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ONE. INDIAN. GIRL. Chetan Bhagat is the author of six bestselling TIME magazine named him as one of the most influ. Siyavula's open Mathematics Grade 12 textbook. Learner's book and teacher's guide (PDF). Learner's book (ePub). Reference material. Maths formulas. Open textbooks offered by Siyavula to anyone wishing to learn maths and science. PDF (CC-BY-ND). Mathematics Grade Read online. Textbooks. English.

Everything Maths Grade 12 Pdf

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Siyavula textbooks: Grade 12 Maths. Collection Editor: PDF generated: October 29, . In mathematics many ideas are related. We saw Because we are making monthly payments, everything needs to be in months. Everything Maths. Mathematics is commonly thought of as being about numbers but mathematics is actually a language! Mathematics is the language that. Just browse to the on-line version of Everything Maths on your mobile phone, tablet or computer. To read it off-line you can download a PDF or e-book version.

Applying the definition Find the value of: We generally use the "common" base, 10, or the natural base, e. The number e is an irrational number between 2. It comes up surprisingly often in Math- ematics, but for now suffice it to say that it is one of the two common bases.

Natural Logarithm The natural logarithm symbol In is widely used in the sciences. The natural logarithm is to the base e which is approximately 2.

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While the notation log 10 a; and log e x may be used, log 10 x is often written log x in Science and log e x is normally written as ln x in both Science and Mathematics. So, if you see the log symbol without a base, it means log It is often necessary or convenient to convert a log from one base to another. An engineer might need an approximate solution to a log in a base for which he does not have a table or calculator function, or it may be algebraically convenient to have two logs in the same base.

Logarithms can be changed from one base to another, by using the change of base formula: Normally a and 6 are known, therefore log,, a is normally a known, if irrational, number. For example, change log 2 12 in base 10 is: Change of Base Change the following to the indicated base: VMgiq at www.

These laws are based on the exponential laws and are summarised first and then explained in detail. Logarithm Law 7: Logarithm Law 2: From the work done up to now, it is also useful to summarise the following facts: Useful to know and re- member 1. Firstly, we need to relate x and y to the base a. Then from Equation 2.

Start with calculating the left hand side: Therefore, log Logarithm Law 3: Logarithm Law 4: Logarithm Law 5: The derivation of this law is identical to the derivation of Logarithm Law 5 and is left as an exercise. Logarithm Law 6: Simplify the following: VMgjI at www. Try to write any quantities as exponents can be written as 5 3.

Step 2: Final Answer We cannot simplify any further. The final answer is: Try to write any quantities as exponents 8 can be written as 2 3.

Determine which laws can be used. We can use: Step 4: Final Answer The final answer is: Now it is much easier to solve these equations by using logarithms. By applying Law 5, you will be able to use your calculator to solve for x.

Example 4: Then the aim is to make the unknown quantity i. For example, the equation is solved by moving all terms with the unknown to one side of the equation and taking all constants to the other side of the equation 2 X. Identify the base with x as an exponent There are two possible bases: Step 3: Apply the log laws to make x the subject of the equation. Substitute into the original equation to check answer.

There are formulae that deal with earthquakes, with sound, and pH-levels to mention a few. To work out time periods is growth or decay, logs are used to solve the particular equation.

Example 6: How long will it take for the city to triple its size? For this example n represents a period of 2 years, therefore the n is halved for this purpose. Exercise 1. If the initial population is 5 , how long will it take for the population to reach ? You can afford to save R per month. How long will it take you to save R20 ?

Olbr 2. Determine the final answer In this case we round up, because 7 years will not yet deliver the required R30 The investment need to stay in the bank for at least 8 years. Chapter 2 End of Chapter Exercises 1.

Show that lo Sa ;. Without using a calculator show that: Simplify, without the use of a calculator: Simplify to a single number, without use of a calculator: Evaluate without using a calculator: Find the value of log 27 3 3 without the use of a calculator.

