MATHEMATICS: PAPER II MARKING GUIDELINES

Transcription

1 GRADE 10 IEB STANDARDISATION PROJECT NOVEMBER 01 MATHEMATICS: PAPER II MARKING GUIDELINES Time: hours 100 marks These marking guidelines are prepared for use by examiners and sub-examiners, all of whom are required to attend a standardisation meeting to ensure that the guidelines are consistently interpreted and applied in the marking of candidates' scripts. The IEB will not enter into any discussions or correspondence about any marking guidelines. It is acknowledged that there may be different views about some matters of emphasis or detail in the guidelines. It is also recognised that, without the benefit of attendance at a standardisation meeting, there may be different interpretations of the application of the marking guidelines. IEB Copyright 01

2 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page of 14 SECTION A QUESTION 1 The mathematics marks obtained by the 9 pupils from a certain school are represented below: Interval 0 x < x < x < x < x < x < x < x < x < x < Frequency (a) What is the modal interval? 60 x < 70 (b) In which interval does the median lie? 60 x < 70 (c) In which interval does the lower quartile lie? 50 x < 60 (d) (e) Give an estimate for the mean mark. You should show working in order to demonstrate your understanding of the process. Express your answer correct to one decimal place mean = = 67, 4 = 9 The actual mean of the data is known to be 66.8% correct to one decimal place. Explain why your estimate in (d) differs from the exact answer. (4) With the grouped data we don t have the actual values so we are using the midpoints of the intervals as our best estimate. [8] IEB Copyright 01

3 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page of 14 QUESTION (a) Indicate whether each of the following statements is TRUE or FALSE. If false, give a neat sketch of a counter example. All rectangles are parallelograms TRUE () The diagonals of a rhombus are equal in length. FALSE AC BD () () If the corresponding sides of two triangles are in the same proportion then the triangles must be equiangular. TRUE (b) In the diagram below ABDC is a parallelogram. BE = DF, B ˆ ˆ 1 = 40, D = 140. Prove that AE = CF E A 1 1 C 40 o 1 B 4 1 D 140 o F 1 ( ) Dˆ = 40 ' s on str. line Bˆ = Dˆ (..) ( ) ( ) ( ) AB = CD opp sides of parallelogram BE = DF given ABE CDF SAS AE = CF congruency (5) [9] IEB Copyright 01

4 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 4 of 14 QUESTION Consider the diagram below: y x (a) Show that ABC is right-angled at B. 1 mab = = mbc = = 8 As mab mbc = 1, AB BC () (b) (c) Determine the coordinates of P and Q, the mid-points of AB and AC respectively. P 1; and Q 5; 0 ( ) ( ) Use analytical methods to show that the line joining P and Q is parallel to BC. () ( 1; ) ( 5; 0) P and Q 0 1 mpq = = 1 5 since mpq = mbc, PQ BC (4) (d) 1 Use analytical methods to prove that PQ = BC. PQ = = 0 = 5 BC PQ IEB Copyright 01 ( ) ( ( )) ( ) ( ( )) = = 80 = = BC (4)

5 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 5 of 14 (e) Determine the coordinates of D if ABCD is a rectangle. D ( 6;8) () [15] QUESTION 4 (a) Consider the diagram below: y θ x P( 4; ) Without the use of a calculator, determine cotθ, expressing your answer as a fraction. 4 cotθ = () determine sinθ, expressing your answer as a fraction. r = + = 4 5 sinθ = 5 () (b) Calculate tanα, if it is given Hint: use a sketch. y 5 sin α = cos 0 1 and α < 5 1 α 1 x 5 tanα = 1 (4) IEB Copyright 01

6 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 6 of 14 (c) Calculate sin 60 + cosec0 + tan 45 without a calculator. Express your answer as a single fraction. sin 60 + cosec0 + tan 45 = = (d) Calculate [ 0 ;90 ] θ without a calculator if sinθ = cos 0 + sin180 cos 60 () sinθ = cos 0 + sin180 cos sinθ = 1+ 0 = θ = 0 () (e) Solve the following equations for θ [ 0 ;90 ] place: sinθ = 0, 4. Give your answers to one decimal θ = 1,5 () cotθ = tan 5 + sin cotθ = 0,58... tanθ = 1,89... θ = 6,1 (4) [18] 50 marks IEB Copyright 01

7 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 7 of 14 SECTION B QUESTION 5 The box and whisker plots for the scores made by two cricketers during the past season are as follows: You do not need any cricketing knowledge for this question other than to know that a high score is better than a low score! (a) Determine the inter-quartile range for player 1. IQR = = 5 () (b) Player makes the following claim: "I score more than x in half of my matches" What is the value of x? x = 5 () (c) Assuming that both players have played a similar number of matches, which player would you select for your team? Justify your answer by referring to statistical concepts you have learnt. I would select player. He is more consistent, smaller inter-quartile range. Has a higher median number of runs. () [6] IEB Copyright 01

