GRADE 11 MATHS TEXTBOOK PDF
Everything Maths Grade 11 - Everything Maths and Science. Pages·· MB·23, Downloads. Everything Maths is not just a Mathematics textbook . Siyavula textbooks: Grade 11 Maths PDF generated: October 29, .. In Grade 10, we worked only with indices that were integers. Open textbooks offered by Siyavula to anyone wishing to learn maths and science. Mathematics Grade Read online Mathematical Literacy Grade
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Visit the Everything Maths and Everything Science mobi sites at:. Visit the Everything Maths and Everything Science websites and download the books. Practise the exercises from this textbook, additional exercises and questions from past exam papers on m. Your dashboard will show you your progress and mastery for every topic in the book and help you to manage your studies. You can use your dashboard to show your teachers.
Google searches. Mathematics is the language that nature speaks to us in. The greater our ability to understand mathematics. More than any other language. In a similar way. Far from being just a cold and abstract discipline. Many of the modern technologies that have enriched our lives are greatly dependent on mathematics.
Mathematics is even present in art. And just as words were not created specifically to tell a story but their existence enabled stories to be told. Civil engineers use mathematics to determine how to best design new structures.
Many of the most sought after careers depend on the use of mathematics. There is in fact not an area of life that is not affected by mathematics. As we learn to understand and speak this language.
The great writers and poets of the world have the ability to draw on words and put them together in ways that can tell beautiful or inspiring stories. Contents 1 Exponents and surds 1. Irrational numbers also include decimal numbers that neither terminate nor recur.
Exercise 1 — 1: The number system Use the list of words below to describe each of the following numbers in some cases multiple words will be applicable: The exponent. Exponents and surds 5. Get this answer and more practice on our Intelligent Practice Service 1. Worked example 1: In the last example we have k 6. Exponents and surds 7.
Rational exponents and surds. A rational number is any number that can be written as a fraction with an integer in the numerator and in the denominator. Exercise 1 — 2: Laws of exponents Simplify the following: Think you got it? Worked example 3: For example. The radicand is the number under the radical symbol. It is also possible for there to be more than one nth root of a number.
Exponents and surds 9. When dealing with exponents. A surd is a radical which results in an irrational number. A radical refers to a number written as shown below.
Simplify the denominator 10 1. Simplify the following and write answers with positive exponents: Use the laws to re-write the following expression as a power of x: Get this answer and more practice on our Intelligent Practice Service 1a.
Rational exponents and surds 1. Exponents and surds The additional laws listed below make simplifying surds easier: It is often useful to write a surd in exponential notation as it allows us to use the exponential laws.
See video: Worked example Chapter 1. By rationalising the denominator. Simplify the following: Expressing the surd in the numerator is the preferred way of writing expressions.
Rationalising the denominator Rationalise the denominator in each of the following: We also need to be able to solve equations that involve surds. Step 4: Step 2: Solve for both factors The zero law states: Step 3: Exercise 1 — 6: Solving surd equations Solve for the unknown variable remember to check that the solution is valid: There are many real world applications that require exponents. Exponential formula. Therefore, in this case: What will the population be in ten years and in one hundred years?
Nqobani invests R into an account which pays out a lump sum at the end of 6 years. If he gets R ,20 at the end of the period, what compound interest rate did the bank offer him?
Give answer correct to one decimal place.
What can city planners expect the population of Johannesburg to be in 13 years time? Abiona places 3 books in a stack on her desk. The next day she counts the books in the stack and then adds the same number of books to the top of the stack. After how many days will she have a stack of books?
If there are initially 45 individual mould cells in the population, determine how many there will be in 19 hours. The number system: Rational exponents and surds: Simplify as far as possible: Re-write the following expression as a power of x: Write as a single term with a rational denominator: Write in simplest surd form: Expand and simplify: Simplify, without use of a calculator: Rationalise the denominator: Prove without the use of a calculator: Simplify completely by showing all your steps do not use a calculator: Equation Inequality Solution Root A quadratic equation is an equation of the second degree.
There are some situations. A mathematical statement that asserts that two expressions are equal. A value or set of values that satisfy the original problem statement.
