Grade 11 Assessment Exemplars

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2 Grade 11 Assessment Eemplars 1 Learning Outcomes 1 and 1.1 Assignment : Functions 5 1. Investigation: Ratios Control Test: Equations, Inequalities, Eponents Project: Finance Eam A: Paper Eam B: Paper 1 19 Learning Outcomes 3 and 4.1 Assignment: Analtical and Transformation Geometr 5. Investigation: Shape and Space 7.3 Control Test: Trigonometr, Mensuration 9.4 Project: Enlargements 31.5 Eam A: Paper 35.6 Eam B: Paper 46 Grade Eemplar Assessments 008

3 Information Sheet: Mathematics b ± b 4ac n = A = P( 1+ i. n) A = P( 1+ i) a n A = P( 1 i) A = P( 1 i. n) n i= 1 1 = n n n i = ( n + 1) i= 1 n i= 1 n ( a + ( i 1 ) d ) = ( a + ( n 1) d ) n i= 1 ar i 1 = n ( r ) n a 1 ; r 1 ar r 1 i= 1 i 1 = a r 1 ; 1 < r < 1 F = n [( 1+ i) 1] i P = [1 (1 + i) i n ] f '( ) lim = h 0 f ( + h) h f ( ) d = ( 1) + ( 1) M ; = m + c = m ( ) 1 1 m 1 = = tanθ 1 m ( ) ( ) a + b = r In ABC ; sin a A b c = = sin( α + β ) = sinα.cos β + cosα. sin β sin B sin C sin( α β ) = sinα.cos β cosα. sin β a = b + c bc. cos A cos( α + β ) = cosα.cos β sinα. sin β 1 area ABC = ab. sin C cos( α β ) = cosα.cos β + sinα. sin β cos α = cos α sin α cos α = 1 sin α cos α = cos α 1 sin α = sinα. cosα ( i = f = i= var = 1 n n n n ) SD = n i= 1 ( ) n n( A) P ( A) = P( A or B) = P( A) + P( B) P( A and B) n( s) Grade Eemplar Assessments 008

4 Instructions and Information Read the following instructions carefull before answering this question paper: 1 This question paper consists of questions. Answer ALL questions. Clearl show ALL calculations, diagrams, graphs, et cetera, which ou have used in determining the answers. 3 An approved scientific calculator (non-programmable and non-graphical) ma be used, unless stated otherwise. 4 If necessar, answers should be rounded off to TWO decimal places, unless stated otherwise. 5 Number our answers correctl according to the numbering sstem used in this question paper. 6 Diagrams are not necessaril drawn to scale. 7 It is in our own interest to write legibl and to present our work neatl. Grade Eemplar Assessments 008

5 Grade 11 Assignment: Functions Marks: 95 Question 1: 1.1 What do ou understand b the asmptote of a function; () 1.1. the ais of smmetr of a function; () the zeros of a function? () 1. g( ) = f ( ) and h( ) f ( ) 1..1 g ( ) is a reflection of ( ) 1.. h ( ) is a reflection of ( ) 1.3 What does it mean if ( ) g( ) =. Write down the equation of the line about which: f (1) f (1) f =? () 1.4 How do ou test whether or not the point ( a; b) lies on the graph of f ( ) =? () 1.5 Write down a formula for the average gradient of a curve g( ) points ( a 1;b 1 ) and ( a ;b ) Question : = between the () [14].1 Sketch the graph of = 4 showing all aes of smmetr, asmptotes, intersection with the aes and an other critical points. (7). Give the equation of;..1 the horizontal asmptote (1).. the ais of smmetr that has a negative gradient ()..3 the graph that would result if ou shifted our sketch up b 4 units () [1] Question 3: 3.1 Sketch the graphs of = f ( ) =, = g( ) = ( 1) and = h( ) = ( 1) on the same sstem of aes. Label each graph, an lines of smmetr or asmptotes that ma eist as well as at least points on each graph. (11) 3. Describe, in words, the effect on the graph of ( ) and d in the equation = p( ) = a + c) + d = f = of the parameters a, c (. (8) [19] Grade Eemplar Assessments 008

6 1 1 Question 4: Given h ( ) = Write down the equation of the asmptote of h (1) 4. Determine the coordinates of the intercepts of h with the and aes (6) Write down the equation of the reflection of h ( ) = in the ais. () 4 1 [9] a Question 5: The sketch below represents the graph of f ( ) = + q + p 5.1 If the line of g is the vertical asmptote of the above function, determine the value of a, p and q and hence the equation of f. (5) 5. What is the equation of the horizontal asmptote? (1) 5.3 What is the equation of the ais of smmetr that has a positive gradient? () [8] Grade Eemplar Assessments 008

7 Question 6: Given o o f ( ) = a sin( b + c) + d and g ( ) = p cos( q + r) + s 6.1 Determine the values of a, b, c, d, p, q, r and s and hence the equations of f and g (4) 6. Read from the sketch, the values of for which ( ) g( ) f = for [ 0 ] 0 ; () [6] Question 7 f ( ) = f g ( ) = + 1 Q P 7.1 Determine the lengths of OA; OB and OC. (4) R E 7. Determine the coordinates of P. (5) 7.3 Calculate the length of RS if S is the turning point. (5) 7.4 Determine the lengths of BD; QB and EC. (6) A O B D 7.5 Write down the equation of: g the reflection of ( ) f in the ais; () C 7.5. the parabola with the same zeros as f ( ), which has been stretched through the ais b a factor of () S 7.6 Calculate the average gradient of f ( ) between S and B. (3) [7] Grade Eemplar Assessments 008

