2022 J1 RVHS H2 Math EOY Revision Package (QP) Ch1-4

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 1 RIVER VALLEY HIGH SCHOOL 2022 JC 1 H2 Mathematics (9758) End-of-Year Revision Package S/N Topic Page Number Questions to be submitted Dates for release Of Solution (on Google Site) 1 Inequalities & Equations 2 3, 4, 7, 8 1 Dec 2 Sequences & Series 6 2, 3, 5, 9, 10, 16, 18, 19 3 Graphing Techniques 17 3, 5, 6, 10, 12 4 Functions 25 1, 3, 4, 5 5 Differentiation and its applications 31 2, 4, 5, 8, 10, 13 15 Dec 6 Integration and its applications 38 1, 2, 5, 7, 8, 9, 12, 13 7 Differential Equations 46 1, 3, 5, 6 Remark: Please note numerical answers are found at the end of each chapter. Instructions: 1. You are advised to complete the Basic Skills questions and revisit the stated tutorial questions before you attempt the revision package . You may want to visit the RV Skill Builder (in the H2 Maths Googlesite) which curates a series of videos of important mathematical skills. 2. You are required to complete, mark your solution and do corrections before submitting your work to your JC 2 Maths tutors by 13 Jan 2023 (Friday). 3. There will be a lecture test during 2023 Week 2 or 3. The topics to be tested will be made known, after the December Term Break. You are expected to review all topics in preparation for the test.

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 2 1 Inequalities & Equations Basic Skills 1. Find the discriminant of the following quadratic expressions. (a) 2 1xx++ (b) 2 45xx+− (c) 2 1xx− + − (d) 22 4 3xx− − + (e) 2 5 29 x x+− (f) 2 63 xx− + + 2. Find the exact roots of the following quadratic equations. (a) 2 3 2 0xx+ + = (b) 2 2 3 0xx− − + = (c) 2 2 1 02 x x− + − = (d) 221 034xx+ − = (e) 2 1xx− + + = (f) 2 6 3 02 x x+ + = 3. Complete the square of the following quadratic expressions. (a) 2 1xx++ (b) 2 45xx+− (c) 231xx+− (d) 22 4 3xx− − + Tutorial Review Tutorial 1 Questions 4 and 7. Revision Questions 1. NYJC Promo 9758/2020/Q2 (i) The first four terms of a sequence nu are given by 1 32.1u = , 2 17u = , 3 0.7u = and 4 7.8u =− . Given that nu is a cubic polynomial in n, find nu in terms of n. [3] (ii) Find the least value of n for which nu is greater than 555. [2] 2. YIJC Promo 9758/2020/Q1 Four families, namely Chan, Lee, Tan and Wong, purchase masks, hand sanitisers and thermometers at a pharmacy. The Tan family made their purchase using a voucher, which entitled them to a 20% discount off the total amount paid. The quantity of each item purchased and the total amounts paid are shown in the following table. Masks (in boxes) Hand sanitisers Thermometers Total Amount paid Chan 4 6 5 $89.30 Lee 2 4 8 $83.30 Tan 3 5 3 $50.28 Wong 5 2 4 Calculate the total amount the Wong family paid. [4]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 3 3. 2017/Prelim/NJC/P2/Q1 There are 3 bike-sharing companies in the current market. For each ride, - bike charges a certain amount per 5 min block or part thereof, - bike charges a certain amount per 10 min block or part thereof and - bike charges a certain amount per 15 min block or part thereof. Rebecca rode each of the bike-sharing companies’ bikes once in each month. The table below shows the amount of time Rebecca clocked for each ride and her total spending for each month. In celebration of the company’s first anniversary, the pricings in February and March 2017 of - bikes are a 5% discount off the immediate previous month’s pricing. Determine which bike-sharing company offers the cheapest rate (without any discount) for a 40-min ride. Justify your answer clearly. [4] 4. 2016/Promo/RI/3 Do not use a calculator in answering this question. Solve the inequality 2 4 1. 23 x xx +  − + + [4] Deduce the solution of (i) 2 42 4 1, 23 x xx +  − + + [2] (ii) 2 4 1. 23 x xx −  +− [2] 5. 2017/Prelim/HCI/P1/Q3 (i) By first expressing 234xx−− in completed square form, show that 234xx−− is always negative for all real values of x . [2] (ii) Hence, or otherwise, without the use of a calculator, solve the inequality ( )( ) 22 2 3 4 1 025 x x x xx − − − −− , leaving your answer in exact form. [4] January 2017 February 2017 March 2017 - bike 25 min 17 min 36 min - bike 30 min 10 min 39 min - bike 15 min 44 min 33 min Total spending $5.70 $5.72 $9.71