Expression An expression is a term or group of terms consisting of numbers. The following are examples of quadratic equations: Equations and inequalities One method for solving quadratic equations is factorisation. This is called the zero product law. Chapter 2. Zero product law Solve the following equations: It is very important to note that one side of the equation must be equal to zero. Method for solving quadratic equations 1.
Always check the solution by substituting the answer back into the original equation. Solve for both roots Chapter 2. Check the solution by substituting both answers back into the original equation Step 5: Worked example 2: There are no common factors Step 2: It is shown here for illustration purposes only. This graph does not form part of the answer as the question did not ask for a sketch. Check the solution by substituting both answers back into the original equation Step 6: Even though the question did not ask for a sketch.
Worked example 4: Determine the restrictions The restrictions are the values for b that would result in the denominator being equal to 0. Multiply each term in the equation by the lowest common denominator and simplify Chapter 2. Check the solution by substituting both answers back into the original equation Step 4: Expand the brackets and simplify 36 2.
Worked example 5: Square both sides of the equation Before we square both sides of the equation. To test the answers.
Squaring an expression changes negative values to positives and can therefore introduce invalid answers into the solution. Step 5: Therefore it is very important to check that the answers obtained are valid.
Exercise 2 — 1: Solution by factorisation Solve the following quadratic equations by factorisation. Completing the square. Completing the square Can you solve each equation using two different methods? Answers may be left in surd form. This simple factorisation leads to another technique for solving quadratic equations known as completing the square.
Use the previous examples as a hint and try to create a difference of two squares. Method 2: We cannot easily factorise this expression. We compare the two equations and notice that only the constant terms are different.
We can create a perfect square by adding and subtracting the same amount to the original equation. Take square roots on both sides of the equation to solve for x. Always remember to include both a positive and a negative answer when taking the square root. Write the left hand side as a difference of two squares. Write the trinomial as a perfect square 40 2. Factorise the equation in terms of a difference of squares and solve for x. In the example above. Method 1: When taking a square root always remember that there is a positive and negative answer.
These roots are rational because 36 is a perfect square. When taking a square root there is a positive and a negative answer. Step 6: Solve for x 42 2. Worked example 7: Therefore divide the entire equation by 2: Solution by completing the square 1. Solve the following equations by completing the square: Quadratic formula. Solve for k in terms of a: The quadratic formula provides an easy and fast way to solve quadratic equations. The method of completing the square provides a way to derive a formula that can be used to solve any quadratic equation.
We then add and subtract so that the equation remains true. Solve for y in terms of p. Check whether the expression can be factorised The expression cannot be factorised. If the expression under the square root sign is less than zero. Finding the roots To determine the roots of f x. This means that the graph of the quadratic function has no x-intercepts.
Solution by the quadratic formula Solve the following using the quadratic formula. This makes the equation simpler and much easier to solve. Determine the restrictions for x The restrictions are the values for x that would result in the denominator being equal to 0. Determine the restrictions for k The restrictions are the values for k that would result in the denominator being equal to 0.
Use values obtained for k to solve for the original variable x 48 2. Exercise 2 — 4: Solve the following quadratic equations by substitution: Multiply by 2: Finding the equation. Given the roots. Find p and the other root. Exercise 2 — 5: Finding the equation 1. Exercise 2 — 6: Mixed exercises Solve the following quadratic equations by either factorisation. Use the quadratic formula to determine the roots of the quadratic equations given below and take special note of: Nature of roots.
The expression under the square root. The discriminant determines the nature of the roots of a quadratic equation. Can you make a conjecture about the relationship between the discriminant and the roots of quadratic equations?
Choose the appropriate words from the table to describe the roots obtained for the equations above. For real roots. Interpret the question For roots to be real and irrational. Note that the graph does not form part of the answer and is included for illustration purposes only. Check both answers by substituting back into the original equation 56 2. Interpret the question For roots to be real and equal. Interpret the question For roots to be real.
Determine the nature of the roots for each of the following equations: Exercise 2 — 7: From past papers 1. HG] 3. HG] 4. Consider the equation: HG] 7. HG] 6. If b and c can take on only the values 1. HG] 5. These are called the critical values of the inequality and they are used to complete a table of signs.