8 Grade 11 Investigation: Ratios Marks: Solve the following equation for in terms of : 3 + = 0 and hence show that the ratio : is 1 : 1 or : 1.. Most paper is cut to internationall agreed sizes: A0, A1, A, A7 with the propert that the A1 sheet is half the size of the A0 sheet and has the same shape as the A0 sheet, the A sheet is half the size of the A1 sheet and has the same shape and so on..1 Eplain what it means that the sheets all have the same shape.. Find the ratio of the length to the breadth of each rectangular piece of paper. All sheets have the same shape An A0 sheet folds into two A1 sheets A0 A1 A1 A A3 A3 A A4 A4 3 The golden rectangle has been recognised through the ages as being aestheticall pleasing. It can be seen in the architecture of the Greeks, in sculptures and in Renaissance paintings. Grade Eemplar Assessments 008

9 3.1 Measure and and hence estimate the golden ratio : 3. The golden rectangle has the propert that when a square the length of the shorter side of the rectangle is cut from it, another rectangle with the same shape is left. A E B AB BC = BC EB D F C The process can be continued indefinitel, producing smaller and smaller rectangles. Using this information, calculate the ratio : in surd form. 3.3 Write the ratio (often called phi: φ ) correct to 3 decimal places. 4. A sequence of rectangles can be built up in the following wa : start with a rectangle made b two identical squares placed net to each other. The net rectangle is formed b adding a square on to the longer side of this rectangle. The process can be continued indefinitel. Investigate the ratio of the length to the breadth of these rectangles. Let the first rectangle have dimensions units b 1 unit. 5. A line segment, AB can be divided in the golden ratio at C in the following wa: 5.1 At B, draw a perpendicular to AB and mark off BD equal to half AB. 5. Join AD. 5.3 On DA mark off DE equal to DB. 5.4 On AB mark off AC equal to AE. 5.5 Now prove that this construction divides AB so that AC:CB=φ Grade Eemplar Assessments 008

10 Grade 11 Test: Equations, Inequalities and Eponents Time: 1 hour Marks: 50 Question 1 Solve for in each of the following; 1.1 ( 4 ) = 1 (4) = (5) = (6) < 5 (5) = 8 Question Solve simultaneousl for and in the following sstem of equations. + = 1 and = 3 3 Question 3 (3) [3] [7] The last nuclear test eplosion was carried out b the French on an island in the south Pacific in Immediatel after the eplosion, the level of Strontium-ninet on the island was 64 times the level considered to be "safe" for human habitation. If the half-life of Strontium-ninet is 8 ears, how long will it take for the island to once again be habitable? (Half-life is the amount of time it takes for half of the amount of a substance to deca.) ( Hint: Let the original amount of Strontium-ninet present be equal to and it then follows that the amount present after 8 ears is ) [8] Grade Eemplar Assessments 008

11 Question 4 Computer A takes 1 minutes more to print the Grade 11 eam papers than does computer B. Used together the computers can print the papers in 8 minutes. Find the time that computer A, used alone, will require to print the papers. [6] Question 5 The two numbers 4 and have the peculiar propert that their difference equals their quotient. That is = and 4 =. There are other numbers with this propert: 4 and 3, 5 and 4 are 3 two other pairs. 5.1 Find a fourth pair of numbers with this propert. () 5. Eplain how pairs of numbers with this propert can be generated. (4) [6] Grade Eemplar Assessments 008

12 Grade 11 Project: Finance Marks: 50 This project is designed to be done b groups of three. Where the number in the class is not divisible b 3, one or two groups must have 4 members and approach 4 companies when tackling the second task described below. It is suggested that each group member approach a different compan, which also needs to be different from the companies approached b others in the group or b other groups. It is suggested that time be allocated to choosing companies so that enough different companies are identified. Your teacher will provide ou with a letter which ou can use to make contact with a compan. 1. Two companies each bought machiner worth R at the same time. Compan A valued the machiner as follows at the end of each of the net three ears: Year 1: R ; Year : R7 500 ; Year 3: R Compan B valued their machiner at the ear ends as follows: Year 1: R ; Year : R ; Year 3: R Each claim that the machiner is depreciating at 15% p.a. 1.1 Investigate and comment on the claim b each compan. 1. Write down the value of the machiner for each compan after n ears.. Write a paragraph on the methods used b three different companies to value depreciating equipment each ear. Compare methods used and suggest reasons for an differences found. Give the name of the compan, the nature of the equipment which is depreciating and the formulae used b each compan to determine the current value of their equipment. Also indicate when the compan plans to replace their depreciating equipment and how the calculate the epected cost to the compan for replacement. Assessment Marks will be awarded as follows: marks 1. 6 marks. 5 marks for calculations 15 marks for eplanation The group members can then all take this mark, or the mark can be multiplied b 3 (or 4 if there are 4 in the group) and different marks can be awarded according to the contribution of each member. The average of the marks in the group must be the mark allocated b our teacher. Grade Eemplar Assessments 008

13 Grade 11 Mathematics Eam Paper 1 Time: 3 hours Marks: 150 Question Solve for in each of the following; (5) ( 1)( + 3) = > 0 (4) = (5) 1. Solve for both and in the sstem of equations below. + = 1 and = 3 3 (7) [1] Question Simplif the following; (3) (3) (3) (5) [11] Question 3 In the sketch below, a new line is drawn each time, from the same verte to the side opposite. This leads to more and more triangles being formed each time. Triangle 1 Triangle Triangle 3 Grade Eemplar Assessments 008