Simplify By using a calculator: Solve the following equation for x: Olbt 2. Olca is Sequences and Series 3. We also look at series, which are the summing of the terms in sequences. VMgjm at www. Arithmetic Sequence An arithmetic or linear sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term For example, 1; 2; 3; 4; 5; 6;.

Common Difference Find the constant value that is added to get the following sequences and write out the next 5 terms. The general arithmetic sequence looks like: Arithmetic Sequence An arithmetic or linear sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term: A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant: If this condi- tion does not hold, the sequence is not an arithmetic sequence.

Plotting a graph of terms in an arithmetic sequence Plotting a graph of the terms of a sequence sometimes helps in determining the type of se- quence involved. Geometric Sequences A geometric sequence is a sequence of numbers in which each new term except for the first term is calculated by multiplying the previous term by a constant value.

This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example. What is influenza? Influenza commonly called "flu" is caused by the influenza virus, which infects the respiratory tract nose, throat, lungs. It can cause mild to severe illness that most of us get during winter time. The main way that the influenza virus is spread is from person to person in respiratory droplets of coughs and sneezes.

This is called "droplet spread". This can happen when droplets from a cough or sneeze of an infected person are propelled generally, up to a metre through the air and deposited on the mouth or nose of people nearby. It is good practise to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu. Assume that you have the flu virus, and you forgot to cover your mouth when two friends came to visit while you were sick in bed. They leave, and the next day they also have the flu.

Let's assume that they in turn spread the virus to two of their friends by the same droplet spread the following day. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner: Each person infects two more people with the flu virus.

Again we can tabulate the events and formulate an equation for the general case: The next day, each one then infects 2 of their friends.

Everything Maths: Grade 12 Mathematics

Now 4 people are newly-infected. Each of them infects 2 people the third day, and 8 people are infected, and so on. These events can be written as a geometric sequence: Note the common ratio 2 between the events.

Recall from the linear arithmetic sequence how the common difference between terms was established. Common Ratio of Geometric Sequence Determine the common ratio for the following geometric sequences: So, if we want to know how many people are newly-infected after 1 days, we need to work out aw: General Equation of Geometric Sequence Determine the formula for the n"'-term of the following geometric sequences: What is the important characteristic of an arithmetic sequence?

Write down how you would go about finding the formula for the n th term of an arithmetic sequence? A single square is made from 4 matchsticks. To make two squares in a row takes 7 matchsticks, while three squares in a row takes 10 matchsticks. All terms in the sequences are integers.

Calculate the values of x and y. For an arithmetic sequence, where a new term is calculated by taking the previous term and adding a constant value, d: Compare this with Equation 3. For quadratic sequences, we noticed the difference between consecutive terms is given by?? Using 3. This means that using a recursive formula when using a computer to work out a sequence would mean the computer would finish its calculations significantly quicker.

Recursive Formula Write the first five terms of the following sequences, given their recursive formulae: The Fibonacci Sequence Consider the following sequence: Each new term is calculated by adding 3.

Hence, we can write down the recursive equation: We call the sum of any sequence of numbers a series. If we only sum a finite number of terms, we get a fin ite series. If we sum infinitely many terms of a sequence, we get an infinite series: It indicates that you must sum the expression to the right of it: The index i increases from m to n in steps of 1.

Some Basic Rules for Sigma Notation 1. What is J2 2? Calculate the value of a if: Olci 3. When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series.

His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1 to Young Karl realised how to do this almost instantaneously and shocked the teacher with the correct answer, If we wish to sum any arithmetic sequence, there is no need to work it out term-for-term.

We will now determine the general formula to evaluate a finite arithmetic series. We start with the general formula 20 3. The sum of an arithmetic series is times its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero.

The common difference of an arithmetic series is 3. Calculate the value of n for which the n th term of the series is 93, and the sum of the first n terms is The sum of n terms of an arithmetic series is 5n 2 — lln for all values of n.

