Grade 10 Tutorials - Maths Excellence

1 Grade 10 Tutorials . LO Topic Page 1 Number patterns and sequences 3. 2 Functions and graphs 6. 2 Algebra and equations 8. 1 Finance 12. 3 Analytical Geometry 14. 3 Transformation 16. 3 Trig / Mensuration 21. 4 Data handling 26. Grade 10 -2- Tutorials Grade 10 Tutorial Number Patterns and Sequences Question 1. Add the next three terms to each number pattern and explain how you calculated these terms: 2; 7; 12; 17; 10; 8; 6; 4; . 1 3 1`. 1 ; 2; 2 ; 3 ;.. 1; 3; 9; 27; . 4 4 2. 1; 1; 2; 3; 5; 8; 13; . Question 2. Write down the next three terms and the general (or nth term) of each pattern: 2; 4; 6; 8; 1; 7; 13; 19; . 1; 4; 9; 16; 25; 21; 17; 13; . 1 1 1 1. x 1; 2 x 2; 3 x 3; 4 x 4;.. ; ; ; ;.. 2 3 4 5. 1 3 3 1 1. ; 1; ; 2;.. 3 ; 3 ; 3 ; 3;.. 2 2 4 2 4.

2 Question 3. How many blocks in the next T? How many blocks in the nth T? Which T has 69 blocks? Grade 10 -3- Tutorials .. How many faces in the next pattern? How many faces in the nth pattern? In which pattern are there 84 faces? How many lines in the pattern with 4 triangles? How many lines are needed for n triangles? How many triangles are formed with 46 lines? Question 4. Figure 1 Figure 2 Figure 3.. How many flowers would be used in the 4th figure? How many flowers would be used in the 10th figure? How many flowers would be used in the n-th figure? Grade 10 -4- Tutorials Question 5. When two people meet, they shake hands, resulting in 1 handshake. If three people met and all shook hands, there would be three handshakes. How many handshakes would there be if 4 people met and shook hands?

3 How many handshakes would there be if 5 people met? Can you generalize this result? Question 6. Your mother has chosen a base pattern for your bathroom floor. The figure below illustrates the pattern she chose. As shown, the pattern is made up of 16 squares, 8 of which are shaded and 8 which are not. Step 1: Base Pattern Duplicates of the same pattern are then added to create Step 2. How many base patterns were added to the original in order to complete Step 2? How many shaded unit squares would you need for Step 2? Each step is accomplished by surrounding the existing figure with copies of the base pattern. How many of the base patterns need to be added to complete Step 3? How many shaded unit squares would you need for Step 3? How many shaded unit squares would you need for Step 6?

4 Write a generalization or rule for determining the number of shaded unit squares that are added in Step n. Grade 10 -5- Tutorials Grade 10 Tutorial Functions and Graphs Question 1. 1. If f ( x) = 2 x and g ( x) = and h( x) = x 2 , answer the following questions;. x Determine the values of the following;. f ( 1) f ( 2). x if f ( x) = 0 g ( 1). g ( 2) x if g ( x) = 2. h( 2) h ( 2). Describe the type of function that is defined in each case. Draw a sketch graph of each of the functions showing all critical points, asymptotes, axes of symmetry and intercepts with the axes. You can use the values in question to assist you if necessary. Each function must be sketched on a separate set of axes. Determine the domain and range of the functions f , g and h.

5 Question 2. Consider the functions s ( x) = x 2 9 and t ( x) = 2 x 6. Sketch the graphs of s and t on the same system of axes, showing ALL. intercepts with the axes and relevant turning points Use your sketch to find the values of x if;. s ( x ) = t ( x). s ( x) > 0. Write down the equation of q if q ( x) results from shifting s ( x) 2 units up. Grade 10 -6- Tutorials Question 3. k Sketched below are the functions g ( x) = b x + c and h( x) = and A, the point of intersection, is ( 1 ; 1 ). x g 2. Find the values of k , c and b h What is the equation of the A(1;1). 1 asymptote of g What is the range of g -2 2. What is the equation of f if f ( x). -1. is the reflection of g ( x) in the y-axis -2. Question 4. Below is a sketch of f ( x) = cos x + q and g ( x) = a sin x 2 y g 1.