14 3.1 Fill in the table on our diagram sheet Sketch number No of internal lines 0 1 Total number of triangles (4) 3. How man internal lines would be added in the n-th sketch? (1) 3.3 Determine how man triangles would there be in the n-th sketch? (5) 3.4 Which sketch would have 153 triangles? (4) [14] Question 4 The diagram shows a sequence of patterns. Each one is made b surrounding the previous pattern (shade black) b squares that are shaded gre. 4.1 On the square grid on our diagram sheet, draw the net pattern () 4. There are two sequences formed. The first, is the number of squares added to each new pattern (the gre squares), the second is the total number of squares making up the pattern. Write down the first si terms of each sequence (6) 4.3 Find a formula for the n th term in each sequence. (4) [1] Grade Eemplar Assessments 008

15 Question After just ears, a laptop computer is one third it s original value. Assuming reducing balance depreciation, what was the annual rate of depreciation? (5) 5. Sindiswa is investing her mone with a local investment compan that is offering her an interest rate of 14% p.a compounded quarterl. Her colleague, Linda, is investing hers with a professional bank that is offering her 1,5% p.a compounded monthl. Calculate who will receive a better return. (6) 5.3 Bron deposits R500 into a bank account and makes no withdrawals for 8 ears. At the end of the fifth ear he deposits an additional R100. If the interest rate for the first 4 ears is 8% p.a compounded quarterl and 9,5% p.a compounded semi-annuall for the remaining four ears, what will have accrued in the account at the end of the eighth ear. (7) [18] Question 6 Below, the following two functions are sketched; f ( ) = 3 and f g g ( ) = Find the zeroes of both functions. (5) 6. Determine, using an method, the coordinates of the minimum value of f (4) S 6.3 ST is drawn such that it is perpendicular to the ais. If the length of ST = 4, find the coordinates of S. (4) T 6.4 Find the average gradient of the curve f between = 0 and = 3 (3) 6.5 Give the equation of k if k () results from shifting f () a 4 3 unit to the left. () 6.6 Give the equation of j if j () results from shifting g () 1 unit up. () [0] Grade Eemplar Assessments 008

16 Question 7 4 Consider the equation h ( ) = Determine the root of h ( ) = 0 (3) 7. What is the value of h (0)? (3) 7.3 Write down the equation/s representing the asmptote/s of this function () 7.4 Draw a sketch of h () showing clearl all intercepts with aes and an asmptotes that ma eist. If there are an lines of smmetr, indicate these on the sketch. (6) Question 8 [14] The graphs below are the functions p( ) = a. b c and s( ) = b c s p What is the value of c () 8. If the curve of () 8.3 If p () cuts the ais at s passes through the point ( ;1) 8.4 Determine the equation of r if () 1, find the value of b. (3) =, find the value of a. (3) 3 p is shifted units to the right to create r () () [10] Grade Eemplar Assessments 008

17 Question 9 1 f g The sketch represents the functions f ( ) = a cosb and g ( ) = csin( + d) for the interval [ 60;360] Determine the values of a, b, c and d (6) 9. Read off the graph the value of for which f ( ) = g( ) (1) 9.3 Write down the equation of h if o h ( ) = g( 10) () 9.4 What is the equation of p if p () is the reflection of f () in the line = 0 (1) [10] Question 10 A chocolate factor produces both chocolate coated raisons as well as chocolate coated peanuts. In a da, the can produce a maimum of 100kg s of raisons and 15kg s of peanuts. The are mied together and sold in two different packets. The Nuts about Nuts packet has one third raisons and two thirds peanuts. The other packet, Half-n-Half, has equal quantities of both.packets of Nuts about Nuts sell at a profit of R4 per kilogram while the Half-n-Half sells at a profit of R5 per kilogram. Let there be kg s of Half-n-Half produced in a da and kg s of Nuts about Nuts Give, in terms of and, the mathematical constraints that must be satisfied each da. (4) 10. Write down a function for the Profit (P) to be made () 10.3 Illustrate the constraints graphicall, on the grid paper provided. Clearl indicate the feasible region. (8) 10.4 Use our graph to determine; the values of and that will ensure maimum profit. (4) the profit earned according to the values found in () [0] Grade Eemplar Assessments 008

18 Diagram Sheet Question 3 Question 4 Sketch number No of internal lines 0 1 Total number of triangles Question 10 Grade Eemplar Assessments 008

19 Grade 11 Mathematics Eam Paper 1 Time: 3 hours Marks: 150 Question Solve for in each of the following (in simplest surd form where applicable): = 1 (4) (4) = (5) 1. Solve for both and in the sstem of equations below. + 6 = 0 and = 0 (7) [0] Question Simplif the following;.1 3 ( 3 6) + + (Answer in simplest surd form) (3). Solve for : 3. = 0, 375 (3) 8.3 The mass of a certain microorganism is ( 5 10 )g. How man organisms are there in a population with a total mass of 0,5 g? (4) Question 3 [10] 3.1 Nthabi is running a small business. She has just bought equipment for R She decides to depreciate the equipment at 0% p.a. on the straight line basis. When will she write the equipment off? () 3.1. Nthabi changes her mind and depreciates the equipment at 5% p.a. on the reducing balance. Calculate the value of the equipment after 5 ears. Give our answer correct to the nearest Rand. (4) 3. Which is the better investment offer: 10,8% p.a. compounded dail (use 365 das in a ear) or 10,3% p.a. compounded monthl? (6) Grade Eemplar Assessments 008

