IB Questionbank Test



1a. [3 marks] The volume of a hemisphere, V, is given by the formulaV = ,where S is the total surface area.The total surface area of a given hemisphere is 350?cm2.Calculate the volume of this hemisphere in cm3.Give your answer correct to one decimal place.Markscheme? OR?? ? ?(M1)Note: Award (M1) for substitution of 350 into volume formula.?= 473.973…? ? (A1)?= 474 (cm3)? ??(A1)(ft)? ?(C3)?Note: The final (A1)(ft) is awarded for rounding their answer to 1 decimal place provided the unrounded answer is seen.?[3 marks] 1b. [1 mark] Write down your answer to part (a) correct to the nearest integer.Markscheme474 (cm3)? ? ? (A1)(ft) (C1)Note: Follow through from part (a).?[1 mark] 1c. [2 marks] Write down your answer to part (b) in the form a × 10k , where 1 ≤ a < 10 and k?∈?.Markscheme4.74 × 102?(cm3)? ? ?(A1)(ft)(A1)(ft)? ?(C2)?Note: Follow through from part (b) only.Award (A0)(A0) for answers of the type 0.474 × 103.?[2 marks] 2a. [2 marks] Consider the following statements.p : it can go wrongq : it does go wrongWrite down in symbolic form:If it does not go wrong then it cannot go wrong.Markscheme?q ? ?p? ? ? (A1)(A1) (C2)Note: Award (A1) for two negations seen, (A1) for correct antecedent and consequent on either side of an implication.?[2 marks] 2b. [2 marks] Write down in words the argument p ? q.Markschemeif it can go wrong then it does go wrong ? ?(A1)(A1) (C2)Note: Award (A1) for “if…then” seen, (A1) for correct antecedent and consequent.?[2 marks] 2c. [2 marks] Write down in words the inverse of p ? q.Markschemeif it cannot go wrong then it does not go wrong ??(A1)(A1)(ft) (C2)Note: Award (A1) for “if…then” seen, (A1)(ft) for their correct antecedent and consequent. Follow through from part (b).?[2 marks] 3a. [2 marks] Consider the curve y = 5x3 ? 3x.Find .Markscheme15x2 ? 3? ? ? (A1)(A1) (C2)Note: Award (A1) for 15x2, (A1) for ?3. Award at most (A1)(A0) if additional terms are seen.?[2 marks] 3b. [2 marks] The curve has a tangent at the point P(?1, ?2).Find the gradient of this tangent at point P.Markscheme15?(?1)2 ? 3? ? ? (M1)Note: Award ?(M1) for substituting ?1 into their?.?= 12? ? ?(A1)(ft) (C2)Note: Follow through from part (a).?[2 marks] 3c. [2 marks] Find the equation of this tangent. Give your answer in the form y = mx + c.Markscheme(y?? (?2)) = 12 (x?? (?1))? ? ?(M1)OR?2 = 12(?1) + c? ???(M1)Note: Award ?(M1) for point and their gradient substituted into the equation of a line.?y = 12x + 10? ? ?(A1)(ft) (C2)Note: Follow through from part (b).?[2 marks] 4a. [2 marks] The diagram shows a triangle defined by the points A(3 , 9), B(15 , 6) and C(5 , 3).Calculate the gradient of the line AC.Markscheme? ? ??(M1)Note: Award (M1) for correct substitution into gradient formula.= ?3? ? ? ??(A1) (C2)?[2 marks] 4b. [2 marks] Determine, giving a reason, whether angle AC?B is a right angle.Markschemegradient of CB = 0.3? ? ??(A1)and since?, angle ACB is not a right angle? ? ? ?(A1)(ft) (C2)?Note: Award (A1)(ft) only if the justification is supported by a numerical calculation. Follow through from part (a).If distance formula and Pythagoras’ theorem are used, AB = 12.3693…, AC = 6.32455…, BC = 10.4403….Award (M1) for summing the squares of their AC and BC, (A1) for all three lengths correct and showing that 149 (AC2?+ BC2) ≠ 153 (AB2).If distance formula and cosine rule are used, award (M1) for correct substitution of their calculated lengths into cosine rule, (A1) for all three lengths correct and showing that AC?B = 91.7357...(°) ≠ 90(°).?[2 marks] 4c. [2 marks] The straight line, L, is parallel to BC and passes through A.Find the equation of L.Give your answer in the form ax + by + d = 0, where a, b and d are integers.Markschemey?? 9?= 0.3?(x?? 3)? OR? 9 = 0.3?(3) + c? ? ? ?