GRADE 11 PHYSICAL SCIENCES Siyavula and Free High School Science Text contributors . For off-line reading on your PC, tablet, iPad and site you can download a digital erned by the laws of physics but you probably aren't. Siyavula's open Physical Sciences Grade 11 textbook. Learner's book and teacher's guide (PDF). Learner's book (ePub). Reference material. Physics. Download our open textbooks in different formats to use them in the way that suits you. Click on Mathematics Grade 11 Mathematical Literacy Grade
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Siyavula's open Physical Sciences Grade 11 textbook, chapter 2 on Newton'S Laws. GRADE 11 PHYSICAL SCIENCES. VERSION 1 CAPS. WRITTEN BY Siyavula and Free High School Science Text contributors . websites and download the books. .. of the posts at the physics level an advanced high school student could. Siyavula Textbooks. Grade 4 – 6 Natural Sciences and Technology workbooks: Learning Grade 10 – 12 Mathematics and Physical Sciences textbooks They have everything you expect from your regular printed school textbook, but come For a start, you can download or read them on-line on your mobile phone.
Francois Toerien; Ren Toerien; Dr. Dawn Webber; Michelle Wen; Dr. Alexander Wetzler; Dr. Spencer Wheaton; Vivian White; Dr.
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Well structured, impactful Corporate Social Investment CSI has the ability to contribute positively to nation building and drive positive change in the communities. MMIs commitment to social investment means that we are constantly looking for ways in which we can assist some of South Africas most vulnerable citizens to expand their horizons and gain greater access to lifes opportunities.
This means that we do not view social investment as a nice to have or as an exercise in marketing or sponsorship but rather as a critical part of our contribution to society. The merger between Metropolitan and Momentum was lauded for the complementary fit between two companies. This complementary fit is also evident in the focus areas of CSI programmes where Metropolitan and Momentum together cover and support the most important sectors and where the greatest need is in terms of social participation.
Metropolitan continues to make a difference in making sure that HIV AIDS moves away from being a death sentence to a manageable disease. Metropolitans other focus area is education which remains the key to economic prosperity for our country.
Momentums focus on persons with disabilities ensures that this community is included and allowed to make their contribution to society. Orphaned and vulnerable children are another focus area for Momentum and projects supported ensure that children are allowed to grow up safely, to assume their role along with other children in inheriting a prosperous future.
Visit the Everything Maths and Everything Science mobi sites at: m. 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, parents, universities or bursary institutions what you have done during the year.
Mathematics is the language that nature speaks to us in. As we learn to understand and speak this language, we can discover many of natures secrets. Just as understanding someones language is necessary to learn more about them, mathematics is required to learn about all aspects of the world whether it is physical sciences, life sciences or even finance and economics.
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. In a similar way, one can draw on mathematics to explain and create new things. Many of the modern technologies that have enriched our lives are greatly dependent on mathematics. And just as words were not created specifically 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.
Determine the length of the line segment P Q. 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. Exercise 5 — 2: Domain and range Give the domain and range for each of the following functions: Chapter 5. Exercise 5 — 3: Intercepts Determine the x. Alternative form for quadratic equations: The turning point is p. The minimum value of f x is q. The maximum value of f x is q. Exercise 5 — 4: Turning points Determine the turning point of each of the following: Axis of symmetry 1.
Determine the axis of symmetry of each of the following: Write down the equation of a parabola where the y -axis is the axis of symmetry. Mark the intercepts. State the domain and range of the function. Give the domain and range of the function. Determine the intercepts. State the domain and range Domain: This gives the points 1. Step 1: Use calculations and sketches to help explain your reasoning. Discuss the two different answers and decide which one is correct. Shifting the equation of a parabola Carl and Eric are doing their Mathematics homework and decide to check each others answers.
Work together in pairs. A shift to the right means moving in the positive x direction. Homework question: Determine the new equation of the shifted parabola 1. If the parabola is shifted 1 unit to the right. Writing an equation of a shifted parabola The parabola is shifted horizontally: The parabola is shifted vertically: The parabola is shifted 3 units down. The parabola is shifted 1 unit to the right. If the parabola is shifted 3 units down.
Draw a sketch of each of the following graphs: Sketch graphs of the following functions and determine: Draw the following graphs on the same system of axes: Exercise 5 — 6: Sketching parabolas 1. Finding the equation of a parabola from the graph If the intercepts are given. Exercise 5 — 7: Finding the equation Determine the equations of the following graphs. The average gradient between any two points on a curve is the gradient of the straight line passing through the two points.