Quadratic inequalities. Complete a table of signs We must determine where each factor of the inequality is positive and negative on the number line: An inequality can therefore be solved graphically using a graph or algebraically using a table of signs to determine where the function is positive and negative.
A rough sketch of the graph The graph below does not form part of the answer and is included for illustration purposes only. A graph of the quadratic helps us determine the answer to the inequality. So we can write the same inequality in different ways and still get the same answer. Remember that if we multiply or divide an inequality by a negative number.
When working with an inequality in which the variable is in the denominator, a different approach is needed. Always remember to check for restrictions.
Solving the inequality It is very important to recognise that we cannot use the same method as above to solve the inequality. If we multiply or divide an inequality by a negative number, then the inequality sign changes direction. We must rather simplify the inequality to have a lowest common denominator and use a table of signs to determine the values that satisfy the inequality. Complete a table of signs.
Hm... Are You a Human?
Step 7: Solve the following inequalities and show each answer on a number line: Draw a sketch of the following inequalities and solve for x: Simultaneous linear equations can be solved using three different methods: For solving systems of simultaneous equations with linear and non-linear equations, we mostly use the substitution method. Graphical solution is useful for showing where the two equations intersect.
In general, to solve for the values of n unknown variables requires a system of n independent equations. By doing this we reduce the number of equations and the number of variables by one. These are the coordinate pairs for the points of intersection as shown below. Simultaneous equations. Make y the subject of the second equation. Check that the two points satisfy both original equations Step 5: These are the coordinate 2 4 pairs for the points of intersection as shown below.
Check that the two points satisfy both original equations. These are the coordinate 2 pairs for the points of intersection as shown below. Solve the following systems of equations algebraically. Leave your answer in surd form, where appropriate. Solve the following systems of equations graphically. Check your solutions by also solving algebraically. Solving word problems requires using mathematical language to describe real-life contexts. Problem-solving strategies are often used in the natural sciences and engineering disciplines such as physics, biology, and electrical engineering but also in the social sciences such as economics, sociology and political science.
To solve word problems we need to write a set of equations that describes the problem mathematically. Examples of real-world problem solving applications are:. How long will it take both of them to paint a room together? Set up a system of equations. Liboko builds a rectangular storeroom.
It rains half as much in July as it does in December. Find the integers if their total is Tlali can paint a room in 2 hours. Mohato and Lindiwe both have colds. Problem solving strategy 1. Assign variables to the unknown quantities. Solve for the variables using substitution. The difference of two numbers is 10 and the sum of their squares is If it rains y mm in July. Translate the words into algebraic expressions by rewriting the given information in terms of the variables.
EMBFY 4. Find the two numbers. If the diagonal of the room is and the perimeter is 80 m. How old is Bongani? Check the solution. If Lindiwe sneezes x times. Simple word problems Write an equation that describes the following real-world situations mathematically: Read the problem carefully. Zane can paint a room in 4 hours. What is the question and what do we need to solve for?
The product of two integers is If a cup of cappuccino costs R 3. Erica paid R Word problems. Determine the amount of the proposed increase.
If this is done then a single membership would cost of a family 7 membership. The gym is considering increasing all membership fees by 5 the same amount.
Identify the unknown quantity and assign a variable Let the amount of the proposed increase be x. Multiply the equation through by the lowest common denominator and simplify 1. If both taps are opened. Working on its own. Convert all units of time to be the same First we must convert 1 hour. If he wants an average score of If the length is twice the breadth.
Exercise 2 — Kevin has played a few games of ten-pin bowling. Then use the equation to answer the following questions: If Sheila were to walk half as fast as she is currently walking. Kevin scored 80 more than in the second game. Distance is measured in meters and time is measured in seconds. Find the equation that describes the relationship between time and distance.
Tsilatsila builds a fence around his rectangular vegetable garden of 8 m2. The table below lists the times that Sheila takes to walk the given distances. When an object is dropped or thrown downward. In the third game. A wooden block is made as shown in the diagram. The total surface area of the block is cm2. The length of the block is y. The ends are right-angled triangles having sides 3x. Determine the correct equation that was on the board.
