Practice Test 1 [90 marks]

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Practice Test 1 [90 marks] The lengths of trout in a

represented in the following histogram. 1a.

(A2) Award (A2) for all correct entries, for 3 correct

1 Practice Test 1 [90 marks] The lengths of trout in a fisherman s catch were recorded over one month, and are represented in the following histogram. 1a. Complete the following table. (A2) Award (A2) for all correct entries, for 3 correct entries. 1b. State whether length of trout is a continuous or discrete variable. continuous (C1)

2 1c. Write down the modal class. 60 (cm) < trout length 70 (cm) (C1) Accept equivalent notation such as (60, 70] or ]60, 70]. Award (A0) for (incorrect notation). 1d. Any trout with length 40 cm or less is returned to the lake. Calculate the percentage of the fisherman s catch that is returned to the lake Award for their 4 divided by their 22. = 18.2 ( ) (ft) Follow through from their part (a). Do not accept The first three terms of a geometric sequence are u 1 = 486, u 2 = 162, u 3 = 54. 2a. Find the value of r, the common ratio of the sequence OR Award for dividing any u n+1 by u n. 1 = (0.333, ) 3 2b. Find the value of n for which u n = 2.

3 1 486( ) n 1 = 2 3 Award for their correct substitution into geometric sequence formula. n = 6 (ft) Follow through from part (a). Award (A0) for u 6 = 2 or u 6 with or without working. 2c. Find the sum of the first 30 terms of the sequence. S 30 = 486( ) Award for correct substitution into geometric series formula. = 729 (ft) 2 The equation of line L 1 is y = x a. Write down the gradient of L (C1) Point P lies on L 1 and has x-coordinate 6. 3b. Find the y-coordinate of P.

4 2 y = ( 6) 2 3 Award for correctly substituting 6 into the formula for L 1. (y =) 2 Award (A0) for ( 6, 2) with or without working. The line L 2 is perpendicular to L 1 and intersects L 1 when x = 6. 3c. Determine the equation of L 2. Give your answer in the form ax + by + d = 0, where a, b and d are integers. 3 gradient of L 2 is (ft) 2 Follow through from part (a). 3 2 = ( 6) + c 2 3 OR y 2 = (x ( 6)) 2 Award for substituting their part (b), their gradient and 6 into equation of a straight line. 3x 2y + 22 = 0 (ft) Follow through from parts (a) and (b). Accept any integer multiple. 3 Award (A0) for y = x

5 The function f is of the form f(x) = ax + b + c x, where a, b and c are positive integers. Part of the graph of y = f(x) is shown on the axes below. The graph of the function has its local maximum at ( 2, 2) and its local minimum at (2, 6). 4a. Write down the domain of the function. (x R), x 0 (A2) Accept equivalent notation. Award (A0) for y 0. Award for a clear statement that demonstrates understanding of the meaning of domain. For example, D : (, 0) (1, ) should be awarded (A0). 4b. Draw the line y = 6 on the axes. (C1) line. The command term Draw states: A ruler (straight edge) should be used for straight lines ; do not accept a freehand y = 6 4c. Write down the number of solutions to f(x) = 6.

6 2 (ft) (C1) Follow through from part (b)(i). 4d. Find the range of values of k for which f(x) = k has no solution. 2 < k < 6 Award for both end points correct and for correct strict inequalities. Award at most (A0) if the stated variable is different from k or y for example 2 < x < 6 is (A0). A triangular postage stamp, ABC, is shown in the diagram below, such that AB = 5 cm,b^ac = 34,A^BC = 26 and A^CB = a. Find the length of BC. BC 5 = sin34 sin120 Award for substituted sine rule formula, for correct substitutions. BC = 3.23 (cm) ( (cm)) Find the area of the postage stamp. 5b.

