American Community Schools Department of Mathematics

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1 American Community Schools Department of Mathematics June, 2016 Re: Summer Mathematics Review Packet Every year the math department (all teachers JK-12) prepares review packets for all grade levels in the elementary school and all courses in the Middle School and the High School. In the beginning of this year and previous years, we received positive input from parents and students pertaining to this assignment. We are eager to present to you our review packets (attached) this year and we hope your child will spend stimulating and productive time completing this summer assignment. Although it is important for children to rest in the summer months, it is equally important that children continue to read and do math problems. In September your child s teacher will record this assignment as complete or incomplete. It is a requirement to complete and therefore students will be held accountable for its completion. Furthermore, to ensure that this packet was in their possession this summer we ask that you sign below confirming receipt of the packet. In September Academy and Middle School teachers will request the return of the packet with parental signature. A signature is not necessary for next year s elementary school students. Solutions are provided (wherever deemed necessary) so that a child may assess him or herself along the way. The material attached consists of problems relating to this year s taught curriculum. Children attending Early Childhood (JK and K) will be receiving age-appropriate hands-on activity ideas to complete during the summer. The math department (JK-12) wishes everyone a restful and enjoyable summer. Best regards, Dr. Tsokos Division Chair Ms. Andrikopoulos- Mathematics Coordinator Parent Signature: Student Name: To prepare for study in IB Mathematical Studies SL Year 2 course

2 Summer Assignment IB Mathematical Studies Preparing students entering IB Mathematical Studies SL Year 2 Paper 1 Practice Questions 1. Given p = x z y, x = 1.775, y = 1.44 and z = 48, calculate the value of p. Barry first writes x, y and z correct to one significant figure and then uses these values to estimate the value of p. (i) Write down x, y and z each correct to one significant figure. (ii) Write down Barry s estimate of the value of p. (c) Calculate the percentage error in Barry s estimate of the value of p. (Total 6 marks) 2. The table below shows the frequency distribution of the number of dental fillings for a group of 25 children. Number of fillings Frequency q 4 1 Find the value of q. Use your graphic display calculator to find (i) (ii) (iii) the mean number of fillings; the median number of fillings; the standard deviation of the number of fillings. (4) (Total 6 marks) 1

3 Summer Assignment IB Mathematical Studies 3. The graph of the quadratic function f(x) = 3 + 4x x 2 intersects the y-axis at point A and has its vertex at point B. Find the coordinates of B. Another point, C, which lies on the graph of y = f(x) has the same y-coordinate as A. (i) Plot and label C on the graph above. (ii) Find the x-coordinate of C. (Total 6 marks) 4. A room is in the shape of a cuboid. Its floor measures 7.2 m by 9.6 m and its height is 3.5 m. diagram not to scale Calculate the length of AC. Calculate the length of AG. (c) Calculate the angle that AG makes with the floor. (Total 6 marks) 2

4 Summer Assignment IB Mathematical Studies 5. A teacher earns an annual salary of USD for the first year of her employment Her annual salary increases by 1750 USD each year. Calculate the annual salary for the fifth year of her employment. She remains in this employment for 10 years. Calculate the total salary she earns in this employment during these 10 years. (Total 6 marks) 6. The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. Complete the tree diagram below. Calculate the probability of rain tomorrow. (Total 6 marks) 3

Summer Assignment IB Mathematical Studies 7. The diagram shows the straight lines L 1 and L 2. The equation of L 2 is y = x. Find (i) the gradient of L 1 ; (ii) the equation of L 1.

5 Summer Assignment IB Mathematical Studies 7. The diagram shows the straight lines L 1 and L 2. The equation of L 2 is y = x. Find (i) the gradient of L 1 ; (ii) the equation of L 1. Find the area of the shaded triangle. (Total 6 marks) 8. The seventh term, u 7, of a geometric sequence is 108. The eighth term, u 8, of the sequence is 36. Write down the common ratio of the sequence. (1) Find u 1. The sum of the first k terms in the sequence is (c) Find the value of k. (Total 6 marks) 4