Determine the common difference. The third term of an arithmetic sequence is —7 and the 7 th term is 9. Determine the sum of the first 51 terms of the sequence. A" 1 More practice CwJ video solutions Qfj or help at www. Olcj 2. Olcq 3. This is the sequence of squares of the integers: Derivation of the Finite Squared Series We will now prove the formula for the finite squared series: We can write out each term of a geometric sequence in the general form: If we subtract 3.

VMgjq at www. Given the geometric sequence 1; —3; 9;. The eighth term of a geometric sequence is The third term is Find the sum of the first 7 terms. The ratio between the sum of the first three terms of a geometric series and the sum of the 4" 1 -, 5"'- and 6"'-terms of the same series is 8: Determine the common ratio and the first 2 terms if the third term is 8.

In this section, we consider what happens when we add infinitely many terms together. You might think that this is a silly question - surely the answer will be oo when one sums infinitely many numbers, no matter how small they are?

The surprising answer is that while in some cases one will reach oo like when you try to add all the positive integers together , there are some cases in which one will get a finite answer. If you don't believe this, try doing the following sum, a geometric series, on your calculator or computer: You might think that if you keep adding more and more terms you will eventually get larger and larger numbers, but in fact you won't even get past 1 - try it and see for yourself!

We denote the sum of an infinite number of terms of a sequence by Y. If a series does not converge, then we say that it diverges. When r, the common ratio, is strictly between —1 and 1, i. There is also a formula for working out the value to which the series converges. Let's start off with Formula 3. See video: VMgly at www. Given the geometric series: The sum to infinity of a geometric series with positive terms is 4 and the sum of the first two terms is 2.

Find a, the first term, and r, the common ratio between consecutive terms.

A" 1 More practice CrJ video solutions f? J or help at www. A new soccer competition requires each of 8 teams to play every other team once. The midpoints of the opposite sides of square of length 4 units are joined to form 4 new smaller squares. This midpoints of the new smaller squares are then joined in the same way to make even smaller squares. This process is repeated indefinitely. Calculate the sum of the areas of all the squares so formed. Thembi worked part-time to download a Mathematics book which cost R29, On 1 February she saved Rl,60, and everyday saves 30 cents more than she saved the previous day.

So, on the second day, she saved R1. After how many days did she have enough money to download the book? Consider the geometric series: B in terms of n. A certain plant reaches a height of mm after one year under ideal conditions in a greenhouse.

During the next year, the height increases by 12 mm. Show that the plant will never reach a height of more than mm. Michael saved R during the first month of his working life.

A man was injured in an accident at work. He receives a disability grant of R4 in the first year. This grant increases with a fixed amount each year. After how many years does his expenses exceed his income? The Cape Town High School wants to build a school hall and is busy with fundraising.

Manuel, an ex-learner of the school and a successful politician, offers to donate money to the school. Having enjoyed mathematics at school, he decides to donate an amount of money on the following basis.

He sets a mathematical quiz with 20 questions. For the correct answer to the first question any learner may answer , the school will receive 1 cent, for a correct answer to the second question, the school will receive 2 cents, and so on. The donations 1; 2; 4: Calculate Give your answer to the nearest Rand a The amount of money that the school will receive for the correct answer to the 20 th question.

Calculate the values of n for which the n th term of the series is 93 and the sum of the first n terms is The first term of a geometric sequence is 9, and the ratio of the sum of the first eight terms to the sum of the first four terms is Find the first three terms of the sequence, if it is given that all the terms are positive.

Find k and m if both are positive. The second and fourth terms of a convergent geometric series are 36 and 16, respec- tively. Find the sum to infinity of this series, if all its terms are positive. Eva uate: Find the second term. Find p if: Find the integer that is the closest approximation to: If it does, work out what it converges to: Find the 10 th term. The powers of 2 are removed from the set of positive integers 1; 2; 3; 4; 5; 6; Observe the pattern below: The following question was asked in a test: Here are some of the students' answers: Who is correct?

