## Revision

Remember to show all the steps in your working.

### Whole numbers

1. Write all the numbers from the cloud in the table below, and place a tick in all the column(s) of the type of numbers they are. The first number has been completed for you:

 Number value Number system Real numbers Natural numbers Integers Rational numbers Irrational numbers -3 âœ” âœ” âœ”
2. The Ndlovu family is travelling to the Kruger National Park on holiday. Here is a summary of their journey:

 Time Odometer reading (km) Description 06:12 123 564 Leave home 08:32 123 785 Stop for breakfast and petrol 09:18 123 785 Leave petrol station 11:34 124 011 Stop for toilet break 11:51 124 011 Leave petrol station 13:32 124 175 Reach Kruger gate
1. Calculate the length of time the journey took, in hours. Give your answer as mixed number.

2. Calculate the average speed of the journey, correct to one decimal place.

3. A car travelling at an average speed of 110 km/h takes $$2 \frac{1}{4}$$ hours to complete a journey. If the return journey needs to be completed in 2 hours, calculate the average speed that must be maintained.

4. If four tins of bully beef cost R75,80, how much money would seven similar tins cost?

5. A farmer has enough chicken feed to feed 300 hens for 20 days. If he buys 100 more hens, how long would the same amount of chicken feed last before it runs out?

6. How long will it take R5 000 invested at 7,2% simple interest p.a. to grow to R5 900?

7. Chardonnay wishes to buy a new TV set costing R7 499. She does not have enough money and so needs to buy it on hire purchase. The store requires a 10% deposit and then equal monthly payments of R$$x$$ for 2 years. If the simple interest charged on the account is 15%, calculate the value of $$x$$.

8. How much interest will Tebogo get on R12 500 deposited for 21 months in a bank account that provides 5,3% compound interest per annum?

### Integers

All the questions in this section should be answered without using a calculator.

1. Write a number in each box to make the calculations correct:

1. $$☐ + ☐ = -34$$

2. $$☐ + ☐ = -34$$

2. These questions show sequences of numbers. Fill the correct values in the boxes:

1. 18; 10; 2; ☐

2. 2; -10; 50; ☐

3. -6 386; -6 392; -6 398; ☐

3. This question shows a number line in which the missing number is halfway between the other two numbers. Fill the correct value into the box:

4. Calculate the following:

1. $$28 - (-15)$$

2. $$(-5)(12)(-7)$$

3. $$5 + 5 \times -6$$

4. $$\frac{(\sqrt{81})(-2)^{3}}{-(-3)^{2}}$$
5. $$\frac{(-3)^{2} \sqrt[3]{216}}{(-9)(-3)}$$

5. Augustus ruled the Roman Empire from 27 BC to AD 14. For how many years did he rule?

### Fractions

All the questions in this section should be answered without using a calculator.

1. Simplify the following:

1. $$\sqrt{\frac{36}{81} x^8}$$

2. $$\frac{5}{2}x^{2} - \frac{5}{4}x^{2}$$
3. $$(\frac{3}{4}xy^{3})( \frac{4}{9}y)$$
2. Simplify the following:

1. $$\frac{4 x^{10}}{8x^5}$$
2. $$\frac{5}{x} - \frac{1}{x}$$
3. $$\frac{5x}{6y^2}\times\frac{3y}{15x}$$
4. $$\frac{x+2}{4z^2}\div\frac{4(x+2)}{2z^3}$$

### The decimal notation for fractions

All the questions in this section should be answered without using a calculator.

1. Calculate the following:
1. 27,49 - 6,99
2. $$0,03 \times 1,4$$
3. $$1,44 \div 0,012$$
2. Simplify the following:
1. $$\sqrt{0.04 x^{16}}$$
2. $$3.5x^{2} - 4,6 x^{2}$$
3. $$(1,2 x^{2}y^{3})(5y x^{2})$$
3. Simplify the following:
1. $$\frac{0.2 x^{15}}{0.01x^5}$$
2. $$\frac{0.45}{x} - \frac{1.35}{x}$$
3. $$\frac{0,5 x^{3}}{4.5y^2} \times \frac{3y}{2,5x}$$
4. $$\frac{2,5 x^{3}}{2y^2} \div \frac{0,5x}{0,03y^6}$$

### Exponents

All the questions in this section should be answered without using a calculator, unless otherwise specified in the question.

