2023 HCI Promotional Examination Revision Package

2023 Promotional Examination Revision Package S/N Topics Page 1 Sequences and Series 2 2 Graphs and Transformations 16 3 Inequalities and Systems of Linear Equations 28 4 Functions 34 5 Differentiation and Its Applications 42 6 Integration and Its Applications Skill Set 56 Practice Questions 64 7 Vectors Skill Set 80 Practice Questions 97 8 Practice Papers 2021 RI Promotional Examination (Modified) 108 2022 RI Promotional Examination (Modified) 115 2021 HCI Promotional Examination 122 2022 HCI Promotional Examination 128 Hwa Chong Institution (College)

2 | P a g e Hwa Chong Institution 1 Sequences and Series 1. (A Level N84/P1/Q1) (a) The sum, , of the first terms of an arithmetic progression is given by . Given also that = 6 and =11, (i) Find the values of and , [3] (ii) Deduce, or find otherwise, an expression for the th term and the value of the common difference. [3] (b) Find the set of values of lying in the interval such that the sum to infinity of the geometric series is greater than 2. [4] 2. (NJC14/Promo/Q11) (a) The sum of the first n terms of a sequence 1 2 3, , ,...u u u is given by 111 n n cS kk      , where c and k are non-zero real constants, with k > 1. (i) Show that n nu ck  for all positive integers n. [2] (ii) Prove that the sequence  nu is a geometric progression. Hence explain why the sum to infinity of the geometric progression exists, and evaluate it in terms of c and k. [4] (b) A patient is administered a 500 mg dose of Drug A on the first day. Each subsequent day, the dosage of Drug A is reduced by 20 mg. (i) Find the total amount of Drug A (in mg) administered at the end of 2 weeks. [2] Another patient is administered a 500 mg dose of Drug B every 6 hours. At the end of each 6-hour period, 32% of the amount of drug present at the start of the 6-hour period remains. (ii) Find the amount of Drug B (in mg) present immediately after the patient takes the 3rd dose. [2] (iii) An overdose occurs when the amount of Drug B exceeds 735 mg. Calculate the maximum number of doses the patient can take to avoid overdosing. [3] nS n 2 nS pn qn 3S 5S p q n  11 22     21 sin sin ...   Hwa Chong Institution (College)

3 | P a g e Hwa Chong Institution 3. (RI14/Promo/Q6) The annual wage of a certain occupation offered by Company P begins with an initial amount of $9500 and increases by $400 every year for 14 years till it reaches $15100 and remains constant at $15100 thereafter. Let 1 2 3() np n a a a a     , where ia denotes the annual wage in the ith year offered by Company P for i = 1, 2, 3, …, n. (i) Show that 2200 9300 for 15,() 15100 for 15, n n npn n A n     where A is a constant to be determined. [4] (ii) The annual wage of the same occupation offered by Company Q begins with the same initial amount of $9500 but it increases at a fixed rat