A shrub of height cm is planted. At the end of the first year, the shrub is cm tall. Thereafter, the growth of the shrub each year is half of its growth in the previous year. Show that the height of the shrub will never exceed cm. Nominal and effective interest rates were also described.

Since this chapter expands on earlier work, it would be best if you revised the work done in Grades 10 and 1 1, When you master the techniques in this chapter, you will be able to assess critically how to invest your money when you start working and earning. And when you are looking at applying for a bond from a bank to download a home, you will confidently be able to get out the calculator and work out how much you could actually save by making additional repayments.

This chapter will provide you with the fundamental concepts you will need to manage your finances. VMgmf at www. Remember that P is the initial amount, A is the current amount, i is the interest rate and n is the number of time units number of months or years.

So if we invest an amount and know what the interest rate is, then we can work out how long it will take for the money to grow to the required amount. Now that you have learnt about logarithms, you are ready to work out the proper algebraic solution. If you need to remind yourself how logarithms work, go to Chapter 2 on Page 4.

The basic finance equation is: Solving for n: It is very easy to derive any time you need it. It is simply a matter of writing down what you have, deciding what you need, and solving for that variable. After an unknown period of time our account is worth R4 , For how long did we invest the money? How does this compare with the trial and error answer from Chapter??. Determine how to approach the problem We know that: Write final answer The R3 was invested for 2 years.

Adding these together gives you the amount needed to afford all three payments and you get R2 , You would have had exactly the right amount of money to do that obviously! You can check this as follows: Amount at Time i. Of course, for only three years, that was not too bad. But what if I asked you how much you needed to put into a bank account now, to be able to afford R a month for the next 1 5 years. If you used the above approach you would still get the right answer, but it would take you weeks!

There is - I'm sure you guessed - an easier way! This section will focus on describing how to work with: This is an example of a present value annuity. This is an example of a future value annuity. Sequences and Series W emcag Before we progress, you need to go back and read Chapter 3 from Page 19 to revise sequences and series.

In summary, if you have a series of n terms in total which looks like this: Given that we are in the finance section, you would be right to guess that there must be some financial use to all this. Here is an example which happens in many people's lives - so you know you are learning something practical.

Let us say you would like to download a property for R , so you go to the bank to apply for a mortgage bond. The bank wants it to be repaid by annually payments for the next 20 years, starting at end of this year. At the end of the 20 years the bank would have received back the total amount you borrowed together with all the interest they have earned from lending you the money.

You would obviously want to work out what the annual repayment is going to be! Let X be the annual repayment, i is the interest rate, and M is the amount of the mortgage bond you will be taking out. Time lines are particularly useful tools for visualising the series of payments for calculations, and we can represent these payments on a time line as: Time line for an annuity in arrears of X for n periods.

The present value of all the payments which includes interest must equate to the present value of the mortgage loan amount. Mathematically, you can write this as: Not only would you probably get bored along the way, but you are also likely to make a mistake. Naturally, there is a simpler way of doing this! You can rewrite the above equation as follows: We just find this a useful method because you can get rid of the negative exponents - which can be quite confusing! As an exercise - to show you are a real financial whizz - try to solve this without substitution.

It is actually quite easy. Now, the item in square brackets is the sum of a geometric sequence, as discussion in section 3. So we can write: On the other hand, if I know I will have only R30 per year to repay my bond, then how big a house can I download? That is easy. The bad news is that R does not come close to the R you wanted to pay!


The good news is that you do not have to memorise this formula. In fact , when you answer questions like this in an exam, you will be expected to start from the beginning - writing out the opening equation in full, showing that it is the sum of a geometric sequence, deriving the answer, and then coming up with the correct numerical answer. He has viewed a flat which is on the market for R , and he would like to work out how much the monthly repayments would be.

He will be taking out a 30 year mortgage with monthly repayments. If you are not clear on this, go back and revise Section??. Solve the problem Now it is easy, we can just plug the numbers in the formula, but do not forget that you can always deduce the formula from first principles as well!

