Practice Book. Practice. Practice Book

Transcription

1 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Exam CAPS

2 Grade 10 MATHEMATICS PRACTICE TEST ONE Marks: Fred reads at 300 words per minute. The book he is reading has an average of 450 words per page. 1.1 Find an expression for the number of pages that Fred has read after x hours. (4) 1. How many pages would Fred have read after 3 hours? (1). The next two questions are based on the expression 6 37xxy Factorise the expression. (). Find the value of y if x =. (1).3 For what value/s of x will y = 0? () 3. The sum of two numbers is 5. Their product is 3. Find the sum of the squares of the two numbers by answering the following questions. 3.1 Expand to complete the following: yx )( (1) 3. If the two numbers mentioned above are x and y, then write down the equations for the sum and the product of the two numbers. (1) 3.3 Substitute the information given above into your answer for 3.1 and hence determine the sum of the squares of the two numbers. (Hint: make sure to include both sides of the identity.) (3) 4. Factorise the following expressions: 4.1 x xy y 463 (3) xx 6 () xx () 6 1 p (3)

3 5. Solve for x: 5.1 x 4 x 3 3 (3) 5. (1 )(1 ) 3 5xxxx (4) 5.3 x 1 1 x (3) 6. Study the graph of xfbelow )( and answer the questions that follow. 6.1 What is the range of xf?)( (1) 6. If )( tan kxf, find the value of k. () 6.3 For what value/s of x is xfincreasing? )( ()

4 7. Study the graph below and answer the questions that follow. 7.1 What is the period of xf?)( (1) 7. Write down the equation of (xf ). () 7.3 What is the maximum value of xf?)( (1) 7.4 Which one of the following statements is correct? (Write down only the correct letter.) a) xfis )( not symmetrical about any line. b) xfis )( symmetrical about the x-axis. c) xfis )( symmetrical about the y-axis. d) xfis )( symmetrical about the line y = x. (1) 8. Find the missing term of each of the following sequences: 8.1 3;;?;; 7;; 9 (1) 8. 6;;?;; 4;; 48 (1) 8.3 1;; ;; 4;; 7;;?;; 16 (1) 8.4 1;; 3;; 7;;?;; 1 (1) p?;; ;; 3 q pq (1) [TOTAL: 50 marks] 3

5 Grade 10 MATHEMATICS PRACTICE TEST TWO Marks: Consider a function of the form f x ax b.) ( 1.1 Determine the coordinates of the turning point of xfin )( terms of a or b. () 1. Depending on the values of a and b, the turning point could be either a maximum or a minimum. If the turning point is a minimum, write down the possible values of a and b. (3) 8. Consider the functions xf )( 4 and xxg )(.4 x.1 Sketch ( and ) xgxfon )( the same set of axes. Label all intercepts with the axes, asymptotes and turning points. (4). There is one value that xgcan )( take on that xfcannot. )( Write down this value. (1) 3. Refer to the graph below and answer the questions that follow. The functions drawn 6 below are: xf )( k and xxg )(. x 3.1 Find the value of k. () 3. Find the coordinates of point A. (3) 3.3 Write down the domain of (xf ). () 4

6 3.4 Find the coordinates of point B. (1) 3.5 Find the y coordinate of point C (which is directly above point B). () 4. The function xx ) is given. f ( 4.1 Sketch the graph of xfshowing )( all intercepts with the axes and other important points. (5) 4. What is the range of xf?)( () 4.3 For what value/s of x is xf?0)( () 4.4 What will the equation of xfbecome )( if the graph is shifted down by 3 units? (1) 5. Use your knowledge of quadrilaterals to answer the following questions. 5.1 Below are pairs of parallelograms. If you are only given information about their diagonals, in which pair(s) can you distinguish between the two parallelograms? a) a rhombus and a rectangle b) a square and a rhombus c) a kite and a trapezium d) a rectangle and a square () 5. Match each definition with the correct figure. If a definition applies to more than one figure, then choose the figure that it describes the best. You may only use each definition once. (Write the number of the figure and the letter of the definition you do not have to rewrite the whole definition.) Figure Definition (i) square A a quadrilateral with diagonals that bisect at 90 (ii) rhombus B a quadrilateral with one pair of parallel sides (iii) kite C a quadrilateral with a 90 corner angle and four equal sides (iv) trapezium D a quadrilateral with equal adjacent sides (8) 5

7 6. For each of the following, determine whether the statement is true or false. If false, correct the statement. 6.1 Both pairs of opposite sides of a kite are parallel. () 6. The diagonals of a rectangle bisect at 90. () 6.3 The adjacent sides of a rhombus are equal. () 6.4 A trapezium has two pairs of parallel sides. () 6.5 A square is a rhombus with a 90 corner angle. () [TOTAL: 50 marks] 6