(A1)(ft)Note: Follow through from part (b). Award (A1)(ft) for their gradient and given point A substituted into equation of a line. If the gradient of BC was not calculated for use in part (b), then the substituted gradient must be 0.3 to award (A1).?3x?? 10y + 81 = 0?(or any integer multiple)? ? ??(A1)(ft) (C2)Note: Award (A1)(ft) for writing their equation in the form?ax?+?by?+?d?= 0 with integral coefficients. Follow through within part (c).?[2 marks] 5a. [1 mark] Bella throws a ball from the top of a wall onto flat horizontal ground.The path of the ball is modelled by the quadratic curve y = 3 + 4x ? x2, where x represents the horizontal distance the ball is thrown and y represents the height of the ball above the ground. All distances are measured in metres.The wall lies along the y-axis. The curve intersects the y-axis at point A and has its vertex at point B.Write down the height in metres from which the ball was thrown.Markscheme3?(m)? ? ? ?(A1) (C1)?[1 mark] 5b. [3 marks] Calculate the maximum height, above the ground, reached by the ball.Markscheme(x = )? ? ? (M1)Note: Award (M1) for correct substitution into vertex formula.?OR0 = 4 ? 2x? ? ? ?(M1)Note: Award (M1) for correctly differentiating and equating to zero.?x = 2? ? ? ?(A1)(y = )? 7?(m)? ? ? ? (A1)(ft) (C3)Note: The final (A1)(ft) follows from their stated x-value, provided y > their (a).?[3 marks] 5c. [2 marks] Find the horizontal distance from the base of the wall to the point at which the ball hits the ground.Markscheme3 + 4x?? x2?= 0? ? ? ?(M1)Note: Award (M1) for substituting y = 0 into the equation of the curve.x = 4.65?(m)? (4.64575…)? ? ?(A1)?(C2)?[2 marks] 6a. [2 marks] The marks achieved by students taking a college entrance test follow a normal distribution with mean 300 and standard deviation 100.In this test, 10?% of the students achieved a mark greater than k.Find the value of k.Markscheme? ? (M1)Note: Award (M1) for diagram that shows the correct shaded area and percentage, k has to be greater than the mean.ORAward (M1) for P(mark > k) = 0.1 or P(mark ≤ k) = 0.9 seen.?428? (428.155…)? ? ? (A1) (C2)?[2 marks] 6b. [2 marks] Marron College accepts only those students who achieve a mark of at least 450 on the test.Find the probability that a randomly chosen student will be accepted by Marron College.Markscheme ??(M1)Note: Award (M1) for diagram that shows the correct shaded area and the value 450 labelled to the right of the mean.ORAward (M1) for P(mark ≥ 450) seen.?0.0668? (0.0668072…, 6.68?%, 6.68072…?%)? ? ? (A1) (C2)?[2 marks] 6c. [2 marks] Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.Markscheme? ?? ? ? ??(M1)Note: Award (M1) for 0.0228 (0.0227500…) seen. Accept 1 ? 0.97725.?= 0.341? ?(0.340532…,?34.1?%, 34.0532…%)? ? ??(A1)(ft) (C2)Note: Follow through from part (b), provided answer is between zero and 1.?[2 marks] 7a. [2 marks] The amount of yeast, g grams, in a sugar solution can be modelled by the function,g(t) = 10 ? k(c?t) for t ≥ 0where t is the time in minutes.The graph of g(t) is shown.The initial amount of yeast in this solution is 2 grams.Find the value of k.Markscheme2 = 10???k(c0) ? ?(M1)Note: Award (M1) for substituting 2 and 0 into the function.?(k =) 8? ? ??(A1) (C2)?[2 marks] 7b. [3 marks] The amount of yeast in this solution after 3 minutes is 9 grams.Find the value of c.Markscheme9?= 10???8(c?3)? ? ? ? ?(M1)Note: Award (M1) for substituting their k, 9, and 3 into the function.?c?3?= 0.125? OR??c3 = 8??OR ?c?3?=?? ? ? ??(M1)Note: Award (M1) for isolating "c3" or "c?3" from all constants in their equation.?OR0?=?9 ? 10 + 8(c?3)?? ? ? ??(M1)Note: Award (M1) for setting one side of their equation to 0.?OR? ? ? ? ?(M1)Note: Award (M1) for a sketch of the graph for their equation.?(c = ) 2? ? ? ? ? ? ??(A1)(ft) (C3)Note: Follow through from part (a).?