20 3.3 John inherits R and decides to invest the mone so that he can finance weddings for his two daughters. The one gets married eactl one ear later and he withdraws R to pa for this event. If interest of 10% p.a. compounded quarterl is applicable for two ears and this rate then changes to 9,5% p.a. compounded monthl, calculate how much he will have available when his other daughter gets married after five ears (four ears after the first daughter). (8) [0] Question 4 In the sketch below a 4 b 4 game board is illustrated. The aim of the game is to move the disc in the bottom left hand block into the top right hand block in the smallest number of moves. It was found that for this board, the smallest number of moves is 1. The same game can be plaed on bigger or on smaller boards. The following are the smallest number of moves on the given sized boards: b 3 b 3 4 b 4 5 b 5 Smallest number of moves What is the smallest number of moves on a 6 b 6 board? () 4. What is the smallest number of moves on an n b n board? (4) 4.3 Lindiwe sas the smallest number of moves for an n b n board is an + b where a and b can be determined from the equations: 5 = a + b and 13 = 3a + b Is she correct? Justif our answer full. (6) Question 5 [10] Consider the following patterns of diamond shapes: 5.1 How man diamonds( ) are there in the net pattern? () 5. How man diamonds are there in the nth pattern? (4) 5.3 Which pattern has 960 diamonds? (4) [10] Grade Eemplar Assessments 008

21 Question 6 Given the functions = f ( ) = 1 ( + 1) + and = g( ) = 6 : 6.1 Write down the co-ordinates of the turning point of f () 6. Calculate the roots of the equation ( ) = 0 f (4) 6.3 Write down the equation of the ais of smmetr of f. (1) 6.4 Sketch the graphs of = f ( ) and g( ) 6.5 Determine the values of for which ( ) g( ) = on the same sstem of aes (4) f (4) 6.6 Describe in words the difference between shape of = f ( ) and f ( ) = () [17] Question 7 1 Sketched below are the functions = f ( ) = and = g( ) =.. A and B are intercepts of the two graphs with the -ais and C is the point of intersection of the two graphs. DEF is parallel to the -ais with D and E on the two graphs. ^ 7.1 Determine the distance AB. (3) g f 7. Given that OF = 3 units, determine the average gradient between the points 7..1 A and E and () A B O C D E F > 7.. B and D and hence (1) 7..3 determine which curve is steeper between = 0 and = 3 (1) 7.3 Determine the co-ordinates of C () 7.4 Write down the equation that results if = 1 is shifted 1 unit to the right. () 7.5 Write down the equation if = 1 is shifted 1 unit down. () [13] Grade Eemplar Assessments 008

22 Question 8 3 = 1 Sketched below are the graphs of p ( ) and q ( ) = Calculate the co-ordinates of A and B. (4) 8. Write down the equation of the horizontal p () asmptote of ( ) ^ C 8.3 Write down the domain of p ( ) () O B > 8.4 Show that ( ) = q( ) p and state the significance of this fact to the sketch. (3) D 8.5 Determine the co-ordinates of D if CD = 4 where CD is perpendicular to the -ais. (5) A [16] Question The sketch below represents the function f ( ) = a cos( b + c) for the interval [ 180 ;180 ] 9.1 Determine the values of a, b and c (4) 9. On the diagram sheet, plot the graph of g = sin (4) ( ) 9.3 Write down the equation of the reflection of f ( ) about the ais. () 9.5 Read from our graph the values of [ 180 ] 0 ; Write down the equation of the reflection of g ( ) about the ais. () for which f ( ) = g( ) (3) [15] Grade Eemplar Assessments 008

23 Question 10 The sketch below represents the feasible region ABCDEF of a linear programming problem. 8 A F 6 B E 4 C D 5 10 > 10.1 Two of the constraints are 1 6 and Write down the inequalities that represent the other constraints. (5) 10. The point ( 5; p) Q q;3 is not in the feasible region. Write down one possible value of p and one possible value of q. () P lies in the feasible region and the point ( ) 10.3 Find the values of and for which the objective function O1 = has a minimum value. () 10.4 Write down the minimum value of O 1 (1) 10.5 Find the values of and for which O = 3 4 is maimised. () Write down the maimum value of O (1) 10.7 Marlene sas that O3 = maimises at more than one point. Is she right? Eplain. (4) [17] Grade Eemplar Assessments 008

24 Diagram Sheet Name: Question Grade Eemplar Assessments 008

25 Grade 11 Assignment: Analtical and Transformation Geometr Marks: 50 A metacog ma be seen as something between a mind map and a summar. It is the wa ou choose to order our knowledge and understanding of a particular topic. Using our notes, tet book and an other reference source, develop a metacog that ou can use as a learning and revision aid for analtical and transformation geometr. Use the rubric provided as a guide to what ou should include. Once ou have developed our metacog, ou will be required to use it to answer a series of questions. RUBRIC [35] INCLUSION OF IMPORTANT CONTENT ARRANGEMENT OF CONTENT INCLUSION OF FORMULAS EXPLANATION OF FORMULAS DERIVATION OF FORMULAS USE OF DIAGRAMS APPLICATION OF KNOWLEDGE Substantial amounts of important content have been omitted. No attempt made to group and sequence content. Formulas have not been included. Formulas have not been included. Formulas have not been included. No attempt made to include diagrams. No attempt made to include applications of knowledge. Ke aspects of content have been omitted. Little attempt made to group and sequence content. Fewer than half the necessar formulas have been included. Elements of formulas seldom eplained. Derivation of formulas seldom noted. Inadequate attempt made to include diagrams. Inclusion of applications of knowledge mostl inadequate. Most important content has been included. Either the sequencing or grouping of content is not logical. Most of the necessar formulas have been included. Eplanation of elements of formulas omitted in several instances. Derivation of formulas not noted in several instances. Diagrams not alwas included resulting in confusion about content. Applications of knowledge are mostl sufficient. Almost all important content has been included. Most content is both logicall grouped and sequenced. One or two necessar formulas have been omitted. One or two omissions in eplanation of elements of formulas. One or two omissions in noting derivation of formulas. Diagrams included, but on occasions, either unclear or not relevant. Applications of knowledge are included but either contain repetition or are not sufficient in one or two instances. All important content has been included. Content has been both logicall grouped and sequenced. All necessar formulas have been included. The elements of all formulas have been clearl eplained. Derivation of all formulas has been clearl noted. Diagrams included. Clearl drawn and relevant. Ke and different applications of knowledge are included. Grade Eemplar Assessments 008