8 Grade 10 MATHEMATICS PRACTICE TEST THREE Marks: Which of the following accounts would the best investment? Assume that you have R1 000 to invest for 3 years. a) Zebra Bank offers 8% per annum compounded monthly. b) Giraffe Savings offers 8,% per annum compounded yearly. c) Rhino Investments offers 8,4% per annum simple interest. (5). The points A(x;;1), B( 1;;4), C and D are shown on the Cartesian plane below..1 If the gradient of AB is 3, show that x =. (3) 1. If the gradient of AD is, show that D is the point (0;;). (3).3 If D is the midpoint of AC, find the coordinates of C. (4).4 Determine whether ABC is equilateral, isosceles or scalene. Show all of your working. (5) 7

9 3. Use the diagram below to answer the questions that follow. 3.1 Write down an expression for: a) tan (1) b) tan (1) AB tan 3. Use your answers in 3.1 to prove that. (3) BC tan 3.3 Hence, if AC = 6 units,,76 and 39,97, find the length of AB. (4) 3.4 Use your calculator to find the value of sin( 3 ), correct to two decimal places. (1) 4. Sometimes statistics can be misleading. Use your understanding of statistics to answer the following questions. 4.1 A car salesperson says, I sold five cars last week. That s an average of one car every day. That means that I m going to sell 0 cars this month. Do you agree with his logic? Give a reason for your answer. (3) 4. A study was done to see if a new skin cream could make wrinkles disappear. It was tested on six women while they were visiting a health spa and over 80 % reported that their skin felt smoother. Do you think the results of this study are reliable? Give at least two reasons for your answer. (5) 4.3 The average life expectancy in a certain country is around 70 years. Does that mean that nobody will live to be 100? () 5. Determine whether each of the following statements is true or false. If the statement is false, explain or give a counter example to prove that the statement is false. 5.1 The diagonals of a trapezium are never equal. () 5. A square is a rhombus with a 90 angle. () 5.3 A rhombus is the only quadrilateral with adjacent sides that are equal. () 5.4 The diagonals of a kite always bisect at 90. () 5.5 The diagonals of a rhombus always bisect each other. () [TOTAL: 50 marks] 8

10 Grade 10 MATHEMATICS PRACTICE TEST FOUR Marks: Simplify the following expressions as far as possible: xx () 1. 1 x xx (3) 3 7. Bernard inherited a flat in England that belonged to his grandmother. He decided to sell it and use the money to buy a house in South Africa. Below are the exchange rates at the time of the sale: Cross rates Rand (R) Pound ( ) 1 Rand (R) = 1 R14,46 1 Pound ( ) = 0, The flat was sold for How many Rands is this? (). Would Bernard want a strong Rand or a weak Rand? Give a reason for your answer. ().3 Refer to the table of cross rates. Describe the mathematical relationship between the two numbers 14,46 and 0,069. (1) 3. The diagram below shows squares of increasing sizes. With each extra layer of small squares we add, we build a bigger square. In the second layer, we add 3 small squares. In the third layer, we add 5 small squares. 3.1 How many tiles will there be in total if we have n layers of small squares? () 9

11 3. How many small squares will be added on in layer 5? (1) 3.3 Write down an expression for the number of tiles added on in layer n. (3) 3.4 Study the pattern carefully and use the relationship between the layers and the whole area to find the value of the following sum to terms: () 3.5 Use your answer to 3.4 to find the value of the following sum to terms: () 4. Use the figure below to answer the questions that follow. 4.1 Find the midpoint of AC. () 4. Use midpoints to prove that ABCD is a parallelogram. (3) 4.3 Prove that ABCD is NOT a rhombus in two different ways: a) using sides (3) b) using diagonals (3) 4.4 Prove that ABCD is not a rectangle. (4) 10

12 5. Your friend Nandi is working on a homework exercise. She is getting very frustrated because her answers do not seem to make any sense. In the two triangles below, she is trying to solve for x. Explain why her answers do not make sense in each case. (5) 6. Your favourite soccer team is changing its kit. The new kit will be a striped shirt and plain shorts. The team colours are blue and white. The stripes and the background colour of the shirt must be different (i.e. white with blue stripes or blue with white stripes). 6.1 Write down the different possible colour combinations for the team kit. () 6. What is the probability that the stripes on the shirt and the shorts will be the same colour? (3) 7. For two events, A and B, the probability of both occurring is 0, and the probability of neither occurring is 0, If P(A) = 0,6, use a Venn diagram to find P(B). (3) 7. Find P(A or B). () [TOTAL: 50 marks] 11

13 Grade 10 MATHEMATICS PRACTICE TEST ONE MEMORANDUM hour = 60 minutes in one hour, Fred reads = words. Pages per hour = = 40 pages after x hours = 40x (4) 1. Pages read = 40(3) = 10 (1).1 RHS = 6 37 xx 35 = ( 6 5)( xx 7) (). y = 6 37 xxsubstitute 35 x = = 6() 37() 35 = 85 (1).3 0 = ( 6 5)( xx 7) x = 5 or x 7 () 6 1

14 3.1 yx )( = xy y (1) 3. yx = 5 xy = 3 (1) 3.3 yx )( = xy y 5 = x )3(y yx = 19 (3) 4.1 x xy y 463 = x ( yy 3()3 ) = ( 3 )( xy ) (3) xx= 6 ( 5 )( xx 3) () 4.3 xx = xx ) ( () p = (1 )(1 pp ) = (1 )(1 )(1 )(1 ppp ) (3) 13

15 x x 4 1 = = x 3 x 3 3x 1 = 4x 7 x = 1 x = 1 (3) 7 5. ( 1 )(1 = xx ) xx 53 x = 1 x = xx 53 0 = xx = ( 4 1)( xx 1) 1 or x = 1 (4) x = 1 x 7 x 1 = 13 x )7( 7 x 1 x33 = 7 x 1 = 3 3x 4x = 4 x = 1 (3) 14

16 6.1 Ry (1) 6. The tangent graph has been shifted up by units. k = k = () x 90 or x 70 In other words, all values of x between 90 and 70, except for 90, 90 and 70. () (1) 7. y = 3cos x 1 () (1) 7.4 c) (1) (add on each time) (1) 8. 1 (multiply by each time) (1) (add 1, add, add 3, add 4...) (1) (add, add 4, add 6...) (1) 8.5 p q 1 (multiply by each time) (1) pq [TOTAL: 50 marks] 15

17 Grade 10 MATHEMATICS PRACTICE TEST TWO MEMORANDUM 1.1 Turning point occurs at x = 0, and when x = 0, y = b. Thus, the turning point is (0;;b). () 1. If the turning point is a minimum, then the parabola must be U shaped. This means that the coefficient of x must be positive. There is no restriction on the value of b. a > 0 Rb (3).1 (4). 4 (1) 16

18 3.1 Point D = (0;;) (y-intercept of the line y = x + ) The hyperbola has been shifted up by units because y = is now its asymptote. k = () 3. A is the x-intercept of the hyperbola where y = 0. 6 y = x 0 = 6 x 6 x = 6 = x x = 3 Thus, A is the point (3;;0). (3) 3.3 Domain: xrx 0, () 3.4 At B, y = 0, so substitute into y = x +. 0 = x + x = Thus B is the point ( ;;0). (1) 3.5 Point C will have the same x-value as point B because it is directly above it. Since we know the x-value, we can substitute into the equation of the hyperbola to find y. 6 y = x 6 = = 5 () 17

19 4.1 (5) 4. y, Ry () 4.3 x 11, Rx () 4.4 y = x 3 = x 1 (1) 5.1 (a) and (d) () 5. (i) C (ii) (iii) A D (iv) B (8) 18

20 6.1 False, both pairs of adjacent sides of a kite are equal. () 6. False, the diagonals of a rectangle bisect each other, but not necessarily at 90. () 6.3 True () 6.4 False, a trapezium has one pair of parallel sides. () 6.5 True () [TOTAL: 50 marks] 19

21 Grade 10 MATHEMATICS PRACTICE TEST THREE MEMORANDUM 1. The best investment will be the one that has the highest value after three years. Zebra Bank: A = 0, (1 ) = R1 70,4 Giraffe Savings: A = 1 000(1 + 0,08) 3 = R1 66,7 Rhino Investments: A = 1 000(1 + (0,084 3)) = R1 5 Zebra Bank is the best investment. (5) 0

22 .1 m AB = x 41 )1( 3 = 3 x 1 3x + 3 = 3 3x = 6 x = (3). Equation of AD: y = 1 cx Substitute in point A( ;;1). 1 1 = c = c Since D is the y-intercept of AD, D must be the point (0;;). Or answer by inspection. (3).3 Let C be (x;;y). x = 0 x = y 1 = y = 3 C is the point (;;3). Or answer by inspection. (4) 1

23 .4 AB = ( 4 1) ( 1 ( )) = 10 BC = 4( )3 1( ) = 10 AC = ( 3 1) ( ( )) = 0 ABC is an isosceles triangle because it has two equal sides. (5)

