to scale.

Yr6/5/MATAA/HP3/Aug2023 - 4 - 2. [Maximum mark: 28] This question asks you to explore the integral , where . For , let . (a) (i) For , using integration by parts, show that . [4] (ii) Deduce that . [3] (b) (i) For , using integration by parts, show that . [4] (ii) Using (a)(ii) and (b)(i), deduce that . [3] (c) Prove by induction that , . [9] (d) By using the substitution , show that . [5] ()101dnnxx x-ò0n³0n³()101dnnnIx x x=-ò1n³()()1111001d 1 dnnnnxx x x x x---= -òò()111021 dnnnxx x I---=ò1n³()11101d1nnnnIx x xn+-=-+ò()122 1nnnIIn-=+()()2!21 !nnIn=+n+Î2sinxq=128Ip=