6 Summer Assignment IB Mathematical Studies 9. The equation of the line R 1 is 2x + y 8 = 0. The line R 2 is perpendicular to R 1. Calculate the gradient of R 2. The point of intersection of R 1 and R 2 is (4, k). Find (i) the value of k; (ii) the equation of R 2. (4) (Total 6 marks) 10. The weights in kg, of 80 adult males, were collected and are summarized in the box and whisker plot shown below. (c) (d) Write down the median weight of the males. Calculate the interquartile range. Estimate the number of males who weigh between 61 kg and 66 kg. Estimate the mean weight of the lightest 40 males. (1) (1) (Total 6 marks) 5

7 Summer Assignment IB Mathematical Studies metal spherical cannon balls, each of diameter 10 cm, were excavated from a Napoleonic War battlefield. Calculate the total volume of all 75 metal cannon balls excavated. The cannon balls are to be melted down to form a sculpture in the shape of a cone. The base radius of the cone is 20 cm. Calculate the height of the cone, assuming that no metal is wasted. (Total 6 marks) 12. In the diagram, AD = 4 m, AB = 9 m, BC = 10 m, B Dˆ A = 90 and D Bˆ C = 100. diagram not to scale Calculate the size of A Bˆ C. Calculate the length of AC. (Total 6 marks) 6

8 Summer Assignment IB Mathematical Studies 13. In the diagram, B ÂC = 90. The length of the three sides are x cm, (x + 7) cm and (x + 8) cm. diagram not to scale Write down and simplify a quadratic equation in x that links the three sides of the triangle. Solve the quadratic equation found in part. (c) Write down the value of the perimeter of the triangle. (1) (Total 6 marks) 7

9 Summer Assignment IB Mathematical Studies 14. Tony wants to carry out a χ 2 test to determine whether or not a person s choice of one of the three professions engineering, medicine or law is influenced by the person s sex (gender). State the null hypothesis, H 0, for this test. (1) Write down the number of degrees of freedom. (1) Of the 400 people Tony interviewed, 220 were male and 180 were female. 80 of the people had chosen engineering as a profession. (c) Calculate the expected number of female engineers. Tony used a 5 % level of significance for his test and obtained a p-value of correct to 3 significant figures. (d) State Tony s conclusion to the test. Give a reason for this conclusion. (Total 6 marks) 15. In a research project on the relation between the gender of 150 science students at college and their degree subject, the following set of data is collected. Degree Subject Biology Physics Chemistry Gender Male Female Find the probability that a student chosen at random is male; is either male or studies Chemistry; (c) studies Physics, given that the student is male. (Total 6 marks) 8

10 Summer Assignment IB Mathematical Studies Paper 2 Practice Questions 16. The heat output in thermal units from burning 1 kg of wood changes according to the wood s percentage moisture content. The moisture content and heat output of 10 blocks of the same type of wood each weighing 1 kg were measured. These are shown in the table. Moisture content % (x) Heat output ( y) Draw a scatter diagram to show the above data. Use a scale of 2 cm to represent 10 % on the x-axis and a scale of 2 cm to represent 10 thermal units on the y-axis. (4) Write down (i) the mean percentage moisture content, x ; (ii) the mean heat output, y. (c) Plot the point ( x, y ) on your scatter diagram and label this point M. (d) Write down the product-moment correlation coefficient, r. The equation of the regression line y on x is y = 0.470x (e) (f) Draw the regression line y on x on your scatter diagram. Estimate the heat output in thermal units of a 1 kg block of wood that has 25 % moisture content. (g) State, with a reason, whether it is appropriate to use the regression line y on x to estimate the heat output in part (f). (Total 16 marks) 9

11 Summer Assignment IB Mathematical Studies 17. One day the number of customers at three cafés, Alan s Diner (A), Sarah s Snackbar (S) and Pete s Eats (P) was recorded and are given below. 17 were customers of Pete s Eats only 27 were customers of Sarah s Snackbar only 15 were customers of Alan s Diner only 10 were customers of Pete s Eats and Sarah s Snackbar but not Alan s Diner 8 were customers of Pete s Eats and Alan s Diner but not Sarah s Snackbar Draw a Venn Diagram, using sets labelled A, S and P, that shows this information. There were 48 customers of Pete s Eats that day. Calculate the number of people who were customers of all three cafés. There were 50 customers of Sarah s Snackbar that day. (c) Calculate the total number of people who were customers of Alan s Diner. (d) Write down the number of customers of Alan s Diner that were also customers of Pete s Eats. (1) (e) Find n[(s P) A ]. (Total 11 marks) 10