8 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 8 of 14 QUESTION 6 Determine the length of EF to the nearest meter if the following lengths are given. CG is m CD is 7 m GF is 695 m DF and CE are straight lines. You are also given that D ˆ = E. ˆ E D 7 1 G C 695 F You should show all calculations. In ' s DGC and EGF Dˆ = Eˆ given 1 ( ) ( ) Gˆ = Gˆ vert. opp. ' s rd ( ) Cˆ = Fˆ of ( ) DGC EGF AAA EF GF = ( ' s) DC GC GF DC EF = = = 75m GC [6] IEB Copyright 01

9 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 9 of 14 QUESTION 7 (a) Sipho and Mary are trying to work out the height of a tree. They are standing m away from the base of the tree. m [Images taken from < and < Sipho has measured the angle of inclination of the top of the tree to be. Sipho says that he can use trigonometry with the given information to calculate the height of the tree. Show the method Sipho might use and calculate the height of the tree in meters to one decimal place. ht of tree tan = ht of tree = tan = 1,6 m () () Mary says that she has not started trigonometry yet but that she can calculate the height of tree using similar triangles. She measures the length of the tree s shadow as 6,7 m. Then she measures her shadow as 77 cm. Mary uses her height of 1,67 m with the other information to correctly calculate the height of the tree. Show the method Mary might use and calculate the height of the tree in meters to one decimal place. tree height Mary ' s height = tree shadow Mary ' s shadow Mary ' s height tree shadow tree height = Mary ' s shadow 1,67 6,7 = 0,77 = 1, 6 m (4) IEB Copyright 01

10 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 10 of 14 (b) Solve for q to one decimal place in the diagram below: p + p q p sin11.55 = q p = qsin11.55 p + tan 1.8 = q q sin tan 1.8 = q q tan 1.8 = qsin q tan 1.8 qsin11.55 = q ( tan 1.8 sin11.55 ) = q = = 10 units tan 1.8 sin11.55 (6) [1] IEB Copyright 01

11 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 11 of 14 QUESTION 8 A bug crawls a distance of 40 units on a straight line in the Cartesian Plane. If the 1 gradient of the line is and the bug ends up at the point T( ; 5), determine the coordinates of the point where it started? Let the horizontal movement be a and the vertical movement be b then: b 1 = a a= b but a + b = 40 ( b) + b = b + b = b = 4 b =± and a =± 6 so, bug crawled down and across 6 ( ) ( ) so, the bug started at 4; or 8; 7 [6] QUESTION 9 4 Given: V = πr V= πrh A large three-dimensional capsule is made up of two hemispheres and a cylinder. The radius of the hemispheres is 4, m. (a) (b) If the overall length of the capsule is l, give an expression for the volume of the capsule in terms of l. 4 V = π 4. + ( l 4.)( π 4. ) If the volume of the capsule is 1 51 m then determine the value of l to the nearest meter = π π 4. l = 4. + l = 0m ( l )( π ) ( π 4. ) () (4) [6] IEB Copyright 01

12 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 1 of 14 QUESTION 10 (a) Give the equations of each of the following graphs: y = sin x () () y = cos x 1 () IEB Copyright 01

13 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 1 of 14 (b) A Ferris Wheel at an amusement park has riders get on at position A, which is metres above the ground. The highest point of the ride is 15 metres above the ground. The ride takes 40 seconds for one complete revolution. The height of a rider above the ground can be modeled by the formula: h θ ) = a cosθ + b θ 0 ;60 ( for [ ] Show that a + b = When the Ferris wheel is on the ground: θ = 0 and h = so = a cos 0 + b a+ b= () () Show that a + b = 15 When the Ferris wheel is at the highest point: θ = 180 and h = 15 so 18 = a cos180 + b a+ b= 15 () IEB Copyright 01

14 GRADE 10 STANDARDISATION PROJECT: MATHEMATICS: PAPER II MARKING GUIDELINES Page 14 of 14 () If a rider makes complete revolutions, determine the amount of time (to the nearest second) spent above 1 metres above the ground. Assume that ( θ ) = 6cosθ + 9 θ 0 ;60 h for [ ] 6cosθ + 9 = 1 6cosθ = 4 cosθ = 1 key = cos = 48, so θ = 11.8 or 8, so, per revolution the rider spends 8, 11,8 40 = 10.7 seconds above 1 m 60 in whole ride, rider spends seconds above 1 m (5) [1] 50 marks Total: 100 marks IEB Copyright 01