1. Write the following numbers in scientific notation:
1. 2 500 001
2. 0,000 304 5
2. Write the following number in "normal" notation: $$9,45 \times 10^{-5}$$.
3. Which of the following numbers is bigger: $$4,7 \times 10^{-9}$$ or $$5,12 \times 10^{-10}$$?
1. $$(5,9 \times 10^{6} ) - (4,7 \times 10^{6})$$
2. $$(5,9 \times 10^{6} ) + (4,7 \times 10^{5})$$
3. $$(7,2 \times 10^{-4} ) \times (2 \times 10^{2})$$
5. Calculate the following, giving your answer as an ordinary decimal number. A calculator may be used:
1. $$(6,3 \times 10^{-4} ) - (1,9 \times 10^{-3})$$
2. $$(5,8 \times 10^{-7} ) \div (8 \times 10^{-11})$$
6. Simplify the following, leaving all answers with positive exponents:
1. $$3^{-2}$$
2. $$2^{7} \times 6^{-3} \times 3^{2}$$
3. $$\frac{2y^{-3}}{y^3}$$
4. $$(2x^{6}) ^{-3}$$
5. $$(2x^{7}) (2,5x^{-8})$$
6. $$(-3a^{2}bc)^2 ( -5ac^{-2})$$
7. $$\frac{(2d^2e)^2}{(4d^{-3}e^2)^{-1}}$$
7. Solve the following equations:
1. $$3 \times 3^{x} = 81$$
2. $$2^{x+1} = 0,125$$
3. $$4^{x} +10 = 74$$

### Patterns

1. Create a sequence that fits this description: the first term is negative, and each successive term is obtained by squaring the previous term and then subtracting 10.
2. Write down the first four terms of your created sequence.

3. For each of the following sequences, (i) write in words the rule that describes the relationship between the terms in the sequence, and (ii) use the rule to extend the sequence by three more terms:
1. -5; -2; 10; -20;...
2. -4,5; -6,25; -8;...
4. In this question you are given the rule by which each term of the sequence can be found. In all cases, n is the position of the term. Determine the first three terms of each of the sequences:
1. $$3 - 5n$$
2. $$2n ^{2} - 3n + 1$$
1. Write down the rule by which each term of the sequence can be found (in a similar format to those given in question 3, where n is the position of the term): -15; -12; -9; ...
2. Use this rule to find the value of the 150th term of the sequence.
5. Determine the pattern and then write the missing values in the table below:
 Position in sequence 1 2 3 4 5 10 Value of the term 2 5 10 17 226
6. The picture below shows a pattern created by matchsticks.

1. Draw your own series of matchstick patterns in which there is a common difference between each pattern. It must be different to all the matchstick patterns shown in Chapter 6 and this chapter, and should contain the first three matchstick patterns in the series.
2. Write in words the rule that describes the number of matchsticks needed for each new pattern.
3. Use the rule to determine the missing values in the table below, and fill them in:
 Number of the pattern 4 5 6 7 50 Number of matches needed

### Functions and relationships

1. Use the given formula to calculate the values of t, given the values of p:

2. Use the given formula to calculate the missing input values, p, and output values, t.

1. Consider the values in the table below:
 $$x$$ -2 -1 0 1 4 12 $$y$$ -4 -1 2 5 65
1. Write, as an algebraic formula, the rule for finding the y-values in the table. The formula is in the form $$y = ax + b$$, where $$a$$ and $$b$$ are integers.
2. Use the rule to determine the missing values in the table, and fill them in.
2. Consider the graph shown below:

1. Complete the following table by reading off the coordinates of points on the graph:
 $$x$$ -3 -2 -1 0 1 2 $$y$$
2. Write down an algebraic formula for the graph, in the form $$y =$$...
3. Complete the flow diagram below to represent the relationship shown on the graph:

### Algebraic expressions

1. Simplify as far as possible:
1. $$(2x^{2} - 4x^{2}) ^{3}$$
2. -$$2x^{2} (5x^{3} - 3x^{2} + 2x - 5)$$
3. $$(4b^{2} - 7b^{2} (5b^{-2} + 3b^{-1} - 7)$$
4. $$\frac{18x^2 - 12x + 2}{6x}$$
5. $$(2x+5)(3x-1)$$
6. $$(4a - 3)^{2}$$
7. $$\frac{6x^3 -2(3x)(4x) + x}{4x^2}$$
2. Simplify as far as possible:
1. $$4(a-2b) - 5(3b + a)$$
2. $$5 +2(x^2 +5x +3)$$
3. $$3x(2x^2 -3x + 4) - 3(5 - 2x)$$
4. $$(a + 3b - 2c) - (4a + b - c) - (2b - c +3a)$$
5. $$4(3x^2 + x - 4) - (x + 3)^{2}$$