Average gradient. A 5. What happens to the average gradient as A moves away from B? The gradient at a point on a curve is the gradient of the tangent to the curve at the given point. At the point where A and C overlap. What happens to the average gradient as A moves towards B?
This line is known as a tangent to the curve. What is the average gradient when A overlaps with B? Find the average gradient between two points P a.
Determine the average gradient between P 2. Explain what happens to the average gradient if Q moves closer to P. Given a curve f x with two points P and Q with P a. We can write the equation for average gradient in another form.
Calculate the average gradient between P 2. When the point Q overlaps with the point P. The average gradient between P and Q is: Determine the gradient of the curve at point A.
Draw a sketch of the function and determine the average gradient between the points A. Exercise 5 — 9: Consider the following hyperbolic functions: Hyperbolic functions. Complete the table to summarise the properties of the hyperbolic function: The value of a affects the shape of the graph and its position on the Cartesian plane.
The value of q also affects the horizontal asymptotes. The value of p also affects the vertical asymptote. Discovering the characteristics For functions of the general form: Exercise 5 — Domain and range Determine the domain and range for each of the following functions: Asymptotes Determine the asymptotes for each of the following functions: The asymptotes indicate the values of x for which the function does not exist.
For the standard and shifted hyperbolic function. Complete the following for f x and g x: Compare f x and g x and also their axes of symmetry. Axes of symmetry 1. What do you notice? In order to sketch graphs of functions of the form. The vertical asymptote is also shifted 2 units to the right. Determine the domain and range Domain: Determine the value of a To determine the value of a we substitute a point on the graph.
Examine the graph and deduce the sign of a We notice that the graph lies in the second and fourth quadrants. Sketching graphs 1.
Siyavula textbooks: Grade 11 Physical Science
Illustrate this on your graph. Draw the graphs of the following functions and indicate: Exponential functions. The value of a affects the shape of the graph and its position relative to the horizontal asymptote. Determine the asymptote The asymptote of g x can be calculated as: Asymptote Give the asymptote for each of the following functions: Mark the intercept s and asymptote. We need to solve for p and q. Use the x-intercept to determine p Substitute 2. Mixed exercises 1.
Label this function as j x. For the diagrams shown below. On separate systems of axes. Find the equation for each of the functions shown below: The sine function. Revision On separate axes. For each function also determine the following: Use your sketches of the functions above to complete the following table: The effects of k on a sine graph 1. If k is negative. Negative angles: For each function determine the following: For each function. To draw a graph of the above equation.
Sketch the following graphs on separate axes: The sine function 1. The cosine function. For each function in the previous problem determine the following: The effects of k on a cosine graph 1. Calculating the period: For each graph determine: The effects of p on a cosine graph 1.
Determine the minimum turning point At the minimum turning point. The cosine function 1. Determine the value of p. Determine the value of a. Two girls are given the following graph: The tangent function. The effects of p on a tangent graph 1. Determine the asymptotes The standard tangent graph. The tangent function 1. Determine the equation for each of the following: Give one of the solutions. Indicate the turning points and intercepts on the diagram.
Hyperbolic functions: Standard form: Exponential functions: Average gradient: Parabolic functions: Tangent functions: Shifted form: Cosine functions: Sine functions: What is the new equation then? Give the equation of the new graph originating if: The coordinates of the turning point are 3. Suppose the hyperbola is shifted 3 units to the right and 1 unit down.
What are the coordinates of the turning point of the shifted parabola?
The columns in the table below give the y -values for the following functions: Match each function to the correct column. Using your knowledge of the effects of p and k draw a rough sketch of the following graphs without a table of values. Trigonometry Solving equations Worked example 1: Identify the opposite and adjacent sides and the hypotenuse Step 2: Determine the value of b Always try to use the information that is given for calculations and not answers that you have worked out in case you have made an error.
Finding an angle Worked example 2: In the calculation below. Determine the angle of inclination of the string correct to one decimal place.
Determine the following angles correct to one decimal place: In ABC. Chapter 6.
Siyavula textbooks: Grade 10 Physical Science
Kite 22 The perpendicular 3. Given the diagram below. Will the ladder reach the window? Simplify the following without using a calculator: The height of an open window is 9 m from the ground. This enables us to solve equations and also to prove other identities. Quotient identity Investigation: Quotient identity 1. Examine the last two rows of the table and make a conjecture. Complete the table without using a calculator. Trigonometric identities allow us to simplify a given expression so that it contains sine and cosine ratios only.
Square identity 1. Draw a sketch and prove your conjecture in general terms. Make a conjecture. Trigonometric identities.
Use a calculator to complete the following table: Here are some useful tips for proving identities: Write the expression in terms of sine and cosine only We use the square and quotient identities to write the given expression in terms of sine and cosine and then simplify as far as possible.