26 USING YOUR RUBRIC, ANSWER THE FOLLOWING QUESTIONS: (No assistance ma be given or an other source used in order to answer the questions.) 1. Write down the formula for calculating the distance between two points.. Which theorem is used to derive this formula? 3. Eplain how ou would use the distance formula to prove that two triangles are congruent. 4. Write down the formula for calculating the midpoint of a line joining two points. 5. Eplain how ou would use the midpoint formula to prove that a quadrilateral is a parallelogram. 6. What is the difference between gradient and inclination? 7. Eplain how ou would use gradient to determine whether or not three points were collinear. 8. If two lines are perpendicular, what is the relationship between their gradients? 9. When ou tr to calculate the gradient of a line using the inclination, our calculator shows an error message. How do ou interpret this? 10. Write down the general equation of a straight line passing through two points. 11. If ou know the equation of one side of a parallelogram, what additional information do ou require in order to calculate the equation of the opposite side? 1. The point A(4;1) is mapped onto A' using the rule (;) (+1;-). Where is the point A' in relation to A? 13. The point B' is the reflection of B(0:-) about the line =. What are the coordinates of B'? 14. Eplain the transformation that is being used if (;) (-;-). 15. Polgon A'B'C'D' is an enlargment through the origin b a factor of of polgon ABCD. Describe the relationship between AB and A'B'. [15] Grade Eemplar Assessments 008

27 Grade 11 Investigation: Marks: 50 P P' R' Q R Refer to the figure above. PQR is a right-angled triangle, with the right-angle ling on the origin and coordinates P(0;6), Q(0;0), R(3;0). PQR has been rotated about the origin so that R' lies on the hpotenuse PR. 1 Use an method to determine the angles of PQR and P QR, and the angle of rotation of PQR. Show all calculations. (6) Develop a conjecture about the relationship between the angle of rotation and P. () 3 Prove our conjecture. (4) 4 Investigate the conditions under which the conjecture is true and under which conditions it is no longer true. Write up our conclusions, with proofs where possible. 5 Epand the investigation to include the following situations and write up our results: P P Q Isosceles R Q Equilateral R 6 What happens when PQR is scalene? Grade Eemplar Assessments 008

28 Assessment Guidelines Questions 1, and 3 will be marked according to a memorandum, with marks as indicated. The remainder of the investigation will be marked using the grid below: How well have ou communicated our ideas and discoveries? You should assume that the person marking our work has not seen the problem before. You should include diagrams, -D models, accurate constructions (as appropriate) to clarif our communication. How well ou have thought about the problem. Is there evidence that ou have investigated the problem? Have ou considered different possibilities? Have ou made an attempt to etend our thoughts and ideas beond the obvious? Has our investigation led ou to make an conjectures about the problem? Have ou attempted to prove or disprove these conjectures? Have ou successfull proved or disproved the conjectures? Q Q Q PRESENTATION 0 1 Grade Eemplar Assessments 008

29 Grade 11 Test: Trigonometr and Mensuration Time: 1 hour Marks: 50 Question Simplif the following epressions without the use of a calculator and show ALL the calculations: o cos38 sin 315 o o cos 5 sin18 o (3) 1.1. sin ( ). sin ( 90 + ) tan( ) + sin. cos ( 90 ) (5) 1. If 3 sinα = and 180 < α < 70, determine the value of: tan α sin α. () 1.. sin α + cos α () 1..3 Determine the numerical value of α. (1) 1.3 Show that: cos 1+ cos = sin tan (5) [18] Question.1 If 3 sin θ = 1,347, calculate cos (θ - 15 ) for θ (0 ; 180 ) (4). Determine the general solution for : sin. (cos 1) = 0 (6) [10] Grade Eemplar Assessments 008

30 Question 3 In the diagram below : AB = 3 units ; AD = 9,5 units ; A ˆ = o 11 ; o C BD ˆ = BDˆ C = 67 3 B 67 o C A 11 o 9,5 67 o D 3.1 Show, b calculation, that BD = 10,98 units () 3. Hence calculate the perimeter of ABCD. (6) 3.3 Calculate the area ABCD (4) [1] Question 4 A time-capsule is made of a cone and a clinder and is filled with goods for remembrance. Its dimensions are as given in the sketch. 4.1 Calculate the volume of the capsule. (5) 0 cm 50 cm 4. Calculate the total surface area of the capsule. (5) 15 cm [10] Grade Eemplar Assessments 008

