2021 Telematics Gr 12

Directorate: Curriculum FET TELEMATIC SCHOOLS PROJECT 2021 MATHEMATICS Grade 12

Telematics Mathematics Grade 12 Learner Material 2 March to October 2021 Dear Grade 12 Learner In 2021 there will be 10 live Telematics sessions, 5 of these sessions are in English and 5 in Afrikaans. Only one of the five sessions is indicated for grade 12. I would however like to recommend that you attend all the Telematics sessions as you will benefit from all of them. This is why on the next page the dates, times and topics for all sessions are included. This workbook contain material for: 1. Grade 12 session on number patterns (17 March) 2. Grade 11 session on the quadratic number pattern (20 April) 3. Grade 11 session on Financial Mathematics (27 July) Please note that you will be assessed on all the grade 11 content in grade 12. Please email, school@sun.ac.za with your name, surname, email and school name to register to obtain a username and password. You will then be able to access all previously recorded Telematics sessions and materials on other Topics from the Telematics Schools Project website, https://schools.sun.ac.za/ . Please note that accessing these recordings from this website is zero-rated. At school you should receive a book called “Grade 12 Tips for Success”. In it you will have a breakdown of the weighting of the various Topics in Mathematics. Ensure that you download a QR reader, this will enable you the scan the various QR codes. The “MATHEMATICS EXAMINATION GUIDELINE”, can be downloaded by scanning the adjacent QR code or by clicking on, http://bit.ly/EXAM_GUIDE . This examination guideline document contains important information regarding your end of year examination paper as well the acceptable abbreviations to Geometry theorems which you are to use when answering Geometry questions. At the start of each Telematics lesson, the presenters will provide you with a summary of the important concepts and together with you will work though the exercises. You are encouraged to come prepared, have a pen and enough paper (ideally a hard cover exercise book) and your scientific calculator with you. You are also encouraged to participate fully in each lesson by asking questions and working out the exercises, and where you are asked to do so, sms or e-mail your answers to the studio. Remember:” Success is not an event, it is the result of regular and consistent hard work”. GOODLUCK, Wishing you all the success you deserve! Copyright reserved

Telematics Mathematics Grade 12 Learner Material 3 March to October 2021 2021 Mathematics Telematics Program Day Date Time Grade Subject Topic Term 1: 13 Jan - 26 March Wednesday 03 March 15:00 – 16:00 10 Mathematics Algebra Thursday 04 March 15:00 – 16:00 10 Wiskunde Algebra Wednesday 17 March 15:00 – 16:00 12 Mathematics Number Patterns Thursday 18 March 15:00 – 16:00 12 Wiskunde Getalpatrone Term 2: 13 April – 25 June Tuesday 20 April 15:00 – 16:00 Quadratic 11 Mathematics Number Patterns Wednesday 21 April 15:00 – 16:00 Kwadratiese 11 Wiskunde Getalpatrone Term 3: 20 July – 01 October Tuesday 27 July 15:00 – 16:00 11 Mathematics Finance Wednesday 28 July 15:00 – 16:00 11 Wiskunde Finansies Term 4: 12 October - 8 December Wednesday 20 October 16:00 – 17:00 11 Mathematics Exam Preparation Thursday 21 October 16:00 – 17:00 Eksamen 11 Wiskunde Voorbereiding Copyright reserved