Solve for x: Find one set of 3 8. Give your answer correct to two decimal places. Nature of roots Roots are non-real Roots are real and equal Roots are real and unequal: End of chapter exercises 1.
Saskia and Sven copy it down incorrectly. Solve for x in terms of p by completing the square: Calculate the value of p. Abdoul got stuck along the way. His attempt is shown below: After doing some research. Solve the following systems of equations graphically: A stone is thrown vertically upwards and its height in metres above the ground at time t in seconds is given by: Now deduce that the total cost.
Solve the following quadratic equations by either factorisation. Leave your answer in surd form. Determine the value of k and the other root. Solve for y: Solve for t: You can check this by substituting values for n: Different types of series are studied in Grade The common or constant difference d is the difference between any two consecutive terms in a linear sequence.
In this chapter.
Successive or consecutive terms are terms that directly follow one after another in a sequence. The general term can be used to calculate any term in the sequence. In Grade 11 we study sequences only. A mathematical expression that describes the sequence and that generates any term in the pattern by substituting different values for n.
A statement. EMBG3 A sequence does not have to follow a pattern but when it does. Linear sequence EMBG4 A sequence of numbers in which there is a common difference d between any term and the term before it is called a linear sequence. Determine the common difference To calculate the common difference. Chapter 3. Number patterns This pattern can also be expressed in words: Term value Tn 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 Pattern number n Notice that the position numbers n can be positive integers only.
Drawing a graph of the pattern We can also represent this pattern graphically. Linear sequences 1. Calculate how many desks are in the ninth row. T15 and T The general term is given for each sequence below. Write down the next three terms in each of the following sequences: Calculate the missing terms. Study the dotted-tile pattern shown and answer the following questions..
Determine the value of n.. Determine the value of n. Quadratic sequences.. Quadratic sequences 1. Now study the number of blank tiles tiles without dots and answer the following questions: Determine the number of blank tiles. This sequence has a constant difference between consecutive terms. In other words. Any sequence that has a common second difference is a quadratic sequence. General case If the sequence is quadratic.
Exercise 3 — 2: Quadratic sequence A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant. Determine the second difference between the terms for the following sequences: Consider the following example: Quadratic sequences. Continuing the sequence. So the sequence will be: Finding the next two terms in the sequence The next two terms will be: Is this a linear or a quadratic sequence?
Plot a graph of Tn vs n. Determine the general term Tn for the sequence. Plot a graph of Tn vs n 3 2 Use the general term for the sequence. We can. So if there are four different teams in a group. How many matches would be played if there are 6 teams in each group? D and E. Determine the number of matches played if there are 4 teams in a group Let the teams from four different countries be A. How many matches would be played if there are 5 teams in each group?
AE BC. AD BC. BE CD. C and D. Each country in a group must play every other country in the group once. E and F. Determine the number of matches played if there are 6 teams in a group Let the teams from six different countries be A. Determine the general formula of the sequence. BF CD. CF DE. Consider the sequence We examine the sequence to determine if it is linear or quadratic: AF BC. Calculate the common second difference for each of the following quadratic sequences: Exercise 3 — 3: Determine whether each of the following sequences is: Given 3.
Is she correct? For each of the following patterns. Given the following sequence: Is he correct? Given the pattern: For each of the following sequences. Cubes of volume 1 cm3 are stacked on top of each other to form a tower: Given the following pattern of blocks: A quadratic sequence has a second term equal to 1.
D 3 d Hence. Challenge question: There are 15 schools competing in the U16 girls hockey championship and every team must play two matches — one home match and one away match. In this chapter we deal with the equation of a straight line. The gradient of a line is determined by the ratio of vertical change to horizontal change. Analytical geometry Determine the length of the line segment P Q.
NCERT Solutions For Class 11 Maths
Determine the mid-point T x. Points on a straight line The diagram shows points P x1. Show that the line passing through R 1. Assign variables to the coordinates of the given points Let the coordinates of P be x1.
Write down the mid-point formula and substitute the values T x. D C A — Both pairs of opposite sides are equal and parallel. Chapter 4. Assign values to x1. Determine D x. From the sketch we expect that point D will lie below C. B and C: Consider the given points A.