31 Grade 11 Project: Enlargements Marks: Section A The centre of enlargement is the origin for all these eamples. Definition: Scale Factor - is the number ou multipl the original lengths b to get the lengths on the enlargement. Each grid has an -ais from 1 to 15, and a -ais from to 15. Question A' 1.1 Use the grid and complete: A ( ; ) B ( ; ) A' ( ; ) B' ( ; ) 6 1. What do ou notice about the co-ordinates of the images A' and B'? 4 A B' 1.3 Calculate the length of AB and A'B'. B 1.4 What do ou notice? What is the scale factor for this enlargement? Question 1 A'.1 Use the grid and complete: A ( ; ) B ( ; ) A' ( ; ) B' ( ; ) What do ou notice about the co-ordinates of the images A' and B'? 6 A.3 Calculate the length of AB and A'B'. 4 B'.4 What do ou notice? B What is the scale factor for this enlargement?.6 Calculate the area for each triangle..7 What is the area scale factor? Grade Eemplar Assessments 008

32 Question 3 14 A' D' 3.1 Use the grid and complete: A ( ; ) B ( ; ) A' ( ; ) B' ( ; ) 1 3. What do ou notice about the co-ordinates? Calculate the length of CB and AC. 8 6 A C' D B' 3.4 Calculate the length of C'B' and A'C' 3.5 What do ou notice? 4 C B 3.6 What is the scale factor for this enlargement? 3.7 Calculate the area of both rectangles What is the area scale factor? Question Write down conjectures about the coordinates, lengths of sides and area of A' B' C' D' in question Do ou think this will alwas be true for an enlargement through the origin? Question Put each shape below through the enlargement described. All enlargements are through the origin. Draw the enlargements on the grids provided. Use our conjecture to determine the co- ordinates of the enlargements. Scale factor = 3 Scale factor = Did our conjecture give the correct answers? Eplain. Grade Eemplar Assessments 008

33 Question 6 Let s tr and prove this conjecture! Use the grid provided. A' The scale factor is k. (Hint: Prove that BC k = B'C') A B' C' B(;) C( + BC ; ) Question 7 The cube and the circle below show enlargements or reductions but have not been drawn to scale. In each case, write down the scale factor and calculate. (The volumes of the cubes and the areas of the circles are given.) 7.1 P Q S R P / Q / M 51 cm 3 N S / 64 cm 3 R / M / N / T K T / / K / cm π 49 cm π / 7.3 Write down an conjectures that ou notice about the above enlargements. Grade Eemplar Assessments 008

34 Section B Note: Finding the centre of Enlargement To find the centre of enlargement join A to A, B to B and C to C etending them if necessar. The point where these lines intersect is the centre of enlargement Question 1 All the diagrams below show an enlargement or reduction. Find the position of the centre of enlargement or reduction and state the value of the scale factor A A' D G D' G' B C E' F' E F B' C' P' Q' P Q E' F' F E H H' S R S' R' P Q G P' Q' G' S S' R R' Question Determine a general formula for a scale factor k and centre of enlargement (a ; b) Grade Eemplar Assessments 008

35 Grade 11 Mathematics Eam Paper Time: 3 hours Marks: 150 Question 1 In the diagram below, L(-5; -), M(-1; -6) and K(5; 4) are the vertices of KLM in a Cartesian plane. Determine: K( 5 ; 4 ) 1.1 N, the midpoint of MK (3) 1. the gradient of LM (3) 1.3 the length of LM (leave the answer in simplest surd form) L( -5 ; - ) (3) 1.4 equation of the LM (4) 1.5 the equation of the line parallel to LM passing through N. (4) M( -1 ; -6 ) 1.6 the inclination of LM (3) [0] Question In the diagram below, A(4 ; 3), B ( ; 7), C (- ; 5) and D (0 ; 1) are four points in a Cartesian plane. B( ; 7 ).1 Show that CA= BD (5) C( - ; 5 ). Show that the coordinates of M, the midpoint of BD, are (1 ; 4) (3) A( 4 ; 3 ).3 Prove that AM BD (5).4 Prove that A, M and C are collinear (3) D( 0 ; 1 ).5 State, giving a reason, which tpe of quadrilateral ABCD is. () [18] Grade Eemplar Assessments 008

36 Question 3 The diagram alongside shows ABC with its transformation A / B / C /. C' 8 A 3.1 Write down the coordinates of A and A / () 3. Describe the above transformation. () 3.3 If A // B // C // is the rotation of ABC through 180º, sketch A // B // C // using the diagram sheet provided. (3) A' B' B C 3.4 State the coordinates of verte C //. () 3.5 Write down the general coordinates of the transformation as described in question 3.3. () Question 4 [11] 4.1 The diagram below shows quadrilateral PRMT with P(3 ; 5) on the Cartesian plane. 6 P Sketch, using a scale factor of 4, the enlargement of PRMT through the origin using the grid on the diagram sheet. (4) 4 R 4.1. Write down the coordinates of R / and T / on the sketch. () T M 5 4. If K(3 ; 5) is rotated through the origin in a clockwise direction through an angle of 90 o, write down the coordinates of K /, the image of K. () 4.3 ABC has coordinates A( ; 6), B( 3 ; 4) and C(1 ; 8 ) and its transformation A / B / C / 3 1 has coordinates A(1 ; 3), B ; and C ; 4. Describe the transformation. (3) [11] Grade Eemplar Assessments 008