Example 3: You have savings of R10 which you intend to use for a deposit. How much would your monthly mortgage payment be if you were considering a mortgage over 20 years. Determine what is given and what is required The following is given: Determine how to approach the problem We are considering monthly mortgage repayments, so it makes sense to use months as our time period.

Once we have converted 20 years into months, we are ready to do the calculations! First we need to calculate M, the amount of the mortgage bond, which is the download price of property minus the deposit which Sam pays up-front.

Solve the problem 15 4. Show Me the Money Now that you've done the calculations for the worked example and know what the monthly repayments are, you can work out some surprising figures. That is more than double the amount that you borrowed! This seems like a lot. However, now that you've studied the effects of time and interest on money, you should know that this amount is somewhat meaningless.

The value of money is dependant on its timing. Nonetheless, you might not be particularly happy to sit back for 20 years making your Rl ,48 mortgage payment every month knowing that half the money you are paying are going toward interest. But there is a way to avoid those heavy interest charges. It can be done for less than R extra every month. So our payment is now R2 Making this higher repayment amount every month, how long will it take to pay off the mortgage? The present value of the stream of payments must be equal to R the present value of the borrowed amount.

So we need to solve for n in: Can you see what is happened? You surely know by now that the difference between the additional R48 ,74 that you have paid and the R76 ,20 interest that you have saved is attributable to, yes, you have got it, compound interest! In the above section, we had a few payments, and we wanted to know what they are worth now - so we calculated present values. But the other possible situation is that we want to look at the future value of a series of payments.

Maybe you want to save up for a car, which will cost R45 - and you would like to download it in 2 years time. You need to work out how much to put into your bank account now, and then again each month for 2 years, until you are ready to download the car. Can you see the difference between this example and the ones at the start of the chapter where we were only making a single payment into the bank account - whereas now we are making a series of payments into the same account?

This is a sinking fund. So, using our usual notation, let us write out the answer. Make sure you agree how we come up with this. Because we are making monthly payments, everything needs to be in months. So let A be the closing balance you need to download a car, P is how much you need to pay into the bank account each month, and i r2 is the monthly interest rate.

Check back to the start of the chapter if this is not obvious to you by now. If you draw a timeline you will see that the time between the first payment and when you download the car is 24 months, which is why we use 24 in the first exponent.

Again, looking at the timeline, you can see that the 24" 1 payment is being made one month before you download the car - which is why the last exponent is a 1. Always check that you have got the right number of payments in the equation. Check right now that you agree that there are 24 terms in the formula above. So, now that we have the right starting point, let us simplify this equation: This is not a rule you have to memorise - you can see from the equation what the obvious choice of X should be.

Let us re-order the terms: BUT and it is a big but we need a monthly interest rate. Do not forget that the trick is to keep the time periods and the interest rates in the same units - so if we have monthly payments, make sure you use a monthly interest rate! You have taken out a mortgage bond for R to download a flat. How much money must be invested now to obtain regular annuity payments of R5 per month for five years?

Answer to the nearest hundred rand. This section aims to allow you to use these valuable skills to critically analyse investment and loan options that you will come across in your later life. This way, you will be able to make informed decisions on options presented to you. At this stage, you should understand the mathematical theory behind compound interest.

However, the numerical implications of compound interest are often subtle and far from obvious. Recall the example 'Show Me the Money' in Section 4. For an extra payment of R29 a month, we could have paid off our loan in less than 14 years instead of 20 years. This provides a good illustration of the long term effect of compound interest that is often surprising. In the following section, we'll aim to explain the reason for the drastic reduction in the time it takes to repay the loan.

We are not considering the repayments individually. Think about the time you make a repayment to the bank. There are numerous questions that could be raised: Since you are paying off the loan, surely you must owe them less money, but how much less?

We know that we'll be paying interest on the money we still owe the bank. When exactly do we pay interest? How much interest are we paying? The answer to these questions lie in something called the load schedule. We will continue to use the earlier example. There is a loan amount of R We worked out that the repayments should be Rl , Consider the first payment of Rl ,48 one month into the loan.