12 Summer Assignment IB Mathematical Studies 18. Some of the customers in each café were given survey forms to complete to find out if they were satisfied with the standard of service they received. Pete s Eats Alan s Diner Sarah s Snackbar Total Dissatisfied Satisfied Total One of the survey forms was chosen at random, find the probability that the form showed Dissatisfied ; the form showed Satisfied and was completed at Sarah s Snackbar; (c) the form showed Dissatisfied, given that it was completed at Alan s Diner. A χ 2 test at the 5 % significance level was carried out to determine whether there was any difference in the level of customer satisfaction in each of the cafés. (d) Write down the null hypothesis, H 0, for the χ 2 test. (1) (e) Write down the number of degrees of freedom for the test. (1) (f) Using your graphic display calculator, find 2 calc. (g) State, giving a reason, the conclusion to the test. (Total 12 marks) 11

13 Summer Assignment IB Mathematical Studies 19. Pauline owns a piece of land ABCD in the shape of a quadrilateral. The length of BC is 190 m, the length of CD is 120 m, the length of AD is 70 m, the size of angle BCD is 75 and the size of angle BAD is 115. diagram not to scale Pauline decides to sell the triangular portion of land ABD. She first builds a straight fence from B to D. Calculate the length of the fence. The fence costs 17 USD per metre to build. Calculate the cost of building the fence. Give your answer correct to the nearest USD. (c) Show that the size of angle ABD is 18.8, correct to three significant figures. (d) Calculate the area of triangle ABD. (4) She sells the land for 120 USD per square metre. (e) Calculate the value of the land that Pauline sells. Give your answer correct to the nearest USD. (Total 14 marks) 12

14 Summer Assignment IB Mathematical Studies 20. A geometric sequence has 1024 as its first term and 128 as its fourth term. Show that the common ratio is 2 1. Find the value of the eleventh term. (c) Find the sum of the first eight terms. (d) Find the number of terms in the sequence for which the sum first exceeds (Total 10 marks) 21. Consider the arithmetic sequence 1, 4, 7, 10, 13, Find the value of the eleventh term. The sum of the first n terms of this sequence is 2 n (3n 1). (i) Find the sum of the first 100 terms in this arithmetic sequence. (ii) The sum of the first n terms is 477. Show that 3n 2 n 954 = 0. Using your graphic display calculator or otherwise, find the number of terms, n. (6) (Total 8 marks) 13

15 1. p = Note: Award for correctly substituted equation for p. 7 = , (A1) (C2) 4 (i) x = 2, y = 1, z = 50 (A1) 99 (ii) p = Note: Follow through from part (i), irrespective of whether working is shown. Note: If 2 s.f. used throughout part (i) award (A1)(ft) for 1.78 or 1.8. (A1)(ft) (C2) (c) Note: Award for correctly substituted % error formula. Note: Follow through from parts and. = 13.1% (A1)(ft) (C2) Notes: % sign not required. Do not accept 13.1% If 100 missing and incorrect answer, award (M0)(A0). If 100 missing and answer incorrectly rounded, award (A1)(AP). [6] 2. q = 25 ( ) Note: Award for subtraction from 25 of all values from the table. = 5 (A1) (C2) (i) 2.2 (A2)(ft) (C2) Note: Award for use of mean formula with correct substitution. Follow through from part, irrespective of whether working is shown. (ii) 2 (A1) (C1) 1

16 (iii) 1.39 (A1)(ft) (C1) Note: Follow through from part, irrespective of whether working is shown. Award (A1)(AP) for [6] 3. 4 x = 2 x = 2 (A1) OR dy = 4 2x dx x = 2 (A1) (2, 7) or x = 2, y = 7 (A1) (C3) Notes: Award (A1)(A0) for 2, 7 without parentheses. (i) C labelled in correct position on graph (A1) (C1) 2