[3 marks] 7c. [1 mark] Write down the maximum amount of yeast in this solution.Markscheme10 (grams)? ? ? ?(A1) (C1)[1 mark] 8a. [1 mark] The histogram shows the lengths of 25 metal rods, each measured correct to the nearest cm.Write down the modal length of the rods.Markscheme3? ? ?(A1) (C1)?[1 mark] 8b. [3 marks] Find the median length of the rods.Markschememedian is 13th position? ? ? (M1)CF: 2, 6, 14, 20, 23, 25? ? ? ?(M1)median = 3? ? ? (A1) (C3)?[3 marks] 8c. [1 mark] The upper quartile is 4?cm.Calculate?the lower quartile.Markscheme2.5? ? ? ?(A1) (C1)?Note: Award (A1)(ft) if the sum of their parts (c)(i) and (c)(ii) is 4.?[1 mark] 8d. [1 mark] Calculate the interquartile range.Markscheme1.5? ? ? ?(A1)(ft)? (C1)?Note: Award (A1)(ft) if the sum of their parts (c)(i) and (c)(ii) is 4.?[1 mark] 9a. [2 marks] Harry travelled from the USA to Mexico and changed 700 dollars (USD) into pesos (MXN).The exchange rate was 1 USD = 18.86 MXN.Calculate the amount of MXN Harry received.Markscheme700 × 18.86? ? ? (M1)Note: Award (M1) for multiplication by 18.86.= 13?200 (13?202) (MXN)? ? ? (A1) (C2)?[2 marks] 9b. [2 marks] On his return, Harry had 2400 MXN to change back into USD.There was a 3.5 % commission to be paid on the exchange.Calculate the value of the commission, in MXN, that Harry paid.Markscheme2400 × 0.035? ? ? ?(M1)Note: Award (M1) for multiplication by 0.035.= 84 (MXN) ??? (A1) (C2)?[2 marks] 9c. [2 marks] The exchange rate for this exchange was 1 USD = 17.24 MXN.Calculate the amount of USD Harry received. Give your answer correct to the nearest cent.Markscheme? ? ??(M1)Note: Award (M1) for dividing 2400 minus their part (b), by 17.24. Follow through from part (b).= 134.34 (USD)? ? ? ?(A1)(ft) (C2)Note: Award at most (M1)(A0) if final answer is not given to nearest cent.?[2 marks] 10a. [1 mark] Abhinav carries out a χ2 test at the 1?% significance level to determine whether a person’s gender impacts their chosen professional field: engineering, medicine or law.He surveyed 220 people and the results are shown in the table.State the null hypothesis, H0, for this test.Markschemegender and chosen profession are independent? ? ? ?(A1) (C1)Note: Accept there is no association between chosen profession and gender. Accept “not dependent”. Do not accept “not related” or “not correlated” or “not influenced”, or “does not impact”.?[1 mark] 10b. [2 marks] Calculate the expected number of male engineers.Markscheme? ?? ? ? ?(M1)Note: Award (M1) for correct substitution into expectation formula.= 45? ? ? ?(A1) (C2)?[2 marks] 10c. [2 marks] Find the p-value for this test.Markscheme0.0193 (0.0192644…)? ? ??(A2) (C2)?[2 marks] 10d. [1 mark] Abhinav rejects H0.State a reason why Abhinav is incorrect in doing so.Markscheme0.0193 > 0.01? (1%)? ? ?? (A1)(ft)ORthe p-value is greater than the significance level? (1%)? ? ? ? ?(A1)(ft) (C1)?Note: A numerical value in (c) must be seen to award the (A1)(ft). Follow through from part (c), only if it is > 0.01.Accept a correct answer from comparing both the numerical value of the Χ2 statistic and the numerical value of the critical value: 7.89898… < 9.21.?[1 mark] 11a. [1 mark] The table shows the first five terms of three sequences: un , vn and wn.State which sequence is?arithmetic.Markschemewn ? ?? (A1) (C1)??[1 mark] 11b. [1 mark] State which sequence is geometric.Markschemeun ? ?? (A1) (C1)?[1 mark] 11c. [2 marks] Find the exact value of the 11th term of the geometric sequence.Markscheme10?(2)11?1? ? ? (M1)?Note: Award (M1) for correct substitutions into geometric sequence formula.?= 10?240? ? ? (A1)(ft) (C2)?Note: Exact answer only. Accuracy rules do not apply in this question part; do not accept the 3 sf answer of 10?200.?[2 marks] 11d. [2 marks] Find the sum of the first 20 terms of the arithmetic sequence.Markscheme? OR? ? ? ?