7 1 (5)( )sin 26 2 (ft) Award for substituted area of a triangle formula, for correct substitutions. = 3.54 (cm 2 ) ( (cm 2 )) (ft) Follow through from part (a). Arthur and Jacob dream of owning a speedboat that costs euros (EUR). Arthur invested x EUR in an account that pays a nominal annual interest rate of 3.6%, compounded monthly. After 18 years he will have EUR in the account. 6a. Calculate the value of Arthur s initial investment, x. Give your answer to two decimal places = PV(1 + ) Accept x instead of PV. Award for substitution into compound interest formula, for correct substitution. OR N = 18 I% = 3.6 FV = (±) P/Y = 1 C/Y = 12 Award for C/Y = 12 seen, for all other correct entries. OR N = 216 I% = 3.6 FV = (±) P/Y = 12 C/Y = 12 Award for C/Y = 12 seen, for all other correct entries. PV =

8 Jacob invested 9000 EUR for n years. The investment has a nominal annual interest rate of 3.2% and is compounded quarterly. After n years, the investment will be worth EUR. 6b. Find the value of n = 9000(1 + ) 4 n Award for substitution into compound interest formula and equating to , for correct substitution. OR I% = 3.2 PV = (±)9000 FV = ( ) P/Y = 1 C/Y = 4 Award for C/Y = 4 seen, for all other correct entries. OR I% = 3.2 PV = (±)9000 FV = ( ) P/Y = 4 C/Y = 4 Award for C/Y = 4 seen, for all other correct entries. n = 43 The graph of a quadratic function has y-intercept 10 and one of its x-intercepts is 1. The x-coordinate of the vertex of the graph is 3. The equation of the quadratic function is in the form y = ax 2 + bx + c. 7a. Write down the value of c. 10 (C1) Accept (0, 10). 7b. Find the value of a and of b. [4 marks]

9 3 = b 2a 0 = a(1) 2 + b(1) + c 10 = a(6) 2 + b(6) + c 0 = a(5) 2 + b(5) + c Award for each of the above equations, provided they are not equivalent, up to a maximum of. Accept equations that substitute their 10 for c. OR sketch graph showing given information: intercepts (1, 0) and (0, 10) and line x = 3 y = a(x 1)(x 5) Award for (x 1)(x 5) seen. a = 2 b = 12 (ft) (ft) (C4) Follow through from part (a). If it is not clear which is a and which is b award at most (A0)(ft). [4 marks] 7c. Write down the second x-intercept of the function. 5 (C1) In the Canadian city of Ottawa: 97% of the population speak English, 38% of the population speak French, 36% of the population speak both English and French. 8a. Calculate the percentage of the population of Ottawa that speak English but not French.

10 97 36 Award for subtracting 36 from 97. OR Award for 61 and 36 seen in the correct places in the Venn diagram. = 61 (%) Accept 61.0 (%). The total population of Ottawa is b. Calculate the number of people in Ottawa that speak both English and French Award for multiplying 0.36 (or equivalent) by = ( ) 8c. Write down your answer to part (b) in the form a 10 k where 1 a < 10 and k Z ( ) (ft)(ft) Award (ft) for 3.55 (3.546) must match part (b), and (ft) Award (A0)(A0) for answers of the type: Follow through from part (b).

11 Consider the following propositions. p: I completed the task q: I was paid 9a. Write down in words q. I was not paid (C1) 9b. Write down in symbolic form the compound statement: If I was paid then I completed the task. q p (C1) 9c. Complete the following truth table. Award for each correct column. 9d. State whether the statements p q and q p are logically equivalent. Give a reason for your answer.

12 yes (ft) as the last two columns of the truth table are the same (R1)(ft) Do not award (R0). Follow through from part (c)(i). Dune Canyon High School organizes its school year into three trimesters: fall/autumn ( F), winter ( W) and spring ( S). The school offers a variety of sporting activities during and outside the school year. The activities offered by the school are summarized in the following Venn diagram. Write down the number of sporting activities offered by the school during its school year. 10a. 15 (C1) Determine whether rock-climbing is offered by the school in the fall/autumn trimester. 10b. no (C1) Accept it is only offered in Winter and Spring. Write down the elements of the set ; 10c. F W

13 volleyball, golf, cycling (C1) Responses must list all three sports for the to be awarded. Write down n(w S). 10d. 4 (C1) Write down, in terms of F, W and S, an expression for the set which contains only archery, baseball, kayaking and surfing. 10e. (F W S) OR F W S (or equivalent) (A2) The company Snakezen s Ladders makes ladders of different lengths. All the ladders that the company makes have the same design such that: the first rung is 30 cm from the base of the ladder, the second rung is 57 cm from the base of the ladder, the distance between the first and second rung is equal to the distance between all adjacent rungs on the ladder. The ladder in the diagram was made by this company and has eleven equally spaced rungs. 11a. Find the distance from the base of this ladder to the top rung.

