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 rate of r % every year. Sam has been offered a job for the same occupation at both Company P and Q. Find the value of r, correct to 2 decimal places, such that the total annual wages earned by Sam if he works for Company P for 15 years is equivalent to the total annual wages earned if he works for Company Q for 12 years. [4] 4. (AJC16/Promo/Q11) A publisher tracks the sales of a new book. It is found that 3n copies of the book are sold in the first week, where n is a positive integer ( 3n  ). In week 2 he sells 13n copies more than in week 1, and in week 3 he sells   1233nn  copies more than in week 2. (i) By finding and simplifying the number of copies sold in each week from week 1 to week 3 in the form  3na , show that the numbers of copies sold in each of the first three weeks form a geometric progression. [3] In fact, starting from week 3, the number of copies sold in each week forms an arithmetic progression with common difference 48 . (ii) Given that the number of copies sold falls to zero in week K, show that 333 nK  . [3] (iii) Hence find the total number of copies sold from week 1 up to and including week K, expressing your answer in terms of n. [3] (iv) The publisher printed 20,000 copies of the book. What is the greatest value of n such that the books will not be out of stock? [3] Hwa Chong Institution (College)

4 | P a g e Hwa Chong Institution 5. (SRJC16/Promo/Q10) In a simple pendulum experiment, an iron bob is suspended by a string from a fixed support. The iron bob is then released from the position A as seen in the diagram, and the angle  made from its swings are recorded until the iron bob comes to a rest at position O. A swing is defined as the complete movement of the iron bob from one side of O to the other. For instance, the first swing would be from right to left, and the second swing from left to right and so on. When the iron bob swings for the first time, the angle  recorded is 80 . In each o f the subsequent swings, the angle  recorded is 3 4 times that of the previous swing. (i) Find the angle recorded at the 7th swing. [2] (ii) At the nth swing, the angle recorded first falls below 3% that of the first swing. Find the sum of the angles made by the iron bob after n swings. [4] The experiment is repeated with the iron bob released from the position B instead, as seen in the diagram. This time, the angle recorded for its first swing i s 120 . In each of the subsequent swings, the angle recorded is 3 less than that of the previous swing and the experiment is stopped after 40 swings. (iii) Find the angle recorded at the 19th swing. [2] (iv) Given that the sum of the angles made by the iron bob in the first n swings is less than 1550 , find the greatest possible value of n. [3] Fixed support String B O Hwa Chong Institution (College)

5 | P a g e Hwa Chong Institution 6. (ACJC18/Promo/Q13) The Koch snowflake is one of the earliest example of a fractal, a pattern that produces a picture which contains an infinite number of copies of itself. The snowflake is constructed by starting with an initial equilateral triangle, then recursively adding layers of equilateral triangles as follows: (I) divide each line segment into three segments of equal length, (II) an equilateral triangle is drawn with the middle segment from step (I) as its base and pointing outward, as shown below. The diagrams below show how the first, second and third layers of triangles are added to the initial triangle. + = middle third segment of line first layer of triangles Hwa Chong Institution (College)

6 | P a g e Hwa Chong Institution After a third layer of triangles is added, the snowflake is shown in the following diagram. The Koch snowflake is formed when this process of adding layers of triangles repeats over and over again infinitely. (i) Given that the initial triangle has side of length x cm, by first writing down the area of each triangle in the first and second layers, deduce the area of each triangle in the rth layer. [3] (ii) By first writing down the number o f triangles in the first, second and third layers, deduce the number of triangles in the rth layer. [2] (iii) Show that the total area of triangles in the rth layer is 23 3 4 16 9 r x    [2] (iv) Hence find the total area of the Koch snowf lake, explaining why it is a finite value. [4] + = second layer of triangles Hwa Chong Institution (College)

7 | P a g e Hwa Chong Institution 7. (DHS18/Promo/Q10) To celebrate SG55 in the Year 2020, an athlete has two proposals for a training programme to complete a 55-km run around Singapore. Proposal (1): To run 800 m on day 1, 960 m on day 2 , 1152 m on day 3, and on each successive day, to