First, we can work out how much interest we owe the bank at this moment. We borrowed R a month ago, so we should owe: We call this the interest component of the repayment. This is called the capital component. This is called the capital outstanding. Let's see what happens at the end of the second month. The amount of interest we need to pay is the interest on the capital outstanding. This way, we can break each of our repayments down into an interest part and the part that goes towards paying off the loan.

This is a simple and repetitive process. Table 4. This is called a loan schedule. Now, let's see the same thing again, but with R2 being repaid each year. We expect the numbers to change. However, how much will they change by? As before, we owe Rl in interest in interest. After one month. However, we are paying R2 this time. That leaves R that goes towards paying off the capital outstanding, reducing it to R By the end of the second month, the interest owed is Rl ,69 That's R x i This reduces the amount outstanding to R , Doing the same calculations as before yields a new loan schedule shown in Table 4.

The important numbers to notice is the "Capital Component" column. Note that when we are paying off R2 a month as compared to Rl ,48 a month, this column more than double. In the beginning of paying off a loan, very little of our money is used to pay off the capital outstanding. Therefore, even 19 4. A loan schedule with repayments of Rl ,48 per month. A loan schedule with repayments of R2 per month.

What's more, look at the amount we are still owing after one year i. When we were paying Rl ,48 a month, we still owe R , However, if we increase the repayments to R2 a month, the amount outstanding decreases by over R3 to R , This means we would have paid off over R7 in our first year instead of less than R4 This increased speed at which we are paying off the capital portion of the loan is what allows us to pay off the whole loan in around 14 years instead of the original Note however, the effect of paying R2 instead of Rl ,48 is more significant in the beginning of the loan than near the end of the loan.

It is noted that in this instance, by paying slightly more than what the bank would ask you to pay, you can pay off a loan a lot quicker. The natural question to ask here is: Are they trying to cheat us out of our money? There is no simple answer to this. Banks provide a service to us in return for a fee, so they are out to make a profit.

However, they need to be careful not to cheat their customers for fear that they'll simply use another bank. The central issue here is one of scale. For us, the changes involved appear big. We are paying off our loan 6 years earlier by paying just a bit more a month. To a bank, however, it doesn't matter much either way. In all likelihood, it doesn't affect their profit margins one bit!

Remember that the bank calculates repayment amounts using the same methods as we've been learn- ing. They decide on the correct repayment amounts for a given interest rate and set of terms. Smaller repayment amounts will make the bank more money, because it will take you longer to pay off the loan and more interest will accumulate.

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Larger repayment amounts mean that you will pay off the loan faster, so you will accumulate less interest i. It's a simple matter of less money now or more money later. Banks generally use a 20 year repayment period by default. Learning about financial mathematics enables you to duplicate these calculations for yourself. This way, you can decide what's best for you.

You can decide how much you want to repay each month and you'll know of its effects. A bank wouldn't care much either way, so you should pick something that suits you. They need to get a mortgage for the balance. Determine how to approach the problem Use the formula: A property costs Rl A loan of R 4 is to be returned in two equal annual instalments.

We also saw how we can calculate this using loan schedules. However, there is a significant disadvantage to this method: For example, in order to calculate how much capital is still outstanding at time 12 using the loan schedule, we'll have to first calculate how much capital is outstanding at time 1 through to 11 as well. This is already quite a bit more work than we'd like to do.

Can you imagine calculating the amount outstanding after 10 years time ? Fortunately, there is an easier method. However, it is not immediately clear why this works, so let's take some time to examine the concept. Prospective Method for Capital Outstanding Let's say that after a certain number of years, just after we made a repayment, we still owe amount Y.

What do we know about Yl We know that using the loan schedule, we can calculate what it equals to, but that is a lot of repetitive work. In other words, all the repayments we are still going to make in the future will exactly pay off Y. This is true because in the end, after all the repayments, we won't be owing anything.