37 Question Simplif the following epressions and show ALL the calculations without using a calculator 0 0 cos( 180 ) sin( 90 ) tan ( ) sin( 90 + ) cos( ) sin 63.cos 135.tan sin 40.tan150.cos 7 (6) (7) 5. Prove the following identit: cos ( + tan ) = sin (5) 5.3 Solve for in the following equation: sin 1 = 0 for [0 ; 360 ] (4) 5.4 Determine the general solution of the equation: 4sin 3 = 0. (6) [8] Question 6 Sketched below are the graphs of functions The curves intersect at points O and Q. f ( ) = cos and g ( ) = sin( 60 ) for [ 180 ; 180 ]. 4 P f Q g 6.1 Determine the co-ordinates of the point Q (6). 6. State the range of g if the graph of g undergoes a positive, vertical shift of 1 unit. () 6.3 Write down the new equation of g if it is shifted 60º horizontall to the left. () [10] Grade Eemplar Assessments 008

38 Question In the accompaning figure ABCD is a quadrilateral with AB = 3cm, o and D AC ˆ = 81,8. B ˆ = o 10, BC = 5cm, DC = 8cm D Calculate: AC (4) 8cm 7.1. Dˆ correct to 1 decimal place. (4) A 81,8 C 3cm 10 B 5cm 7. In the figure ABC is an isosceles triangle with b = c Use the figure to show A ( area ABC) that b = () sin A 7.. Now use the cosine rule to a show that b = B C () (1 cos A) Question 8 [1] 8.1 A time-capsule in which mementoes will be saved consists of a clindrical bod with a with a hemisphere on top. The diameter of the clinder is 30 cm and the height of the clinder is 150 cm. Calculate the total volume of the time-capsule. 150 cm 30 cm (5) Grade Eemplar Assessments 008

39 8. A tent manufacturer makes a tent in the shape of a pramid h with a square base. The square ground sheet of the tent is attached to the tent. If slant height, h, is 350 cm and a side of the base is 00 cm, calculate the total surface area of the tent. Question 9 00 cm (5) [10] 9.1 Below are the percentage scores that 15 learners obtained in a Phsical Science Eamination What is the median for the above data? () 9.1. Write down the upper and lower quartiles. () Draw a bo and whisker diagram for the data. () 9. The traffic department investigated where it would be most appropriate to in install speed cameras. As part of their investigation a surve was done of the different speeds of vehicles on a stretch of a national road. The following table shows the results of the surve: SPEED (in km/h) FREQUENCY (Of vehicles) 40 < d < d < d < d < d < d < d CUMULATIVE FREQUENCY 9..1 How man vehicles were observed in the surve? (1) 9.. Complete the cumulative frequenc column using the table on the diagram sheet. () 9..3 Represent the information in the table b drawing an ogive (cumulative frequenc curve) on the grid provided on the diagram sheet. (4) 9..4 Use our graph to determine the median speed. Indicate on our graph using the letter T where ou would read off our answer. (3) [16] Grade Eemplar Assessments 008

40 QUESTION 10 The data below gives the number of homeruns scored b ten Western Province baseball plaers over 5 seasons DATA ( i ) ( i ) n ( i ) i= 1 = 10.1 Calculate the mean number of homeruns. () 10. Complete the above table on the diagram sheet and use it to calculate the variance of the homeruns. (4) 10.3 Now write down the standard deviation. (1) 10.4 What can ou deduce from the standard deviation about the homeruns scored b the ten plaers? () [9] Grade Eemplar Assessments 008

41 QUESTION 11 A car with a 65 l fuel tank capacit used fuel over certain distances travelled as shown in the graph below: Distance - Fuel Consumption Fuel (in l) Distance (in km) 11.1 Draw an approimate line of best fit and find its equation. (4) 11. Use our equation to determine after how man kilometres the tank will be empt. (1) [5] Grade Eemplar Assessments 008

42 Diagram Sheet Question 3 C' 8 A 6 A' 4 B' C B Grade Eemplar Assessments 008

43 Question P 4 T M R Question SPEED (in km/h) FREQUENCY (Of vehicles) 40 < d CUMULATIVE FREQUENCY 60 < d < d < d < d < d < d 00 1 Grade Eemplar Assessments 008

44 9..3 and 9..4 Grade Eemplar Assessments 008

45 Question 10 DATA ( i ) ( i ) n ( i ) i= 1 = Question 11 Distance - Fuel Consumption Fuel (in l) Distance (in km) Grade Eemplar Assessments 008

46 Grade 11 Mathematics Eam Paper Time: 3 hours Marks: 150 Question 1 In the diagram, P(4;0) Q(-;-) R(-5;4) and S(3;7) are the vertices of quadrilateral PQRS which lies in the Cartesian plane. S ( 3 ; 7 ) Determine: 1.1 The gradient of RS. (3) 1. The equation of RS. (4) R ( -5 ; 4 ) 1.3 The inclination of RQ. (3) 1.4 The measure of angle RQP. (4) P( 4 ; 0 ) 1.5 The midpoint of PS. (3) 1.6 The length of PQ (leave our answer in simplest surd form). (3) Question Q ( - ; - ) 1.7 The equation of the line which is parallel to RQ and passes through P. (3) [3] In the diagram, PQ is the diameter of the circle. The equation of PQ is + + = 0..1 Determine the co-ordinates of P, the -intercept of PQ. () Q R M S ( -6 ; - ) P + + = 0. If Q, which lies on the circle and on PQ, is the point (a;4), calculate the value of a. ().3 Show that the co-ordinates of M, the centre of the circle are (-3;1). ().4 Calculate the length of the radius of the circle. Leave our answer in simplified surd form. (3) Grade Eemplar Assessments 008