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 4 6. 2019/Promo/ACJC/P1/Q2 Without using a calculator, solve (i) 61 xx+ , [3] (ii) 61 xx+ . [2] 7. 2019/Promo/RI/P1/Q8 (a) Without using a calculator, solve the inequality 2 2 2 134 xx xx − +− . [4] (b) (i) On the same axes, sketch the graphs of 2 ay x=+ and 2 | |yx=− , where a is a constant such that 12 a . [3] (ii) Hence, or otherwise, solve the inequality 2 2 | |a xx+  − . [2] 8. MI Promo 9758/2020/PU1/Q2 (modified (ii)) (i) Without the use of a calculator, solve the inequality 3 231 xx +− . [4] (ii) Hence solve exactly the inequality ( )( ) 3 1 e 1 2e 3xx  −+ . [2] 9. 2021/Prelim/HCI/P1/Q3 (i) Without using a calculator, solve the inequality 2 2 2 3 1 2 1 2 2 xx x x x + + − + . [4] (ii) Using your answer to part (i), deduce the values of x for the inequality 2 2 2cos 3cos 1 2cos cos 1 2cos 2 xx x x x + + − + , where x−   , leaving your answer in exact form. [3]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 5 10. 2021/Prelim/SAJC/P1/Q4 Sketch the graphs of 2 1 21 , anday y x x a a a −= + =− +− on a single diagram, where a is a positive constant and 12 a , showing all asymptotes and axial intercepts clearly. [4] (i) Using the graphs, solve, in terms of a, 2 1 21. a xx a a a −+ − +− [1] (ii) Hence, solve 2 1 21 1 ax x ax ax a −+ − +− . [3] Answers to Chapter 1: Equations and Inequalities Basic Skills 1 (a) −3 (b) 36 (c) −3 (d) 40 (e) 19/9 (f) 217/9 2 (a) −2 or −1 (b) −3 or 1 (c) 2−2 or 2+2 (d) 11( 3 15) or ( 3 15)44− + − − 2(e) 11 or 1 − 2(f) −6 (repeated root) 3(a) 2 2 131 24x x x + + = + + 3(b) 22 4 5 ( 2) 9x x x+ − = + − 3(c) 2 2 1 133 1 3 6 12x x x + − = + − 3(d) ( ) 222 4 3 5 2 1x x x− − + = − + Revision Questions 1. 321.5 9.6 3.2 37nu n n n= − + + ; least value of n 10= 2. $86.65 3. - bike: $3.26, - bike: $3.36, - bike: $3.42 4. 1 or 3 xx−  (i) 3 or 3xx−  (ii) 1 or 3xx − 5. (ii) 1 6 or 1 6 or 1x x x −  + = 6. (i) 2 or 0 3xx−   (ii) 3 3, 0xx−    7. (a) 4 or 1 2 or 2x x x−    (b)(ii) 0ax−   8. (i) 2x− or 31 2x . (ii) 30 ln 2x   9. (i) 1 21 x−   , (ii) 3x −  − or 3 x  10. (i) 0 or 2x a x   , ii) 11 or 0 2xx a  

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2022 End-of-Year Revision Package 6 2 Sequences & Series Basic Skills 1. Simplify the following expressions: (a) 3 1 84 2 xx x−    (b) 41 21 e e x x − − (c) 2 1 e x x−    (d) ( ) ( )ln 1 ln 2xx− + − (e) ( ) ( ) 3ln 1 ln 1xx+ − + (f) ( ) ( ) ( )2ln ln 2 ln 1x x x− − +