Therefore, the present value of all outstanding future payments equal the present amount outstanding. This is the prospective method for calculating capital outstanding. Let's return to a previous example. Recall the case where we were trying to repay a loan of R over 20 years. A RIO deposit was put down, so the amount being payed off was R In Table 4. Let's try to work this out using the the prospective method. The present value is: This seems to be almost right, but not quite.

We should have got R , We are 8 cents out. However, this is in fact not a mistake. Remember that when we worked out the monthly repayments, we rounded to the nearest cents and arrived at Rl , This was because one cannot make a payment for a fraction of a cent.

Therefore, the rounding off error was carried through. That's why the two figures don't match exactly. In financial mathematics, this is largely unavoidable.

While memorising them is nice there are not many , it is the application that is useful. Financial experts are not paid a salary in order to recite formulae, they are paid a salary to use the right methods to solve financial problems. Definitions EMCAN P Principal the amount of money at the starting point of the calculation i interest rate, normally the effective rate per annum n period for which the investment is made iT the interest rate paid T times per annum, i.

The bank is going to charge him R to set up the account. How much can he expect to get back at the end of the period? Calculate the opening balance required to generate a closing balance of R5 after 2 years. Which of the two answers above is lower, and why? What was the value of the initial deposit?

Suppose Lungelo makes a deposit of X today at interest rate of i for six years. Thabani made his deposit 3 years after Lungelo made his first deposit. If after 6 years, their investments are equal, calculate the value of i and find X. If the sum of their investment is R20 , use the value of X to find out how much Thabani earned in 6 years. Sipho invests R at an interest rate of log l,12 for 5 years. Themba, Sipho's sister invested R at interest rate i for 1 years on the same date that her brother made his first deposit.

If after 5 years, Themba's accumulation equals Sipho's, find the interest rate i and find out whether Themba will be able to download her favourite cell phone after 10 years which costs R2 Repeat this for the case where it is capitalised half yearly i. Every 6 months.

Most of the solutions, relied on being able to factorise some expression and the factorisation of quadratics was studied in detail. This chapter focuses on the factorisation of cubic polynomials, that is expressions with the highest power equal to 3. See introductory video: VMgmo at www. Factor Theorem For any polynomial, f x , for al of f x. Or, more concisely: In other words: If the remainder when dividing f x by x — a is a factor of f x.

Example 2: Conclusion 5. We have seen in Grade 10 that the sum and difference of cubes is factorised as follows: There are many methods of factorising a cubic polynomial. The general method is similar to that used to factorise quadratic equations. If you have a cubic polynomial of the form: For example, find the factors of: We can then use values for a, fo, c and d to determine values for B, D and F. We can re-write 5. If we multiply this out we get: Therefore we can use a trial and error method to find B, D and F.

This can become a very tedious method, therefore the Factor Theorem can be used to find the factors of cubic polynomials. Now, we must find the coefficient of the middle term x. So, the coefficient of the x-term must be 0. Use the factor theorem to confirm that the guess is a root. Then divide the cubic polynomial by the factor to obtain a quadratic. Once you have the quadratic, you can apply the standard methods to factorise the quadratic. There are three factors which we can write as x — a x — b x — c.

So, the coefficient of the x-term must be —1. Use the Factor Theorem to determine all the factors of the following expression: Olej 4. Olem 5. For example: So, the coefficient of the x-term must be —4.

Everything Maths Grade 12

Apply the quadratic formula for the second bracket Always write down the formula first and then substitute the values of a, b and c. Solve for x: Solve for y: Solve for m: Remove brackets and write as an equation equal to zero.

What is the value of ffl. A challenge: In Grade 12 you are expected to demon- strate the ability to work with various types of functions and relations including inverses.

In particular, we will look at the graphs of the inverses of: VMgsz at www. This definition makes complete sense when compared to our real world examples — each person has only one height, so height is a function of people; on each day, in a specific town, there is only one average temperature. However, some very common mathematical constructions are not functions. This relation describes a circle of radius 2 centred at the origin, as in Figure 6.