Note restrictions When working with fractions.
Simplify the left-hand side This is not an equation that needs to be solved. We are required to show that one side of the equation is equal to the other. We can choose either of the two sides to simplify.
Reduce the following to one trigonometric ratio: Prove the following identities and state restrictions where appropriate: Exercise 6 — 2: Trigonometric identities 1. Deriving reduction formulae Investigation: Complete the following reduction formulae: Reduction formula. Determine the value of the following expressions without using a calculator: Write the following in terms of a single trigonometric ratio: We can therefore measure negative angles by rotating in a clockwise direction.
From working with functions. The periodicity of the trigonometric graphs shows this clearly. If k is any integer. Simplify the expression using reduction formulae and special angles 6.
Using reduction formula 1. Sine and cosine are known as co-functions. A and B are complementary angles. The function value of an angle is equal to the co-function of its complement. Simplify the expression using reduction formulae and co-functions Chapter 6. Use the CAST diagram to check in which quadrants the trigonometric ratios are positive and negative.
Co-functions 1. Reduction formulae 1. For convenience. Reduction formulae and co-functions: Write A and B as a single trigonometric ratio: Prove that the following identity is true and state any restrictions: Determine the value of the following. Write the following as a function of an acute angle: Trigonometric equations.
Using reduction formulae. If no interval is given. The periodic nature of trigonometric functions means that there are many values that satisfy a given equation. Determine the reference angle use a positive value. Exercise 6 — 7: Solving trigonometric equations 1. If we do not restrict the solution. Use the CAST diagram to determine where the function is positive or negative depending on the given equation.
Check answers using a calculator. This solution is also correct. This solution is correct. In the second quadrant: Make a substitution To solve this equation. This is because of the periodic nature of the tangent function. Therefore we need only determine the solution: In the third quadrant: Use the CAST diagram to determine the correct quadrants Since the original equation equates a sine and cosine function. Find the general solution for each equation.
Exercise 6 — 8: General solution 1. Find the general solution for each of the following equations: Determine the general solution for each of the following: Exercise 6 — 9: The area rule Investigation: The area rule 1. Consider ABC: Use your results to write a general formula for determining the area of P r q Q p R Chapter 6. In ABC: Use the area rule to calculate the area of ABC Notice that we do not know the length of side a and must therefore choose the form of the area rule that does not include this side of the triangle.
Exercise 6 — The area rule In any P QR: Find the area of P QR given: We now expand the application of the trigonometric ratios to triangles that do not have a right angle: Use your results to write a general formula for the sine rule given P P QR: In ACF: The sine rule In any ABC: Use the sine ratio to express the angles in the triangle in terms of the length of the sides In M SN: This means that we can draw two different triangles with the given dimensions.
Draw In ABC. We call this the ambiguous case because there are two ways of interpreting the given information and it is not certain which is the required solution. In M SP: The two lighthouses are 0.
The lighthouses tell how close a boat is by taking bearings to the boat a bearing is an angle measured clockwise from north. Calculate how far the boat is from each lighthouse. These bearings are shown on the diagram below.
Since we know the distance between the lighthouses and we have two angles we can use trigonometry 6. Solve for unknown angle using the sine rule In ABC: A AB and BC. K Chapter 6. Find the lengths of the sides 2.
Sine rule 1. Find the length of the side 3. Find all the unknown sides and angles of the following triangles: Find the length of the side m. In RST. We therefore investigate the cosine rule: In ABD. Determine BC.
Can you determine BC? Determine CB: A Think you got it? In CHA: Use your results to write a general formula for the cosine rule given P P QR: In CHB: Since h2 is common to both equations we can write: Substituting back we get: How to determine which rule to use: Area rule: It is very important: Solve the following triangles that is. Determine the largest angle in: Sine rule: Cosine rule: The cosine rule 1.
Q a the distance QR b the shortest distance from the lighthouse to the line joining the two ships P Q. Q is a ship at a point 10 km due south of another ship P. W XY Z is a trapezium. The direct distance between Cape Town and Johannesburg airports is km. At this point the plane is km from Johannesburg. From a point Y which is equidistant from X and Z. Find the area of the shaded triangle in terms of x.
Find the area of W XY Z to two decimal places: However the distance cannot be determined directly as a ridge lies between the two points. A surveyor is trying to determine the distance between points X and Z. Determine the value of the following expression without using a calculator: Write the following as a single trigonometric ratio: Without the use of a calculator.
Prove the following identities: Given the equation: At point N on the tower. Find the general solution for the following equations: Prove that In ABC. Collins wants to pave his trapezium-shaped backyard.