14 30 + (11 1) 27 Award for substituted arithmetic sequence formula, for correct substitutions. = 300 (cm) Units are not required. The company also makes a ladder that is 1050 cm long. 11b. Find the maximum number of rungs in this 1050 cm long ladder (n 1) 27 (ft) Award for substituted arithmetic sequence formula 1050, accept an equation, for correct substitutions. n = 38 (ft) Follow through from their 27 in part (a). The answer must be an integer and rounded down. In a school, students in grades 9 to 12 were asked to select their preferred drink. The choices were milk, juice and water. The data obtained are organized in the following table. A χ 2 test is carried out at the 5% significance level with hypotheses: The χ 2 critical value for this test is H 0 : the preferred drink is independent of the grade H 1 : the preferred drink is not independent of the grade Write down the value of x. 12a. 30 (C1)

15 Write down the number of degrees of freedom for this test. 12b. 6 (C1) 12c. Use your graphic display calculator to find the χ 2 statistic for this test ( ) (A2)(ft) Follow through from part (a). Award for truncation to State the conclusion for this test. Give a reason for your answer. 12d. reject (do not accept) H 0 OR accept H 1 (ft) Can be written in words ( ) > 12.6 (R1) Accept χ 2 calc > χ2 crit for the (R1) provided their χ 2 calc value is explicitly seen in their part (c). OR (p =) < (significance level = ) 0.05 (R1) Do not award (R0)(ft). Follow through from part (c). Numerical comparison must be seen to award the (R1).

16 Sara regularly flies from Geneva to London. She takes either a direct flight or a non-directflight that goes via Amsterdam. If she takes a direct flight, the probability that her baggage does not arrive in London is If she takes a non-direct flight the probability that her baggage arrives in London is The probability that she takes a non-direct flight is 0.2. Complete the tree diagram. 13a. Award for each correct pair of probabilities. 13b. Find the probability that Sara s baggage arrives in London (ft) Award (ft) for two correct products of probabilities taken from their diagram, for the addition of their products. = (98.2%, ) (ft) Follow through from part (a).

17 Daniela is going for a holiday to South America. She flies from the US to Argentina stopping in Peru on the way. In Peru she exchanges 85 United States dollars (USD) for Peruvian nuevo sol (PEN). The exchange rate is 1 USD = 3.25 PEN and a flat fee of 5 USD commission is charged. Calculate the amount of PEN she receives. 14a. (85 5) 3.25 Award for subtracting 5 from 85, for multiplying by Award for , for subtracting = 260 (PEN) At the end of Daniela s holiday she has 370 Argentinean peso (ARS). She converts this back to USD at a bank that charges a 4% commission on the exchange. The exchange rate is 1 USD = 9.60 ARS. Calculate the amount of USD she receives. 14b. ( ) 9.6 Award for multiplying by 0.96 (or equivalent), for dividing by 9.6. If division by 3.25 seen in part (a), condone multiplication by 9.6 in part (b). = 37 (USD) A type of candy is packaged in a right circular cone that has volume 100 cm 3 and vertical height 8 cm. Find the radius, r, of the circular base of the cone. 15a.

18 1 100 = πr 2 (8) 3 Award for correct substitution into volume of cone formula. r = 3.45 (cm) ( (cm)) Find the slant height, l, of the cone. 15b. l 2 = ( ) 2 Award for correct substitution into Pythagoras theorem. l = 8.71 (cm) ( (cm)) (ft) Follow through from part (a). Find the curved surface area of the cone. 15c. π Award for their correct substitutions into curved surface area of a cone formula. = 94.6 cm 2 ( cm 2 ) (ft) Follow through from parts (a) and (b). Accept 94.4 cm 2 from use of 3 sf values. International Baccalaureate Organization 2018 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for North hills Preparatory

2015 May Exam. Markscheme. Markscheme. 1a. [2 marks] , where, and. Calculate the exact value of. (M1)

2015 May Exam. Markscheme. Markscheme. 1a. [2 marks] , where, and. Calculate the exact value of. (M1) 2015 May Exam 1a. [2 marks] Calculate the exact value of., where, and. Note:Award for correct substitution into formula. (A1) (C2) Note:Using radians the answer is, award at most (A0). 1b. [2 marks] Give