17 (ii) 3 = 3 + 4x x 2 Note: Award for correct substitution of y = 3 into quadratic. (x =) 4 (A1) (C2) OR Using symmetry of graph x = Note: Follow through from their x-coordinate of the vertex. (x =) 4 (A1)(ft) (C2) [6] 4. Unit penalty applies in parts and AC 2 = Note: Award for correct substitution in Pythagoras Theorem. UP AC = 12 m (A1) (C2) AG 2 = Note: Award for correct substitution in Pythagoras Theorem. UP AG = 12.5m (A1)(ft) (C2) Note: Follow through from their answer to part (c) tan θ = 12 or sin θ = or cos θ = Note: Award for correct substitutions in trig ratio. θ = 16.3 (A1)(ft) (C2) Notes: Follow through from parts and/or part where appropriate. Award (A0) for use of radians (0.284). [6] (5 1)1750 (A1) Note: Award for substituted AP formula, (A1) for correct substitutions. = USD (A1) (C3) Notes: If a list is used, award for recognizing AP, award (A1) for seeing in their list, (A1) for final answer. 3

18 10 (2(45000) + (10 1)(1750)) (A1) 2 Notes: Award for substituted AP formula, (A1)(ft) for correct substitutions. Follow through from their common difference used in part. = USD (A1)(ft) (C3) Notes: Accept If a list is used, award for recognizing sum of AP, (A1) for seeing included in the sum or in a cumulative list. [6] 6. Note: Award (A1) for each correct pair. (A1)(A1)(A1) (C3) (A1)(ft) Notes: Award (A1)(ft) for two consistent products from tree diagram, for addition of their products. Follow through from their tree diagram provided all probabilities are between 0 and 1. = 0.74 (A1)(ft) (C3) [6] 4

19 7. (i) =, (A1) (C2) (ii) y = 1 x + 2 (A1)(ft) (C1) 3 Notes: Follow through from their gradient in part (i). Accept equivalent forms for the equation of a line. area = 61.5 (A1) 2 Note: Award (A1) for 1.5 seen, for use of triangle formula with 6 seen. = 4.5 (A1) (C3) [6] r = Note: Accept (A1) (C1) 7 1 u 1 3 u 1 = = 36 Note: Award for correct substitution in formula for n th term of a GP. Accept equivalent forms. Notes: Accept Follow through from their common ratio found in part. If used from part award (A1)(ft) for an answer of or irrespective of whether working is shown. (A1)(ft) (C2) 5

20 (c) = OR k Notes: Award for correct substitution in the sum of a GP formula, for equating their sum to Follow through from parts and. k Sketch of the function y = Indication of point where y = OR = Note: Award for a list of at least 8 correct terms, for the sum of the terms equated to k = 10 (A1)(ft) (C3) Notes: Follow through from parts and. If k is not an integer, do not award final (A1). Accept alternative methods. If and used award (A1)(ft) for k = 5. If and used award (A0). [6] 9. y = 2x + 8 Note: Award for rearrangement of equation or for 2 seen. m(perp) = 2 1 (A1) (C2) (i) 2(4) + k 8 = 0 Note: Award for evidence of substituting x = 4 into R 1. k = 0 (A1) (C2) 6

21 (ii) y = 2 1 x + c (can be implied) Note: Award for substitution of 2 1 into equation of the line. 0 = 2 1 (4) + c y = 2 1 x 2 (A1)(ft) (C2) OR Notes: Follow through from parts and (i). Accept equivalent forms for the equation of a line. y y 1 = 2 1 (x x1 ) Note: Award for substitution of 2 1 into equation of the line. y = 2 1 (x 4) (A1)(ft) (C2) Notes: Follow through from parts and (i). Accept equivalent forms for the equation of a line. [6] 10. Unit penalty applies in parts and (d) UP 61 kg (A1) (C1) (A1) = 14 (A1)(ft) (C2) Note: Award (A1) for identifying quartiles, (A1)(ft) for correct subtraction of their quartiles. (c) 20 (A1) (C1) 7