We also examine different combinations of geometric objects and calculate areas and volumes in a variety of real-life contexts. Area of a polygon. In ABF: Measurement Banele also uses diagonals of length 60 cm and 1 m.
Vuyo and Banele are having a competition to see who can build the best kite using balsa wood a lightweight wood and paper. Vuyo decides to make his kite with one diagonal 1 m long and the other diagonal 60 cm long. The intersection of the two diagonals cuts the longer diagonal in the ratio 1: O is the centre of the bigger semi-circle with a radius of 10 units. Exercise 7 — 1: Area of a polygon 1. Two smaller semi-circles are inscribed into the bigger one.
She notices that the dimensions of her desk are in the same proportion as the dimensions of her textbook. Which do you think is the better design? Motivate your answer. Chapter 7. Examples of right prisms are given below: A cylinder is another type of right prism which has a circle as its base. Surface area of prisms and cylinders EMBHX Surface area is the total area of the exposed or outer surfaces of a prism.
A solid that is unfolded like this is called a net. The base and top surface are the same shape and size. In order to calculate the surface area of the prism. A triangular prism has a triangle as its base. To calculate the surface area of the prism. When a prism is unfolded into a net.
This is easier to understand if we imagine the prism to be a cardboard box that we can unfold. Below are examples of right prisms that have been unfolded into nets. A triangular prism unfolded into a net is made up of two triangles and three rectangles.
The sum of the lengths of the rectangles is equal to the perimeter of the triangles.
A cylinder unfolded into a net is made up of two identical circles and a rectangle with length equal to the circumference of the circles.
Right prisms and cylinders. A rectangular prism unfolded into a net is made up of six rectangles. A cube unfolded into a net is made up of six identical squares. Calculate the area of the wrapping paper needed to cover the entire box assume no overlapping at the corners.
Determine if this same sheet of wrapping paper would be enough to cover the tin of biscuits. He discovers a full 2 tin of green paint in his garage and decides to paint the tank not the base. Exercise 7 — 2: Calculating surface area 1. Gordon downloads a cylindrical water tank to catch rain water off his roof. Dimensions of the tank: A popular chocolate container is an equilateral right triangular prism with sides of 34 mm.
Calculate the surface area of the box to the nearest square centimetre. The box is mm long. If he uses ml to cover 1 m2. It is measured in cubic units. The volume of a right prism is simply calculated by multiplying the area of the base of a solid by the height of the solid. Will the plastic jug hold 5 of water? Determine the volume of the plastic jug The diameter of the jug is mm. The jug has a diameter of mm and a height of 28 cm. A cylindrical water tank is positioned next to the house so that the rain on the roof runs into the tank.
The diameter of the tank is cm and the height is 2. Exercise 7 — 3: Calculating volume 1. The length of a side of a hexagonal sweet tin is 8 cm and its height is equal to half of the side length. Right pyramids. In other words the sides are not perpendicular to the base. Spheres are solids that are perfectly round and look the same from any direction. Surface area of pyramids. The triangular pyramid and square pyramid take their names from the shape of their base.
Cones are similar to pyramids except that their bases are circles instead of polygons. Calculate the total volume of the building. The height of the building wall is 17 m and the diameter is 26 m. Calculate the total surface area of the building. Finding surface area and volume 1. An ice-cream cone has a diameter of Multiplying a dimension by a constant factor. Calculate the length of arc P. Consider a rectangular prism of dimensions l.
Determine the length of arc M. These relationships make it simpler to calculate the new volume or surface area of an object when its dimensions are scaled up or down. R is the length from the tip of the cone to its perimeter. The new surface area and volume can be calculated by using the formulae from the preceding section.
It is important to see a relationship between the change in dimensions and the resulting change in surface area and volume. Determine the value of R. Below we multiply one. The dad draws up the plans for the new square room of length k metres. Calculate and compare First calculate the area of the square room in the original plan: To achieve this: Show all calculations. The mum looks at the plans and decides that the area of the room needs to be doubled.
The son suggests doubling only the width of the room: Complete the following sentences: Exercise 7 — 5: The effects of k 1. The dad suggests adding 2 m to the length of the sides of the room: Determine the new dimensions of the pool in terms of W remember that the pool must be a cube. Volume is the three dimensional space occupied by an object. Area formulae: Surface area is the total area of the exposed or outer surfaces of a prism. The municipality intends building a swimming pool of volume W 3 cubic metres.The ends are right-angled triangles having sides 3x.
Parallel lines 1. Determine the domain and range Domain: Translate the words into algebraic expressions by rewriting the given information in terms of the variables. In M SP: When an object is dropped or thrown downward.
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