Alternative method: Exercise 4 — 1: Revision 1. Prove that the line P Q. Step 8: Determine the gradient of the line AB if: Calculate the coordinates of the mid-point P x.
Given Q 4. Determine the values of x and y. Determine the length of the line segment between the following points: The different forms are used depending on the information provided in the problem: Explain your answer.
Draw a sketch and determine the coordinates of N x. Given any two points x1. Equation of a line. The two-point form of the straight line equation Determine the equation of the straight line passing through the points: If we are given two points on a straight line.
Exercise 4 — 3: Gradient—point form of a straight line equation Determine the equation of the straight line: Worked example 6: We solve for the two unknowns m and c using simultaneous equations — using the methods of substitution or elimination. Exercise 4 — 4: The gradient—intercept form of a straight line equation Determine the equation of the straight line: Inclination of a line.
This is called the angle of inclination of a straight line. We know that gradient is the ratio of a change in the y -direction to a change in the x-direction: We notice that if the gradient changes.
If we are calculating the angle of inclination for a line with a negative gradient. Determine the gradient correct to 1 decimal place of each of the following straight lines. Determine the angle of inclination correct to 1 decimal place for each of the following: Angle of inclination 1.
Draw a sketch y 2 2. Worked example 8: Worked example 9: Determine the gradient and angle of inclination of the line through M and N Chapter 4. Exercise 4 — 6: Inclination of a straight line 1. Determine the angle of inclination for each of the following: What do you notice about mP Q and mRS?
Parallel lines. Parallel lines 1. Complete the sentence: Determine the equations of the straight lines P Q and RS.
Another method of determining the equation of a straight line is to be given a point on the unknown line. Write the equation in gradient—intercept form We write the given equation in gradient—intercept form and determine the value of m.
Determine the equation of the line CD which passes through the point C 2. Determine whether or not the following two lines are parallel: Determine the equation of the straight line that passes through the point 1.
Exercise 4 — 7: Describe the relationship between the lines AB and CD. If not. Perpendicular lines 1. Perpendicular lines. Determine the equation of the straight line AB and the line CD. What do you notice about these products? Deriving the formula: Write the equation in standard form Let the gradient of the unknown line be m1 and the given gradient be m2.
We write the given equation in gradient—intercept form and determine the value of m2. Determine the unknown gradient Since we are given that the two lines are perpendicular. Use the given angle of inclination to determine gradient Let the gradient of the unknown line be m1 and let the given gradient be m2.
Calculate whether or not the following two lines are perpendicular: Determine the equation of the straight line that passes through the point 2. Determine the equation of the straight line that passes through the point 3. Determine the angle of inclination of the following lines: P R intersects the x-axis at S. Determine the following: Exercise 4 — 9: Determine the equation of the line: Consider the sketch above. The following points are given: Given points S 2.
F GH is an isosceles triangle. Functions can be one-to-one relations or many-to-one relations. Functions allow us to visualise relationships in the form of graphs. As a gets closer to 0. As the value of a becomes smaller. The turning point of f x is above the x-axis.
Quadratic functions. Every element in the domain maps to only one element in the range. A many-to-one relation associates two or more values of the independent variable with a single value of the dependent variable.
As the value of a becomes larger. The turning point of f x is below the x-axis. Functions Exercise 5 — 1: On separate axes. Consider the three functions given below and answer the questions that follow: The effects of a. On the same system of axes.
The effect of q is a vertical shift. Discuss the similarities and differences. The value of a affects the shape of the graph. Describe any differences. The range of f x depends on whether the value for a is positive or negative. Determine the range The range of g x can be calculated from: Every point on the y -axis has an x-coordinate of 0. The x-intercept: Every point on the x-axis has a y -coordinate of 0.
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The next paragraphs provide information on each.Types of trends in groups and periods, different terminology involved for the Systems of unit. When taking a square root always remember that there is a positive and negative answer. A surd is a radical which results in an irrational number. The great writers and poets of the world have the ability to draw on words and put them together in ways that can tell beautiful or inspiring stories. It is important to see a relationship between the change in dimensions and the resulting change in surface area and volume.
Use the x-intercept to determine p Substitute 2. There are many real world applications that require exponents.
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