22 (d) Note: Award for multiplication of midpoints by frequencies. UP = 53 kg (A1) (C2) [6] 11. Unit penalty applies in parts and. 4 75π 5 cm 3 3 Notes: Award for correctly substituted formula of a sphere. Award for multiplying their volume by 75. If r = 10 is used, award (M0)(A1)(ft) for the answer cm 3. UP cm 3 (A1) (C3) 1 π 20 2 h = Notes: Award for correctly substituted formula of a cone. Award for equating their volume to their answer to part. UP h = 93.8 cm (A1)(ft) (C3) Notes: Accept the exact value of Follow through from their part. [6] 12. Unit penalty applies in part. 4 sin ABˆ D their (A Bˆ D) 126 (A1) (C3) Notes: Accept an equivalent trigonometrical equation involving angle ABD for the first. Radians used gives 100. Award at most (A0) if working shown. BD = 8 m leading to 127. Award at most (A0) (premature rounding). 8

23 AC 2 = cos( ) (A1) Notes: Award for substituted cosine formula. Award (A1) for correct substitution using their answer to part. UP AC = 17.0 m (A1)(ft) (C3) Notes: Accept 16.9 m for using 126. Follow through from their answer to part. Radians used gives Award at most (A1)(A0)(ft) if working shown. [6] 13. (x + 8) 2 = (x + 7) 2 + x 2 (A1) Note: Award (A1) for a correct equation. x x + 64 = x x x 2 Note: Award (A1) for correctly removed parentheses. x 2 2x 15 = 0 Note: Accept any equivalent form. (A1) (A1) (C3) x = 5, x = 3 (A1)(ft)(A1)(ft) (C2) Notes: Accept (A1)(ft) only from the candidate s quadratic equation. (c) 30 cm (A1)(ft) (C1) Note: Follow through from a positive answer found in part. [6] 14. Chosen profession is independent of gender. (A1) OR There is no association between gender and chosen profession. (A1) (C1) Note: Do not accept not related, not correlated or not influenced. 2 (A1) (C1) 9

24 (c) OR = 36 (A1) (C2) (d) p-value > 0.05 (R1) Accept H 0 (A1) (C2) Note: Do not award (R0)(A1). [6] = (0.607, 60.6 %, 60.7%) (A1)(A1) (C2) 150 Note: Award (A1) for numerator, (A1) for denominator =,0.74,74% (A1)(ft)(A1) (C2) Note: Award (A1)(ft) for their numerator in +20 provided the final answer is not greater than 1. (A1) for denominator. (c) 16 (0.176, 17.6%) (A1)(A1)(ft) (C2) 91 Note: Award (A1) for numerator and (A1)(ft) for denominator. Follow through from their numerator in provided answer is not greater than 1. [6] 10

25 16. (A1) for correct scales and labels (A3) for all ten points plotted correctly (A2) for eight or nine points plotted correctly (A1) for six or seven points plotted correctly Note: Award at most (A0)(A3) if axes reversed. (A4) (i) x = 42 (A1) (ii) y = 64 (A1) 11

26 (c) ( x, y) plotted on graph and labelled, M (A1)(ft)(A1) Note: Award (A1)(ft) for position, (A1) for label. (d) (G2) Note: Award (G1) for correct sign, (G1) for correct absolute value. (e) line on graph (A1)(ft)(A1) Notes: Award (A1)(ft) for line through their M, (A1) for approximately correct intercept (allow between 83 and 85). It is not necessary that the line is seen to intersect the y-axis. The line must be straight for any mark to be awarded. (f) y = 0.470(25) Note: Award (Ml) for substitution into formula or some indication of method on their graph. y = 0.470(0.25) is incorrect. = 72.0 (accept and 72) (A1)(ft)(G2) Note: Follow through from graph only if they show working on their graph. Accept 72 ±0.5. (g) Yes since 25 % lies within the data set and r is close to 1 (R1)(A1) Note: Accept Yes, since r is close to 1 Note: Do not award (R0)(A1). [16] 12

27 17. (A1) for rectangle and three labelled intersecting circles (A1) for 15, 27 and 17 (A1) for 10 and 8 (A3) 48 ( ) or equivalent = 13 (A1)(ft)(G2) (c) 50 ( ) Note: Award for working seen. = 0 (A1) number of elements in A = 36 (A1)(ft)(G3) Note: Follow through from. (d) 21 (A1)(ft) Note: Follow through from even if no working seen. (e) 54 (A1)(ft)(G2) Note: Award for 17, 10, 27 seen. Follow through from. [11] , 0.333, 33.3% (A1)(A1)(G2) Note: Award (A1) for numerator, (A1) for denominator. 13