There is a simple test to check if a relation is a function, by looking at its graph. This test is called the vertical line test. If it is possible to draw any vertical line a line of constant x which crosses the graph of the relation more than once, then the relation is not a function.

If more than one intersection point exists, then the intersections correspond to multiple values of y lor a single value of x. We can see this with our previous example of the circle by looking at its graph again in Figure 6. We see that we can draw a vertical line, for example the dotted line in the drawing, which cuts the circle more than once.

And just as words were not created specically to tell a story but their existence enabled stories to be told, so the mathematics used to create these technologies was not developed for its own sake, but was available to be drawn on when the time for its application was right. There is in fact not an area of life that is not affected by mathematics.

Many of the most sought after careers depend on the use of mathematics. Civil engineers use mathematics to determine how to best design new structures; economists use mathematics to describe and predict how the economy will react to certain changes; investors use mathematics to price certain types of shares or calculate how risky particular investments are; software developers use mathematics for many of the algorithms such as Google searches and data security that make programmes useful.

But, even in our daily lives mathematics is everywhere in our use of distance, time and money. Mathematics is even present in art, design and music as it informs proportions and musical tones.

The greater our ability to understand mathematics, the greater our ability to appreciate beauty and everything in nature. Far from being just a cold and abstract discipline, mathematics embodies logic, symmetry, harmony and technological progress. More than any other language, mathematics is everywhere and universal in its application. See introductory video by Dr. Mark Horner: VMiwd at www. It has everything you expect from your regular printed school textbook, but comes with a whole lot more.

For a start, you can download or read it on-line on your mobile phone, computer or iPad, which means you have the convenience of accessing it wherever you are. We know that some things are hard to explain in words. That is why every chapter comes with video lessons and explanations which help bring the ideas and concepts to life.

Summary presentations at the end of every chapter offer an overview of the content covered, with key points highlighted for easy revision. All the exercises inside the book link to a service where you can get more practice, see the full solution or test your skills level on mobile and PC. We are interested in what you think, wonder about or struggle with as you read through the book and attempt the exercises. That is why we made it possible for you to use your mobile phone or computer to digitally pin your question to a page and see what questions and answers other readers pinned up.

Everything Maths on your mobile or PC You can have this textbook at hand wherever you are whether at home, on the the train or at school. Just browse to the on-line version of Everything Maths on your mobile phone, tablet or computer. To read it off-line you can download a PDF or e-book version. To read or download it, go to www.

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The short-codes next to the icons allow you to navigate directly to the resources on-line without having to search for them. A V P Q Go directly to a section Video, simulation or presentation Practice and test your skills Ask for help or nd an answer To watch the videos on-line, practise your skills or post a question, go to the Everything Maths website at www.

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See the concepts in action and hear real people talk about how they use maths and science in their work. See video explanation Video: V Video exercises Wherever there are exercises in the book you will see icons and short-codes for video solutions, practice and help.

These short-codes will take you to video solutions of select exercises to show you step-by-step how to solve such problems. See video exercise Video: V You can get these videos by: viewing them on-line on your mobile or computer downloading the videos for off-line viewing on your phone or computer ordering a DVD to play on your TV or computer downloading them off-line over Bluetooth or Wi-Fi from select outlets To view, download, or for more information, visit the Everything Maths website on your phone or computer at www.

At every set of exercises you will see a practice icon and short-code. This on-line practice for mobile and PC will keep track of your performance and progress, give you feedback on areas which require more attention and suggest which sections or videos to look at.Is this graph a function? A loan of R 4 is to be returned in two equal annual instalments. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner: This section will focus on describing how to work with: Arithmetic Sequence An arithmetic or linear sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term For example, 1; 2; 3; 4; 5; 6;.

Think about the time you make a repayment to the bank. Using 7. A brickworks produces "face bricks" and "clinkers". Remaining constraints: