CAPS Z J P [ H T L O [ 4H m a Ex a c t i ce Pr Platinum_Exam Practice Book_COVER_Mathematics_Gr10.indd B k o o 10 e d a r G 2011/09/02 3:16 PM Mathematics Tests.pdf 7/2/2011 4:11:24 PM Grade 10 MATHEMATICS PRACTICE TEST ONE Marks: 50 1.1 1.2 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 4.3 4.4 Fred reads at 300 words per minute The book he is reading has an average of 450 words per page Find an expression for the number of pages that Fred has read after x hours (4) How many pages would Fred have read after 3 hours? (1) The next two questions are based on the expression y x 37 x 35 Factorise the expression (2) Find the value of y if x = 2 (1) For what value/s of x will y = 0? (2) 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 Expand to complete the following: ( x y ) (1) 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) 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) Factorise the following expressions: x xy y (3) x 13 x (2) x x (2) p (3) Mathematics Tests.pdf 7/2/2011 4:11:24 PM 5.1 5.2 5.3 Solve for x: x 3 2 x (1 x)(1 x) x 1 (3) x x (4) 3431 x (3) Study the graph of f (x) below and answer the questions that follow 6.1 6.2 6.3 What is the range of f (x) ? If f ( x) (1) tan x k , find the value of k (2) For what value/s of x is f (x) increasing? (2) Mathematics Tests.pdf 7/2/2011 4:11:24 PM Study the graph below and answer the questions that follow 7.1 7.2 7.3 7.4 8.1 8.2 8.3 8.4 8.5 What is the period of f (x) ? (1) Write down the equation of f (x) (2) What is the maximum value of f (x) ? (1) Which one of the following statements is correct? (Write down only the correct letter.) a) f (x) is not symmetrical about any line b) f (x) is symmetrical about the x-axis c) f (x) is symmetrical about the y-axis d) f (x) is symmetrical about the line y = x (1) Find the missing term of each of the following sequences: 3;; ?;; 7;; 9 (1) 6;; ?;; 24;; 48 (1) 1;; 2;; 4;; 7;; ?;; 16 (1) 1;; 3;; 7;; ?;; 21 (1) p ;; ?;; 1 ;; q pq (1) [TOTAL: 50 marks] Mathematics Tests.pdf 7/2/2011 4:11:24 PM Grade 10 MATHEMATICS PRACTICE TEST TWO Marks: 50 ax b 1.1 Consider a function of the form f ( x) Determine the coordinates of the turning point of f (x) in terms of a or b (2) 1.2 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) 2.1 Consider the functions f ( x) and g ( x) x x 2.2 Sketch f ( x) and g ( x) on the same set of axes Label all intercepts with the axes, asymptotes and turning points There is one value that g (x) can take on that f (x) cannot Write down this value Refer to the graph below and answer the questions that follow The functions drawn below are: f ( x) k and g ( x) x (4) (1) x 3.1 3.2 3.3 Find the value of k (2) Find the coordinates of point A (3) Write down the domain of f (x) (2) Mathematics Tests.pdf 7/2/2011 4:11:24 PM 3.4 3.5 4.1 4.2 4.3 4.4 5.1 Find the coordinates of point B (1) Find the y coordinate of point C (which is directly above point B) (2) The function f ( x) 2 x is given Sketch the graph of f (x) showing all intercepts with the axes and other important points (5) What is the range of f (x) ? (2) For what value/s of x is f ( x) ! ? (2) What will the equation of f (x) become if the graph is shifted down by 3 units? (1) Use your knowledge of quadrilaterals to answer the following questions 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 5.2 (2) 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 90q (ii) rhombus B a quadrilateral with one pair of parallel sides (iii) kite C a quadrilateral with a 90q corner angle and four equal sides (iv) trapezium D a quadrilateral with equal adjacent sides (8) Mathematics Tests.pdf 7/2/2011 4:11:24 PM 6.1 6.2 6.3 6.4 6.5 For each of the following, determine whether the statement is true or false If false, correct the statement Both pairs of opposite sides of a kite are parallel (2) The diagonals of a rectangle bisect at 90q (2) The adjacent sides of a rhombus are equal (2) A trapezium has two pairs of parallel sides (2) A square is a rhombus with a 90q corner angle (2) [TOTAL: 50 marks] Mathematics Tests.pdf 7/2/2011 4:11:24 PM Grade 10 MATHEMATICS PRACTICE TEST THREE Marks: 50 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 8,2% 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 2.1 If the gradient of AB is 3, show that x = –2 2.2 If the gradient of AD is , show that D is the point (0;;2) (3) If D is the midpoint of AC, find the coordinates of C (4) Determine whether 'ABC is equilateral, isosceles or scalene Show all of your working (5) 2.3 2.4 (3) Mathematics Tests.pdf 7/2/2011 4:11:24 PM Use the diagram below to answer the questions that follow 3.1 Write down an expression for: a) tan Į (1) b) tan ș (1) 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 Use your answers in 3.1 to prove that Hence, if AC = 6 units, Į AB BC 22,76q and ș tan Į tanș 39,97q, find the length of AB Use your calculator to find the value of sin( 2ș 3Į), correct to two decimal places (3) (4) (1) Sometimes statistics can be misleading Use your understanding of statistics to answer the following questions 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 20 cars this month.” Do you agree with his logic? Give a reason for your answer (3) 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) The average life expectancy in a certain country is around 70 years Does that mean that nobody will live to be 100? (2) 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 The diagonals of a trapezium are never equal (2) A square is a rhombus with a 90q angle (2) A rhombus is the only quadrilateral with adjacent sides that are equal (2) The diagonals of a kite always bisect at 90q (2) The diagonals of a rhombus always bisect each other (2) [TOTAL: 50 marks] Mathematics Tests.pdf 7/2/2011 4:11:24 PM Grade 10 MATHEMATICS PRACTICE TEST FOUR Marks: 50 1.1 Simplify the following expressions as far as possible: 1.2 x x Mathematics Tests.pdf 15 7/2/2011 4:11:24 PM 6.1 y R 9 (1) 6.2 The tangent graph has been shifted up by 2 units –k = 9 ? k = –2 9 (2) 90q x 90q or 90q x 270q 99 6.3 In other words, all values of x between –90q and 270q, except for –90q, 90q and 270q (2) 7.1 360q 9 (1) 7.2 y = cos x 99 (2) 7.3 9 (1) 7.4 c) 9 (1) 8.1 9 8.2 12 9 8.3 11 9 8.4 13 9 8.5 (add on 2 each time) (1) (multiply by 2 each time) (1) (add 1, add 2, add 3, add 4 ) (1) (add 2, add 4, add 6 ) (1) p 9 q (multiply by each time) pq [TOTAL: 50 marks] 15 (1) Mathematics Tests.pdf 16 7/2/2011 4:11:25 PM 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) 99 (2) 1.2 If the turning point is a minimum, then the parabola must be U shaped This means that the coefficient of x2 must be positive There is no restriction on the value of b a > 0 99 b R 9 (3) 2.1 9999 (4) 2.2 – 4 9 (1) 16 Mathematics Tests.pdf 17 7/2/2011 4:11:25 PM 3.1 Point D = (0;;2) (y-intercept of the line y = x + 2) The hyperbola has been shifted up by 2 units because y = 2 is now its asymptote ? k = 99 (2) 3.2 A is the x-intercept of the hyperbola where y = 0 y = x = 9 x = x = 2x ? x = 9 Thus, A is the point (3;;0) 9 (3) 3.3 Domain: x R, x z 99 (2) 3.4 At B, y = 0, so substitute into y = x + 2 = x + 2 ? x = –2 Thus B is the point (–2;;0) 9 (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 y = = x 9 = 5 9 2 (2) 17 Mathematics Tests.pdf 18 7/2/2011 4:11:25 PM 4.1 99999 y d 99, y R 4.2 (5) (2) 4.3 x 99, x R (2) y = x 4.4 = x 9 (1) 5.1 (a) 9 and (d) 9 (2) 5.2 (i) C 99 (ii) A 99 (iii) D 99 (iv) B 99 (8) 18 Mathematics Tests.pdf 19 7/2/2011 4:11:25 PM 6.1 False, both pairs of adjacent sides of a kite are equal 99 (2) 6.2 False, the diagonals of a rectangle bisect each other, but not necessarily at 90q 99 (2) 6.3 True 99 (2) 6.4 False, a trapezium has one pair of parallel sides 99 (2) 6.5 True 99 (2) [TOTAL: 50 marks] 19 Mathematics Tests.pdf 20 7/2/2011 4:11:25 PM Grade 10 MATHEMATICS PRACTICE TEST THREE MEMORANDUM The best investment will be the one that has the highest value after three years Zebra Bank: A = 000(1 0,08 36 ) 9 12 = R1 270,24 9 Giraffe Savings: A = 000(1 + 0,082)3 = R1 266,72 9 Rhino Investments: A = 000(1 + (0,084 u 3)) = R1 252 9 Zebra Bank is the best investment 9 (5) 20 Mathematics Tests.pdf 21 7/2/2011 4:11:25 PM 2.1 mAB = 1 x (1) 3 99 x 1 = 3x + 3 = –3 3x = –6 x = –2 9 (3) 2.2 Equation of AD: y = x c = c 9 = c Substitute in point A(–2;;1) Since D is the y-intercept of AD, D must be the point (0;;2) 9 Or answer by inspection (3) 2.3 Let C be (x;;y) x2 = 9 ? x = 9 y 1 = 9 ? y = 9 ? C is the point (2;;3) Or answer by inspection (4) 21 Mathematics Tests.pdf 22 7/2/2011 4:11:25 PM 2.4 AB = = 10 9 BC = = (4 1) (1 (2)) (4 3) (1 2) 10 9 AC = = (3 1) (2 (2)) 20 9 'ABC is an isosceles 9 triangle because it has two equal sides 9 (5) 22 Mathematics Tests.pdf 23 7/2/2011 4:11:25 PM 3.1 a) tan Į AB 9 BD (1) b) tan ș BC 9 BD (1) AB = BD tan Į 3.2 ? AB BD tan Į = 99 BC BD tan ș = tan Į 9 tan ș (3) AC = AB + BC ? BC = AC – AB = – AB 9 tan Į AB = tan ș BC 3.3 (from 3.1) BC = BD tan ș and tan 22,76q AB 9 = tan 39,97q AB AB = 0,5 9 AB 2AB = – AB 3AB = ? AB = units 9 (4) 3.4 0,20 9 (1) 23 Mathematics Tests.pdf 24 7/2/2011 4:11:25 PM 4.1 No, 9 an average is not guaranteed to persist 9 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 9 (3) 4.2 No, 9 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 99 Secondly, there are too few people in the test group to make any deductions 99 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, 9 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 9 (2) 24 Mathematics Tests.pdf 25 7/2/2011 4:11:25 PM 5.1 False Diagonals can be equal if opposite sides are equal See below 9 (2) 5.2 True 99 (2) 5.3 False, a kite and a square also have adjacent sides that are equal 99 (2) 5.4 False, diagonals do not necessarily bisect – see below 9 (2) 5.5 True 99 (2) [TOTAL: 50 marks] 25 Mathematics Tests.pdf 26 7/2/2011 4:11:25 PM Grade 10 MATHEMATICS PRACTICE TEST FOUR MEMORANDUM ( x 3)(3 x) = x x 99 1.1 (2) x 1.2 2x 2x = 21x 7(2 x) 3(2 x 1) 99 21 = 21x 14 x x 21 = x3 9 21 (3) 2.1 £240 500 = R240 500 u 14,46 9 = R3 477 630 9 (2) 2.2 Bernard would want a weak Rand relative to the Pound 9 This would mean that he would receive more Rands for each Pound that he earned on the sale 9 (2) 2.3 An inverse or reciprocal relationship (9) exists between the two rates Mathematically: 14,46 0,0692 and 0,0692 14,46 (9) 26 (either description will earn 1 mark) (1) Mathematics Tests.pdf 27 7/2/2011 4:11:25 PM 3.1 n2 99 (2) 3.2 9 (1) Tiles added = 2n – 1 999 3.3 (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 n2 This tells us that the sum of n odd numbers is n2 Sum of 8 000 odd = 0002 9 numbers = 64 000 000 9 (2) 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 8 000 larger Sum to 8 000 = 64 000 000 + 8 000 9 = 64 008 000 9 Note: A general term for the sum of this sequence would be Sn = n2 + n, or Sn = n(n + 1) 27 (2) Mathematics Tests.pdf 28 7/2/2011 4:11:25 PM 4.1 Midpoint AC = · §1 § 1 · ;; ¸ = ¨ ;;1¸ 9 ¨ ¹ ¹ ©2 © 4.2 (2) Midpoint BD = § 1 3· ;; ¸ 9 ¨ ¹ © §1 ©2 · ¹ = ¨ ;;1¸ 9 AC and BD share a midpoint and therefore they bisect each other This means that ABCD is a parallelogram (diagonals bisect) 9 (3) 4.3 a) Using sides, simply prove that adjacent sides are not equal (ABCD is a ||gm) (3 1) (1 4) = 29 9 AB = AD = (3 (3)) (1 (3)) = 40 9 Adjacent sides are not equal and therefore parallelogram ABCD is not a rhombus.9 (3) b) Diagonals of a rhombus bisect at 90q Using gradients: (5) = 9 1 mAC = mBD = (3) = 9 (3) mBD u mAC z –1, so diagonals are not perpendicular Parallelogram ABCD is therefore not a rhombus 9 28 (3) [...]... OLNHWRDSRORJLVHIRUDQ\LQIULQJHPHQWRIFRS\ULJKWVRFDXVHGDQGFRS\ULJKWKROGHUVDUHUHTXHVWHGWR FRQWDFWWKHSXEOLVKHUVLQRUGHUWRUHFWLI\WKHPDWWHU )LUVWSXEOLVKHGLQ ,6%1 www.mml.co.za Platinum_ Exam Practice Book_ COVER _Mathematics_ Gr10.indd 1 2011/09/02 3:16 PM ... 2, add 3, add 4 ) (1) (add 2, add 4, add 6 ) (1) p 9 q (multiply by 1 each time) pq [TOTAL: 50 marks] 15 (1) Mathematics Tests.pdf 16 7/2/2011 4:11:25 PM 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) 99 ... 99 (2) 6.4 False, a trapezium has one pair of parallel sides 99 (2) 6.5 True 99 (2) [TOTAL: 50 marks] 19 Mathematics Tests.pdf 20 7/2/2011 4:11:25 PM 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:... 5.4 False, diagonals do not necessarily bisect – see below 9 9 (2) 5.5 True 99 (2) [TOTAL: 50 marks] 25 Mathematics Tests.pdf 26 7/2/2011 4:11:25 PM Grade 10 MATHEMATICS PRACTICE TEST FOUR MEMORANDUM ( x 3)(3 2 x) = 9 3 x 2 x 2 99 1.1 (2) x 1.2 2x 2x 1 = 3 7 21x 7(2 x) 3(2 x 1) 99... probability of neither occurring is 0,1 If P(A) = 0,6, use a Venn diagram to find P(B) (3) Find P(A or B) (2) [TOTAL: 50 marks] 11 Mathematics Tests.pdf 12 7/2/2011 4:11:24 PM Grade 10 MATHEMATICS PRACTICE TEST ONE MEMORANDUM 1.1 1 hour = 60 minutes ? in one hour, Fred reads 300 u 60 = 18 000 words 9 18 000 Pages per... by inspection (4) 21 Mathematics Tests.pdf 22 7/2/2011 4:11:25 PM 2.4 AB = = 10 9 BC = = (4 1) 2 (1 (2)) 2 (4 3) 2 (1 2) 2 10 9 AC = = (3 1) 2 (2 (2)) 2 20 9 'ABC is an isosceles 9 triangle because it has two equal sides 9 (5) 22 Mathematics Tests.pdf 23 7/2/2011... a parallelogram (3) Prove that ABCD is NOT a rhombus in two different ways: a) using sides (3) b) using diagonals (3) Prove that ABCD is not a rectangle (4) 10 Mathematics Tests.pdf 11 7/2/2011 4:11:24 PM 5 Your friend Nandi is working on a homework exercise She is getting very frustrated because her answers do not seem to make any... 9 2 (2) 17 Mathematics Tests.pdf 18 7/2/2011 4:11:25 PM 4.1 99999 y d 2 99, y R 4.2 (5) (2) 4.3 1 x 1 99, x R (2) y = 2 x 2 2 3 4.4 = 2 x 2 1 9 (1) 5.1 (a) 9 and (d) 9 (2) 5.2 (i) C 99 (ii) A 99 (iii) D 99 (iv) B 99 (8) 18 Mathematics Tests.pdf... 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 (2) Mathematics Tests.pdf 10 7/2/2011 4:11:24 PM 3.2 3.3 3.4 How many small squares will be added on in layer 5? (1) Write down an expression for the number of tiles added on in layer... 4 (4) 5.3 7 x 1 = 3431 x 3 1 x 7 x 1 = (7 ) 9 7 x 1 = 73 3 x ? x – 1 = 3 – 3x 9 4x = 4 ? x = 1 9 (3) 14 Mathematics Tests.pdf 15 7/2/2011 4:11:24 PM 6.1 y R 9 (1) 6.2 The tangent graph has been shifted up by 2 units –k = 2 9 ? k = –2 9 (2) 90q