28 , , 28.3% Note: Award (A1) for numerator, (A1) for denominator. (A1)(A1)(G2) (c) , , 28.6% Note: Award (A1) for numerator, (A1) for denominator. (A1)(A1)(G2) (d) customer satisfaction is independent of café (A1) Note: Accept customer satisfaction is not associated with the café. (e) 2 (A1) (f) (G2) Note: Award (G1)(G1)(AP) for 0.75 or for correct answer incorrectly rounded to 3 s.f. or more, (G0) for 0.7. (g) since χ 2 calc < χ 2 crit (5.991) accept (or Do not reject) H 0 (R1)(A1)(ft) OR Note: Follow through from their value in (e). Accept (or Do not reject) H 0 as p-value (0.686) > 0.05 Notes: Do not award (A1)(R0). Award the (R1) for comparison of appropriate values. (R1)(A1)(ft) [12] 19. Unit Penalty applies in parts and (d) and Financial Penalty applies in parts and (e). BD 2 = (190)(120)cos75 (A1) Note: Award for substituted cosine formula, (A1) for correct substitution. UP = 197 m (A1)(G2) Note: If radians are used award a maximum of (A1)(A0). cost = FP = 3344 USD (A1)(ft)(G2) Note: Accept 3349 from 197. (c) sin( ABD) sin(115) (A1)

29 Note: Award for substituted sine formula, (A1) for correct substitution. = (A1)(ft) = 18.8 (AG) Notes: Both the unrounded and rounded answers must be seen for the final (A1) to be awarded. Follow through from their. If 197 is used the unrounded answer is (d) angle BDA = 46.2 (A1) 70( ) sin(46.2) Area = 2 (A1) Note: Award for substituted area formula, (A1) for correct substitution. UP Area of ABD = 4970 m 2 (A1)(ft)(G2) Notes: If 197 used answer is Notes: Follow through from only. Award (G2) if there is no working shown and 46.2 not seen. If 46.2 seen without subsequent working, award (A1)(G2). (e) FP = USD (A1)(ft)(G2) Notes: Follow through from their (d). 15 r (f) = or equivalent (A1)(A1) 100 Notes: Award (A1) for seen or implied by alternative formula, for substituted CI formula, (A1) for correct substitutions. r = 4.73 Notes: Award G3 for 4.73 with no working. Award G2 for 4.7 with no working. (A1)(ft)(G3) [18] 15

30 r 3 = 128 r 3 = 8 1 or r = r = 2 1 (0.5) (AG) Notes: Award at most (M0) if last line not seen. Award (M0) if 128 is found by repeated multiplication (division) of 1024 by Notes: Award for correct substitution into correct formula. Accept an equivalent method. 1 (A1)(G2) (c) S 8 = OR (A1) Note: Award for substitution into the correct formula, (A1) for correct substitution. (A1) for complete and correct list of eight terms for their eight terms added S 8 = 2040 (A1) (A1)(G2) 16

31 (d) OR n > (ft) Notes: Award for correct substitution into the correct formula for the sum, for comparing to Accept equation. Follow through from their expression for the sum used in part (c). If a list is used: S 15 = S 16 = n = 16 (A1)(ft)(G2) Note: Follow through from their expression for the sum used in part (c). [10] 21. common difference = 3 (may be implied) (A1) u 11 = 31 (A1)(G2) (i) (2 993) ( ) OR (A1)(G2) (ii) n n (3n 1) = 477 OR (2 + 3(n 1)) = n 2 n = 954 3n 2 n 954 = 0 (AG) Notes: Award second for correct removal of denominator or brackets and no further incorrect working seen. Award at most (M0) if last line not seen. 18 (G2) Note: If both solutions to the quadratic equation are seen and the correct value is not identified as the required answer, award (G1)(G0). [8] 17

Y11MST Short Test (Statistical Applications)

2013-2014 Y11MST Short Test (Statistical Applications) [44 marks] Members of a certain club are required to register for one of three sports, badminton, volleyball or table tennis. The number of club members