Telematics Mathematics Grade 12 Learner Material 4 March to October 2021 NUMBER PATTERNS, SEQUENCES AND SERIES NUMBER PATTERNS: Number Patterns are sequence of numbers that follow a particular rule. This rule can be determined in terms of the term number(n) where 𝑇! is the n’th term of the sequence/number pattern. In earlier grades you have encountered various number patterns. In grade 11 you have dealt with the quadratic number pattern. The quadratic number pattern is one having a constant second common difference. In grade 12 the focus is on: 1. Arithmetic and the Geometric number patterns or sequences 2. Arithmetic and Geometric series 3. Sigma Notation. The quadratic number pattens will also be assessed in grade 12. The derivation of the formula to the sum of an Arithmetic Sequence as well as the sum to a Geometric Sequence must be known as it can be assessed. Arithmetic Sequence (Linear Pattern): Geometric Sequence (Exponential Pattern) Arithmetic Sequence is sequence of numbers where A geometric Sequence is a sequence of numbers there is a constant common first difference(d). where the is a constant common ratio(r). Consider the sequence, Consider the sequence, 𝑎 ; 𝑎 + 𝑑 ; 𝑎 + 2𝑑 ; … … … 𝑎 ; 𝑎𝑟 ; 𝑎𝑟 " ; … … … ∴ 𝑑 = 𝑇" − 𝑇# = 𝑇$ − 𝑇" 𝑇" 𝑇$ ∴ 𝑟= = 𝑇# 𝑇" And 𝑇! = 𝑎 + (𝑛 − 1)𝑑 And 𝑇! = 𝑎𝑟 !%# Arithmetic Series Geometric Series Arithmetic Series is the sum of the terms of an Geometric Series is the sum of the terms of a Arithmetic Sequence. Geometric Sequence. 𝑎 + ( 𝑎 + 𝑑) + ( 𝑎 + 2𝑑) + ⋯ … … 𝑎 + (𝑎𝑟) + ( 𝑎𝑟 " ) + … … … The sum of an arithmetic sequence can be The sum of a Geometric sequence can be calculated calculated using the formula: using the formula: 𝑛 &(( ! %#) &(#%( ! ) 𝑆! = (2𝑎 + (𝑛 − 1)𝑑) 𝑆! = Or 𝑆! = and 𝑟 ≠ 1 2 (%# #%( Convergent Geometric Series A geometric series will converge if −1 < 𝑟 < 1. The sum to infinity can be calculated of a Geometric Sequence/Series if −1 < 𝑟 < 1. 𝑎 𝑆* = 1−𝑟 Other Sequence or Series Sigma Notation ( ∑ ) Other patterns can be created by combining, Sigma notation is a very useful and notation for the 1. An Arithmetic with a Geometric Pattern sum of a given number of terms of a sequence. 2. Arithmetic or Geometric with a constant pattern ∑ is the symbol used for.’sum of ’ 3. Arithmetic or Geometric Pattern with some other pattern + Example: : 3𝑘 2 2 2 ,-$ ; ; ; … … .. 1. this is read as the sum of all the terms from the 1×2 2×3 3×4 3rd term to the 5th term of the sequence. 2. It can be expanded as, + : 3𝑘 = 3(3) + 3(4) + 3(5) = 9 + 12 + 15 ,-$ = 36 Copyright reserved

Telematics Mathematics Grade 12 Learner Material 5 March to October 2021 Exercises: 1. Given the sequence 6; 13; 20; … a) Determine a formula for the nth term of the sequence. b) Calculate the 21st term of this sequence. c) Determine which term of this sequence is 97. 2. Consider this number pattern: 8; 5; 2; … a) Calculate the 15th term. b) Determine which term of this sequence is –289. 3. a) Given the arithmetic sequence 1 − p; 2p − 3; p + 5; . . . determine the value of p. b) Determine the values of the first three terms of the sequence. 4. a) Determine the 10th term of the sequence: 3; 6; 12; . . . b) Determine the number of terms in the sequence: 2; 4; 8; . . .; 1024 c) If 5; x; 45 are the first three terms of a geometric sequence, determine the value of x. d) Determine the geometric sequence whose 8th term is 9 and whose 10th term is 25. 5. Determine the sum of the series: 19 + 22 + 25 + . . . + 121 6. The sum of the series 22 + 28 + 34 + . . . is 1870. Determine the number of terms. 7. Given the arithmetic sequence -3; 1; 5; …,393 a) Determine a formula for the nth term of the sequence. b) Write down the 4th, 5th, 6th and 7th terms of the sequence. c) Write down the remainders when each of the first seven terms of the sequence is divided by 3. d) Calculate the sum of the terms in the arithmetic sequence that are divisible by 3. ! 8. The sum of n terms is given by 𝑆! = (1 + 𝑛). Determine 𝑇+ . " 9. 3𝑥 + 1; 2𝑥; 3𝑥 − 7 are the first three terms of an arithmetic sequence. Calculate the value of 𝑥. 10. The first and second terms of an arithmetic sequence are 10 and 6 respectively. a) Calculate the 11th term of the sequence. b) The sum of the first n terms of this sequence is –560. Calculate n. 11. Determine 3 + 6 + 12 + 24 + . . . to 10 terms. 12. If 2 + 6 + 18 + . . . = 728, determine the value of n. 13. Evaluate: ./ :(2𝑛 − 4) !-0 Copyright reserved