24 3.1 a) b) AB tan BD (1) BC tan BD (1) 3. AB = BD.tan and BC = BD.tan (from 3.1) AB BD.tan = BC BD. tan = tan tan (3) 3.3 AC = AB + BC BC = AC AB = 6 AB AB tan = BC tan AB 6 AB = tan,76 tan 39,97 AB 6 AB = 0,5 AB = 6 AB 3AB = 6 AB = units (4) 3.4 0,0 (1) 3

25 4.1 No, an average is not guaranteed to persist. If he were to take his yearly average and apply that to a given week it might be more reliable, but to use a single week s average to try to predict future performance is not wise. In the short run almost anything can happen one could have a good or bad week. It does not make sense to base statistics on a few short-term observations. (3) 4. No, the results are not completely reliable. Firstly, testing the product while the women are at a spa is misleading. The results of the spa treatment can not easily be separated from the results of the face cream. Secondly, there are too few people in the test group to make any deductions. What seems true for six people may not apply on a larger scale. The women might also have responded positively for emotional and psychological reasons. (5) 4.3 No, some people die very young and some people die very old. The highs and the lows balance out. An average does not describe every value in the range. () 4

26 5.1 False. Diagonals can be equal if opposite sides are equal. See below. () 5. True () 5.3 False, a kite and a square also have adjacent sides that are equal. () 5.4 False, diagonals do not necessarily bisect see below. () 5.5 True () [TOTAL: 50 marks] 5

27 Grade 10 MATHEMATICS PRACTICE TEST FOUR MEMORANDUM 1.1 ( 3)(3 = ) 39 xx () 1. 1 xx 1 7( ) 3(xxx 1) x = = = x 3 x 1 x 3 1 (3) = R ,46 = R (). Bernard would want a weak Rand relative to the Pound. This would mean that he would receive more Rands for each Pound that he earned on the sale. ().3 An inverse or reciprocal relationship ( ) exists between the two rates. Mathematically: 1 14,46 1 0,069 and 14, 46 ( ) (either description will earn 1 mark) (1) 0,069 6

28 3.1 n () 3. 9 (1) 3.3 Tiles added = n 1 (3) 3.4 With each layer we add on, we make a bigger square. This means that the sum of n layers (odd numbers) is n. This tells us that the sum of n odd numbers is n. Sum of odd numbers = = () 3.5 This is almost the same as the sequence in 3.4, except each term is 1 larger. This means that the whole sum will be a total of larger. Sum to = = Note: A general term for the sum of this sequence would be S n = n + n, or S n = n(n + 1). () 7

29 Midpoint AC = ;; = 1;; () Midpoint BD = ;; = 1;; AC and BD share a midpoint and therefore they bisect each other. This means that ABCD is a parallelogram (diagonals bisect). (3) 4.3 a) Using sides, simply prove that adjacent sides are not equal. (ABCD is a gm) AB = 3( = )1 1( )4 9 AD = ( 3 ( 3)) ( 1 ( 3)) = 40 Adjacent sides are not equal and therefore parallelogram ABCD is not a rhombus. (3) b) Diagonals of a rhombus bisect at 90. Using gradients: m AC = )5(3 8 = 1 3 m BD = )3(1 4 = )3(4 7 m BD m AC 1, so diagonals are not perpendicular. Parallelogram ABCD is therefore not a rhombus. (3) 8

30 4.4 m AD = )3(3 6 = )3(1 = 3 m DC = )5(3 = 3 5 Since m DC m AD 1, there is no right angle between AD and DC. Since ABCD does not have four right angles, it cannot be a rectangle. (4) 5. Triangle 1 The longest side in a right-angled triangle is always the hypotenuse. In this triangle, the hypotenuse is not the longest side, which is impossible. If we try to solve for x using Pythagoras, we will not be able to find a solution because the triangle does not make sense. Triangle In this triangle, the sum of the angles is not 180 ( = 18 ). This triangle also does not make sense. If we try to use trig ratios to solve for x, we will get a slightly different answer depending on which angle we use. This is because a right-angled triangle can not have a 9 angle and a 63 angle these angles would belong to different triangles, hence the two different answers. (5) 6.1 Blue shirt, white stripes;; blue shorts Blue shirt, white stripes;; white shorts White shirt, blue stripes;; blue shorts White shirt, blue stripes, white shorts () 6. 1 = (3) 4 9

31 7.1 P(B) = 0,3 (3) 7. P(A or B) = 0,4 + 0, + 0,3 = 0,9 (or, use 1 0,1 = 0,9) () [TOTAL: 60 marks] 30

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