47 .5 A line, AB, is drawn through M, parallel to the -ais. If R is the reflection of P about the line AB, determine the co-ordinates of R. ().6 The point S(-6;-) lies on the circumference of the circle. Show that MS is perpendicular to MP. (4).7 Calculate the area of PQS. (4) [19] Question The point W(4;3) lies in the Cartesian plane. Determine the co-ordinates of the image of W under each of the following conditions: W is reflected about the line = () 3.1. W has been rotated about the origin through 90 o in a clockwise direction. () W ( 4; 3) ( ; ) () 3. In the diagram, polgon ABCD has vertices A(8;5), B(5;1), C(7; ) and D(9; 0) 6 A 4 C B D PQRS is a transformation of ABCD and has co-ordinates P(-8;5) Q(-5;1) R(-7;) and S(-9;0). Describe the transformation. () 3.. A B C D is the image of ABCD under the following transformation ( ; ) ( 4; + 1). Draw A B C D on the diagram sheet provided. () 3..3 Describe the above transformation. () Grade Eemplar Assessments 008

48 3..4 The transformation T of the Cartesian plane occurs as follows: A point is rotated about the origin 180 o in the clockwise direction. After this, the point is reduced through the origin b a factor of. If KLMN is the image of A B C D after appling the transformation T, sketch KLMN on the diagram. (4) 3..5 Describe the relationship between the perimeter of A B C D and the perimeter of KLMN. () [18] Question Evaluate: sin 35 cos65 + tan 45 (4) 4. Simplif without the use of a calculator: cos (180 + ) cos( ) tan(180 ) sin( ) + sin(360 ) (8) 4.3 ABC is an triangle. Prove that sin( A + C) = sin B. (3) sin 4.4 Prove the following identit: = tan 3 cos + cos sin (4) 4.5 Solve for in the following equation: 3 tan 3 = 0 for [ 180 ;360 ] (5) 4.6 Given the trigonometric equation 4sin 1 = Factorise the epression 4sin 1 () 4.6. Hence find the general solution of the equation: 4sin 1 = 0 (4) [30] Grade Eemplar Assessments 008

49 Question 5 The sketch below contains the graphs of g ( ) = cos ` and h( ) = sin for [ 180 ;180 ]. The coordinates of I, a point of intersection of the two graphs are 60 ; 3 1 ( ) g( ) = cos h( ) = sin ( ) Use our graph to solve the following equation: cos = sin for [ 180 ;180 ] (3) 5. If g is shifted horizontall 30 degrees horizontall towards the right, what will the new equation of g be? () 5.3 Determine the new co-ordinates of I after g has been shifted. (3) 5.4 If the equation of h is changed to h( ) = sin +1, eplain clearl how the graph of g will change. You must refer to range, amplitude and period. (3) [10] Grade Eemplar Assessments 008

50 Question 6 Patrick has bought a plot of land which has an irregular shape as shown below. He has obtained an old diagram of the land and wishes to work out the length AB of the plot. DC is 11 metres, AD ˆ B is marked and angle BC ˆ E is marked. CE is parallel to DA. B C E D A 6.1 Find the size of angle CBD in terms of and. () 6. Show that 11cos DB = (3) sin( ) 6.3 Now show that 11cos sin AB = () sin( ) 6.4 A ke on the diagram indicates that = 37, 3 and = 4. Use this information to calculate the length of AB. () [9] Question 7 In ABC, AB = c, BC = a, and size of angle B. AC + = a + ac c. Without using a calculator, find the [4] Grade Eemplar Assessments 008

51 Question The Cit Municipalit has ordered special heagonal pramid cones to place in the road to warn drivers of unusual hazards. The cones have to be painted with a red reflective paint. The base of the pramid is a regular heagon consisting of si equilateral triangles with side of 30cm as shown below: Calculate the area of the base of the pramid. (3) 8.1. The heagonal pramid cones are shown in the diagram alongside. The slant height, h, is 65cm. Calculate the surface area of a cone. h (3) 8. A can of aerosol deodorant has a clindrical shape with a hemisphere on top as shown in the diagram below. The diameter of the clinder is 6cm and the height of the clinder is 17cm. Calculate the total volume of the can of deodorant. 6cm 17cm (5) [11] Grade Eemplar Assessments 008

52 Question 9 Below are bo and whisker plots that depict the results of a pollution surve in a Western Cape River conducted from 005 to 007. The surve measured to pollution (total dissolved solids) in milligrams per litre of water TOTAL DISSOLVED SOLIDS (mg/l) Determine the five number summar for 006. () 9. Which ear had the greatest range of results? Eplain our answer. () 9.3 Using the data, comment on the pollution levels in the river over the three ears of monitoring. (3) [7] Question The heights of 10 primar school bos were measured. These heights are given in the table below: Height (cm) ( i ) n i= 1 ( i ) ( i ) Calculate the mean height of the bos. () Complete the table on the diagram sheet and use it to calculate the variance of the heights. (3) Calculate the standard deviation. (1) What conclusion can ou draw about the height of the bos using the standard deviation? (1) Grade Eemplar Assessments 008

53 10. In addition to height, the age of the bos was also recorded and a scatter diagram of this data was produced. This diagram appears below: Age to Height Comparison of Primar School Bos Height (cm) Age (Years) On the diagram sheet, draw an approimate line of best fit. (1) 10.. Use the line to estimate the height of a 9-ear old bo. Indicate on the diagram where ou made this estimate. () Find the equation of the line of best fit. (3) Use our equation of the line of best fit to predict the height of a 16 ear old bo. (1) [14] Grade Eemplar Assessments 008

54 Diagram Sheet Question A 4 C B D Grade Eemplar Assessments 008

55 Question 10 Height (cm) ( i ) ( i ) n i = 1 ( i ) Age to Height Comparison of Primar School Bos Height (cm) Age (Years) Grade Eemplar Assessments 008