6 X -90 -60 -30 0 30 60 90 120 150 180. -1 f -2. Write down the amplitude of f and g What is the range of f What is the period of f Determine the values of a and q What is the equation of h if h( x) is the reflection of g ( x) in the x-axis? Grade 10 -7- Tutorials Grade 10 Tutorial Algebra and Equations Question 1. Use the laws of exponents to simplify the following expressions: x 6 x 2 x 2 (3 p )q 3 p 2. ( x 2 y ) 2. 60 12 2 33 ( xy 3 ) 1. When working with computers, data is measured in powers of 2 as given below: 1 Kilobyte (KB) = 210 bytes , 1 Megabyte (MB) = 210 KB , I Gigabyte (GB) = 210 MB. How many bytes are there in a Megabyte? Give your answer as a power of 2. A memory stick holds 512MB of data. How many bytes is this? Express your answer as a power of 2.

7 If a digital photograph contains 524 288 bytes of data, how many photographs can be stored on a CD? Work in powers of 2 and show all your work. Question 2. Remove brackets and simplify the following expressions: 5a (a 3) 2 x( x + 4) (3 x + 1). (4m 1)(3m + 2) (2 x + 4 y ) 2. 3( x 2) 2 (6 p + 5)(6 p 5). ( y + 7)(5 y 2 + y 3). Question 3. Factorise the following expressions: 15 xy 3 y 1. 4 m 2 25. 3x 2 8 x + 4 2r 2 11r 6. 6 s ( r + 2) 2( r + 2) x + 5 + qx + 5q kx + ky x y Grade 10 -8- Tutorials Question 4. Simplify the following expressions: 24a 3b 2 18 x 5 2 y 3 3x 4 y 2 3 1. +. 6 a 2b 4 xy 6 x 2 2x2 10m 5m 2. 3 x 5x 2 a b a +b +. 5x 3x a 2a Question 5. Check, by substitution, whether or not x = 1 is a solution to each of the following equations: (Show all your work).

8 7 + 2( x 1) = 3 4 x ( x 1)( x + 1) = 0. 4x 1. 3 x( x 1) 2 = 0 = (4 x). 3 3. 8x = 1. 8. Question 6. Solve for x in each of the following equations: 7 x 8 = 27 3 x 18 = 3( x + 7). x 2x 4 2x +1 x 1 5. = =. 3 5 5 8 3 24. 4 x 2 3x = 0 25 x 2 = 0. 12 x 2 16 x + 5 = 0 5 x +1 = 25. x 2 = 54. Question 7. Solve the following inequalities and represent the solution on a number line: 3x 5 1 2( x 1) + 3 > 5 x Grade 10 -9- Tutorials Question 8. Use your calculator and the trial and error method to find an approximate solution (correct to one decimal place). to the following equations. 3 x 2 x 1 = 30 4 x = 44. Question 9. Solve the following simultaneous equations: x 3y = 0. y = x 5 and y = 2 x + 3 3x + y = 5. Question 10. The area of the rectangle in the diagram is 2 x 2 x 3 cm 2.

9 Find the length and breadth of the 2x2 x 3. rectangle in terms of x. For which value(s) of x will the rectangle be a square? Question 11. The cost of operating a taxi includes the wage paid to the driver as well as the cost per kilometre to run the taxi. If a taxi owner pays his drivers R250 per day and the per km cost of running the taxi is , write an equation for the daily cost of operating the taxi. Let C be the daily cost and x be the km travelled in a day. If the taxi travel 234km in one day, what is the cost of operating the taxi for that day? If the cost of taxi operation for a day is R684, how many kilometres did the taxi cover in the day? Grade 10 - 10 - Tutorials Question 12. Set up two equations to represent the following statements: The sum of two numbers is 12.

10 The difference of the two numbers is 7. Draw graphs of these two equations on graph paper and on the same set of axes. Solve the two simultaneous equations using the graphs and write down your solution. Check your answer by substituting your solution into both equations. Question 13. An engineer is testing the emergency stopping time of a lift that is being installed in a high-rise building. The time that the lift takes to stop after the emergency brakes have been applied is given by the equation: x 2 4 x = k , where x is the time in seconds and k is the number of the floor where the brakes were applied. Calculate how long it will take the lift to stop if the brakes are applied on the 12th floor. Grade 10 - 11 - Tutorials Grade 10 Tutorial Finance 1 Erin invests R5000 in a financial institution.