### Equations

1. Solve the following equations:
1. $$4 - 3x = -2$$
2. $$4(2x- 1) = -8$$
3. $$2x + 1 = 3(2x - 1)$$
4. ($$x + 2)(x - 4) = x^{2} + 5x - 1$$
2. Thomas is z years old and Tshilidzi is twice as old as Thomas. The sum of their ages is 42.
1. Write this information in an equation using the variable $$z$$.
2. Solve the equation to find Tshilidzi's age.
3. The base of a triangle is $$(1,5x + 6)$$ cm and the height is 4 cm. The area of the triangle is 24 cm2.
1. Write this information in an equation in $$x$$.
2. Solve the equation to determine the value of $$x$$.
3. What is the length of the base of the triangle?
4. Solve for$$x$$:
1. $$3^{x} = 9$$
2. $$2^{x+1} = 16$$

## Assessment

In this section, the numbers in brackets at the end of a question indicate the number of marks the question is worth. Use this information to help you determine how much working is needed. The total number of marks allocated to the assessment is 75.

1. Gareth completed the following number classification:
 Number value Number system Real numbers Natural numbers Integers Rational numbers Irrational numbers -1,5 âœ” âœ” âœ” $$\sqrt{2}$$ âœ” âœ”

1. Gareth has made some errors. Complete the following table by putting the ticks in the correct boxes: (2)
 Number value Number system Real numbers Natural numbers Integers Rational numbers Irrational numbers -1,5 $$\sqrt{2}$$

2. Explain why you have made the changes you have. (2)
2. Pheto invested R1 500 for 2 years in a bank account. At the end of this period, the initial investment had grown to R1 717,50. What simple interest rate did the bank give him? (Assume that the rate remained unchanged for the entire period.) Give your answer as a percentage. (3)
3. A Benthian changed 2 500 Bendollars to Darsek when he visited the Klingon Empire, and received 2 000 Darsek after the 3% commission had been charged. Determine the Bendollars : Dasek exchange rate and then copy and complete the following sentence: "1 Klingon Darsek = ___ Bendollars". The missing value should be written correct to three decimal places. (3)
4. What is the difference in height between the highest point on the Earth's surface (Mt Everest: 8 848 m above sea level) and the deepest point of the sea (the bottom of the Marianas Trench, 10 994 m below sea level)? (1)
5. Write down two numbers that subtract to give an answer of 21. One of the numbers must be positive and the other negative. (2)
1. What is the value of $$(-1)^{1000 001}$$ ? (1)
2. Explain how you can know the answer in (a) without needing a calculator. (1)
6. Simplify the following, without using a calculator. Show all steps of your working:
1. $$\frac{5}{2}x -\frac{11}{4}x + 1,125x$$
2. $$\sqrt[3]{\frac{0,027 x^7}{316x}}$$
3. $$\frac{0,4x}{10} \times \frac{20x}{0,03} \div \frac{8x^{2}}{5}$$
4. $$\frac{x}{4}+ [8x (x + 1) \times \frac{0,5}{x+1}]$$
7. The diameter of a carbon atom is 0,000000000154 metres. Write this in scientific notation. (2)
8. Simplify the following, leaving all answers with positive exponents:
1. $$3^{-9} \times 3^{4}$$
2. $$\frac{(3d^3e^2)^3}{(2d^{-4}e)^-1}$$
9. Solve for $$x$$: $$9^{2x-3} = 3^{x}$$
10. Consider the following sequence: 6 000; -1 500; 375; ...
1. Extend the sequence by two more terms. (2)
2. Is this the correct rule for the sequence (where n is the position of the term in the sequence): $$6 000(0,25)^{n-1}$$? Explain your answer. (2)
11. The following figure shows a pattern created by matchsticks.

1. Draw the 5th diagram in the pattern alongside the picture above. (2)
2. The first two terms in the sequence created by the number of matchsticks in each pattern is 4; 11. Write down the next three terms in the sequence. (2)
3. Write in words the rule that describes the relationship between the terms in the sequence. (2)
12. Consider the values in the table below:

 $$x$$ -2 -1 0 1 5 16 $$y$$ -10 -3 -2 -1 7 998

1. Write the rule for finding the y-values in the table as an algebraic formula. (2) (Hint: Look at the cubes of the numbers.)
2. Use the rule to determine the missing values in the table, and fill them in. (3)
13. Consider the following graph:

1. Complete the following table by reading off the coordinates of points on the graph: (2)

 $$x$$ -2 -1 0 1 2 3 $$y$$
2. Write down an algebraic formula for the graph in the form $$y = ...$$ (2)
14. Simplify:
1. $$\frac{15+x-5x^2}{5x^2}$$
2. $$(3x+1)(3x-1)$$
3. $$4 - 3(2x+ 3)^{2}$$
15. Solve the following equations:
1. $$x^2 + 5x - 1 - x^2 -x + 3 = 3(x-4)$$
2. $$2(2x+3) = (3x-1)(-2)$$