Telematics Mathematics Grade 12 Learner Material 6 March to October 2021 14. What is the value of m for which: 1 : 5(3),%# = 65 ,-# 15. Consider the sequence: 1 1 1 ; 4 ; ; 7 ; ; 10 ; … … 2 4 8 a) If the pattern continues in the same way, write down the next Two terms in the sequence. b) Calculate the sum of the first 50 terms of the sequence. 16. Calculate 𝑆* if * : 8(4)#%2 2-# 17. Given the series: 3(2𝑥 − 3)" + 3(2𝑥 − 2)$ + 3 (2𝑥 − 3)0 + ⋯. for which vakues of 𝑥 will the series converge? 18. Find the value of 𝑚 if: 1 : 3(2),%# = 93 ,-# 19. For which values of 𝑥 will * :(4𝑥 − 1), ,-# exist. Questions from National Examination Papers: QUESTION 1 DBE/November 2019 1.1 Given the quadratic sequence: 321 ; 290 ; 261 ; 234 ; …. 1.1.1 Write down the values of the next TWO terms of the sequence. (2) 1.1.2 Determine the general term of the sequence in the form Tn  an 2  bn  c . (4) 1.1.3 Which term(s) of the sequence will have a value of 74? (4) 1.1.4 Which term in the sequence has the least value? (2) . . . 5 5 5 1.2 Given the geometric series:    K 8 16 32 1.2.1 Determine the value of K if the series has 21 terms. (3) 5 1.2.2 Determine the largest value of n for which Tn  (4) 8192 [19] Copyright reserved

Telematics Mathematics Grade 12 Learner Material 7 March to October 2021 QUESTION 2 DBE/November 2019 10 10 1 1 2.1 Without using a calculator, determine the value of:  y 3 y  2   y 3 y  1 (3) 2.2 A steel pavilion at a sports ground comprises of a series of 12 steps, of which the first 3 are shown in the diagram below. 1 2 Each step is 5 m wide. Each step has a rise of m and has a tread of m, as 3 3 shown in the diagram below. 2 m 3 5m 1 m 3 The open side (shaded on sketch) on each side of the pavilion must be covered with metal sheeting. Calculate the area (in m 2 ) of metal sheeting needed to cover both open sides. (6) [9] QUESTION 3 DBE/November 2018 3.1 Given the quadratic sequence: 2 ; 3 ; 10 ; 23 ; … 3.1.1 Write down the next term of the sequence. (1) 3.1.2 Determine the nth term of the sequence. (4) 3.1.3 Calculate the 20th term of the sequence. (2) 3.2 Given the arithmetic sequence: 35 ; 28 ; 21 ; … Calculate which term of the sequence will have a value of –140. (3) 3.3 For which value of n will the sum of the first n terms of the arithmetic sequence in QUESTION 2.2 be equal to the nth term of the quadratic sequence in QUESTION 2.1? (6) [16] Copyright reserved

Telematics Mathematics Grade 12 Learner Material 8 March to October 2021 DBE/November 2018 QUESTION 4 1 A geometric series has a constant ratio of and a sum to infinity of 6. 2 4.1 Calculate the first term of the series. (2) 4.2 Calculate the 8th term of the series. (2) n Given:  32  1 k 4.3  5,8125 Calculate the value of n. (4) k 1 20 20 If  32  p , write down  242  1 k k 4.4 in terms of p. k 1 k 1 (3) [11] QUESTION5 DBE/November 2017 5.1 Given the following quadratic number pattern: 5 ; -4 ; -19 ; -40 5.1.1 Determine the constant second difference of the sequence. (2) 5.1.2 Determine the n th term (Tn) of the pattern. (4) 5.1.3 Which term of the pattern will be equal to -25 939? (3) 5.2 The first three terms of an arithmetic sequence are 2k - 7 ; k + 8 and 2k -1 . 5.2.l Calculate the value of the 15 th term of the sequence. (5) 5.2.2 Calculate the sum of the first 30 even terms of the sequence. (4) [18) QUESTION 6 A convergent geometric series consisting of only positive terms has first term a, constant ratio r and nth term, Tn , such that f, 1 � Tn = - . n=3 4 6.1 If T, + T2 = 2, write down an expression for a in terms of r. (2) 6.2 Calculate the values of a and r. (6) [8] Copyright reserved

Telematics Mathematics Grade 12 Learner Material 9 March to October 2021 Quadratic Number Patterns (Grade 11: 20 April 2021) Definition: General term: A quadratic number pattern is formed when the second difference of a sequence 𝑇! = 𝑎𝑛" + 𝑏𝑛 + 𝑐 is a constant value. Example 1: 5; 18; 35; 56; 81; 110 13 17 21 25 29 First difference 4 4 4 4 Second difference. As you can see the second difference is constant therefore this is a quadratic sequence. Example 2: Substitute the first five terms into the general term. 𝑇! 𝑇" 𝑇# 𝑇$ 𝑇% 𝑎+𝑏+𝑐 4𝑎 + 2𝑏 + 𝑐 9𝑎 + 3𝑏 + 𝑐 16𝑎 + 4𝑏 + 𝑐 25𝑎 + 5𝑏 + 𝑐 3𝑎 + 𝑏 5𝑎 + 𝑏 7𝑎 + 𝑏 9𝑎 + 𝑏 2𝑎 2𝑎 2𝑎 The substitution of five terms into the quadratic general formula helps us to find a method on how to find 𝑎, 𝑏 and 𝑐 of the general formula. Let’s see how that works: • The second constant difference is equal to 2𝑎. Therefore, with algebraic manipulation, one can "!" 34!56&!6 89::;(;!3; state that 𝑎 = . " • 𝑇" − 𝑇# = 3𝑎 + 𝑏 Sub in 𝑇# , 𝑇" and 𝑎 ∴ 𝑏 = 𝑇" − 𝑇# − 3𝑎 • 𝑇# = 𝑎 + 𝑏 + 𝑐 Sub in 𝑇# , 𝑎 and 𝑏 ∴ 𝑐 = 𝑇# − 𝑎 − 𝑏 Let’s find the general term of the quadratic sequence in example 1: Given: 5; 18; 35; 56; 81; 110 2!8 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 4 𝑎= ∴𝑎= =2 2 2 𝑏 = 𝑇" − 𝑇# − 3𝑎 ∴ 𝑏 = 18 − 5 − 3(2) = 7 𝑐 = 𝑇# − 𝑎 − 𝑏 ∴ 𝑐 = 5 − 2 − 7 = −4 ∴ 𝑇! = 4𝑛" + 7𝑛 − 4 Copyright reserved

Telematics Mathematics Grade 12 Learner Material 10 March to October 2021 Questions from Examination Papers Question 1 (Gr 11/ November 2018) The quadratic number pattern: 4 ; 𝑝 ; 11 ; 𝑞 ; 22 ; …. has a constant second difference of 1. 1.1 Show that 𝑝 = 7 and 𝑞 = 16 1.2 Determine the general term, 𝑇! , of the quadratic pattern. 1.3 Determine the value of 𝑛 if 𝑇! = 232 1.4 If the sum of two consecutive terms in the pattern is 1227, calculate the difference between these two terms. Question 2 (Gr 11/ November 2017) Given the quadratic pattern: 244 ; 193 ; 148 ; 109 ;……. 2.1 Write down the next term of the pattern. 2.2 Determine a formula for the 𝑛6< term of the pattern. 2.3 Which term of the pattern will have a value of 508? 2.4 Between which TWO consecutive terms of the quadratic pattern will the first difference be 453? 2.5 Show that all the terms of the quadratic pattern are positive. Question 3 (Gr 11/ November 2016) Consider the quadratic pattern: – 9; – 6 ; 1 ; 12 ; 𝑥 ; … 3.1 Determine the value of 𝑥. 3.2 Determine a formula for the 𝑛6< term of the pattern. 3.3 A new pattern, 𝑃! , is formed by adding 3 to each term in the given quadratic pattern. Write down the general term of 𝑃! in the form 𝑃! = 𝑎𝑛" + 𝑏𝑛 + 𝑐 . 3.4 Which term of the sequence found in Question 3.3 has a value of 400? Question 4 (Gr 11/ November 2016) 4.1 A quadratic pattern has a constant second difference of 2 and 𝑇+ = 𝑇#. = 29 4.1.1 Does this pattern have a minimum or maximum value? Justify your answer. 4.1.2 Determine an expression for the 𝑛6< term in the form, 𝑇! = 𝑎𝑛" + 𝑏𝑛 + 𝑐 Question 5 (Gr 11/ November 2015) 5.1 Consider the quadratic pattern: 5 ; 12 ; 29 ; 56 ; … 5.1.1 Write down the NEXT TWO terms of the pattern. 5.1.2 Prove that the first differences of this pattern will always be odd. Question 6 (Gr 11/ November 2015) 6.1 Consider the quadratic pattern: 3 ; 5 ; 8 ; 12 ; … Determine the value of T26 . 6.2 A certain quadratic pattern has the following characteristics: • T1 = p • T2 = 18 • T4 = 4T1 • T3 – T2 = 10 Determine the value of p. Copyright reserved

Telematics Mathematics Grade 12 Learner Material 11 March to October 2021 Financial Mathematics (Gr 11: 27 July2021) Financial terms: • Interest is a fee paid for the use of borrowed money, or money earned on money saved. It is calculated as a percentage of the money borrowed or lent. • Simple interest is the interest on an initial (principal) sum of money. Each year you receive or you are charged the same amount of interest. • Compound interest is also interest on a principal amount P. For each year, the previous year’s final amount becomes the new principal amount. So the interest is calculated on the principal and the interest from the previous year. • Decay or depreciation is when a quantity decreases by a percentage of the amount present. For example, your assets (house, car) and machinery lose value through age and use. • A nominal interest rate is the quoted interest rate. • An effective interest rate is the actual interest rate received. If you are quoted a nominal interest rate of 7% p.a., the resulting effective rate will be different depending on if it is worked out annually, monthly or quarterly. Simple interest: 𝐴 = 𝑃(1 + 𝑛𝑖) Compound interest: 𝐴 = 𝑃(1 + 𝑖 )& Simple decay or depreciation (Straight line Compound decay or depreciation (reducing depreciation): balance depreciation): 𝐴 = 𝑃(1 − 𝑛𝑖) 𝐴 = 𝑃(1 − 𝑖)! Where 𝑃 is the principal (original sum of money invested or borrowed) 𝑖 is the interest (rate per period) 𝑛 is the number of years (no of periods) 𝐴 is the final amount Nominal and effective interest rates: This formula is used to either calculate the nominal or effective interest rate, where 𝑘 is the number of times per year interest is calculated. , 9 !#$%!&' 1 + 𝑖 ;::;369=; = Q1 + , R Question 1 (Gr 11/ November 2018) 1.1 Calculate the effective interest rate per annum if an investment earns interests at a rate of 11,5% p.a., compounded monthly. 1.2 Karabo bought a computer for R4 700. The value of the computer depreciated at a rate of 18% p.a. Using the reducing-balance method, calculate the book value of the computer 4 years after it was bought. Copyright reserved

Telematics Mathematics Grade 12 Learner Material 12 March to October 2021 1.3 Nhlanhla made an initial deposit of R20 000 into an investment account that paid interest at the rate of 7,2% p.a., compounded quarterly. After 2 years the interest rate changed to 7,8% p.a., compounded monthly. Four years after his initial deposit, Nhlanhla withdrew R2 500 from his investment. 1.3.1 Calculate how much Nhlanhla had in this investment account 2 years after the initial deposit was made. 1.3.2 How much will the investment be worth 7 years after the initial deposit was made? Question 2 (Gr 11/ November 2017) 2.1 A company bought machinery costing R80 000. Using the reducing balance method, the machinery had a book value of R20 000 after 5 years. Calculate the rate of depreciation. 2.2 Calculate the effective interest rate if interest is compounded at 5% p.a., compounded quarterly. 2.3 Sipho invested R30 000 for 6 years. The investment earned interest at 12% p.a., compounded monthly for the first two years. There after the interest rate changed to 10,8% p.a., compounded semi-annually for the rest of the period. Calculate the value of the investment at the end of 6 years. (No other transactions were made on the account.) 2.4 Mary deposited R25 000 into a savings account with an interest rate of 18% p.a., compounded monthly. Mary withdrew R8 000 from the account 2 years after depositing the initial amount. She deposited another R4 000 into this account 3 ½ years after the initial deposit. What amount will Mary have 5 years after making the initial deposit in this account? Question 3 (Gr 11/ November 2016) 3.1 A machine costs R25 000 in 2016. Calculate the book value of the machine after 4 years if it depreciates at 9% p.a. according to the reducing balance method. 3.2 The nominal interest rate of an investment is 12,35% p.a., compounded monthly. Calculate the effective interest rate. 3.3 The value of a property increased from R145 000 to R221 292,32 over 6 years. Calculate the average annual rate of increase of the property over 6 years. 3.4 Tebogo made an initial deposit of R15 000 into an account that paid interest at 9,6% p.a., compounded quarterly. Six months later she withdrew R5 000 from the account. Two years after the initial deposit she deposited another R3 500 into this account. How much does she have in the account 3 years after her initial deposit? Question 4 (Gr 11/ November 2015) 4.1 A school buys tablets at a total cost of R140 000. If the average rate of inflation is 6,1% per annum over the next 4 years, determine the cost of replacing these tablets in 4 years' time. 4.2 An investment earns interest at a rate of 7% per annum, compounded semi-annually. Calculate the effective annual interest rate on this investment. 4.3 A savings account was opened with an initial deposit of R24 000. Eighteen months later R7 000 was withdrawn from the account. Calculate how much money will be in the savings account at the end of 4 years if the interest rate was 10,5% p.a., compounded monthly. 4.4 A car costing R198 000 has a book value of R102 755,34 after 3 years. If the value of the car depreciates at r% p.a. on a reducing balance, calculate r. Copyright reserved