(M1)?Note: Award (M1) for correct substitutions into arithmetic series formula.?= 2100? ? ??(A1)(ft) (C2)?[2 marks] 12a. [4 marks] Complete the following truth table.Markscheme(A1)(A1)(ft)(A1)(A1)(ft) (C4)?Note: Award (A1) for each correct column followed through from the respective columns.?[4 marks] 12b. [2 marks] State whether the statement (p ∧ ?q) ? ?(p ∨ q) is a contradiction, a tautology?or neither. Give a reason for your answer.Markschemeneither? ? ? (A1)(ft)since the entries in the final column are not all true and not all false? ? ? ?(R1)? (C2)?Note: Do not award (A1)(R0). Follow through from an incorrect truth table but only if their reasoning is consistent with the final column.Award (R1) only if the final column is clearly identified in the justification.?[2 marks] 13a. [3 marks] Nick has $150?000 in a trust fund. Each year he donates 8?% of the money remaining in his trust fund to charity.Determine the maximum number of years Nick can donate to charity while keeping at least $50?000 in the trust fund.Markscheme? ? ? (A1)(M1)Note: Award (A1) for correct substitution into compound interest formula or correct substitution into the formula for term of a geometric sequence (where r?= 0.92), (M1) for a substituted compound interest formula equated to 50?000 or a substituted term of a geometric sequence formula equated to 50?000.?ORI% =??8? ? ? ? ? ? ? ?(A1)(M1)PV =?±150?000FV =?50?000P/Y = 1C/Y = 1Note: Award (A1) for C/Y = 1 seen, (M1) for other correct entries.?FV and PV must have opposite sign.?OR138?000, 126?960, 116?803.20, 107?458.94, 98?862.23, …? ? ?(M1)t13 = 50?737.96, t14 = 46?678.92? ? ? ? ?(A1)Note: Award (M1) for a list of at least 5 correct terms beginning with 138?000, (A1) for identifying t13 and t14.?n = 13.175713 years? ? ? (A1) (C3)Note: The answer must be an integer.?[3 marks] 13b. [3 marks] Louise invests $200?000 in a bank account that pays a nominal interest rate of 5?%, compounded quarterly, for eight years.Calculate the value of Louise’s investment at the end of this time.Give your answer correct to the nearest cent.Markscheme?? ??(A1)(M1)Note: Award (M1) for substituted compound interest formula, (A1) for correct substitutions.?ORN?= 8? ? ? ? ? ? ? ?(A1)(M1)I% = 5PV =?±200?000P/Y = 1C/Y =?4Note: Award (A1) for C/Y =?4 seen, (M1) for other correct entries.?ORN?= 32? ? ? ? ? ? ? ?(A1)(M1)I% = 5PV =?±200?000P/Y = 1C/Y =?4?Note:?Award?(A1)?for C/Y =?4?seen,?(M1)?for other correct entries.?= ($) 297?626.10? ? ? ?(A1) (C3)Note: Answer must be given correct to two decimal places.?[3 marks]? 14a. [3 marks] A bag contains 5 red and 3 blue discs, all identical except for the colour. First, Priyanka takes a disc at random from the bag and then Jorgé takes a disc at random from the plete the tree diagram.Markscheme? ?(A1)(A1)(A1) (C3)Note: Award (A1) for each correct pair of branches.?[3 marks] 14b. [3 marks] Find the probability that Jorgé chooses a red disc.Markscheme? ? ??(A1)(ft)(M1)Note: Award (A1)(ft) for their two correct products from their tree diagram. Follow through from part (a), award (M1) for adding their two products. Award (M0) if additional products or terms are added.?=?? ?? ? ?(A1)(ft) (C3)Note: Follow through from their tree diagram, only if probabilities are [0,1].?[3 marks] 15a. [1 mark] A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a cuboid with a square base of length 6?cm, as shown in the diagram.The height of the cuboid, x?cm, is equal to the height of the hemisphere.Write down the value of x.Markscheme3?(cm)? ? (A1) (C1)?[1 mark] 15b. [3 marks] Calculate the volume of the paperweight.Markschemeunits are required in part (a)(ii)?? ? ??(M1)(M1)Note: Award (M1) for their correct substitution in volume of sphere formula divided by 2, (M1) for adding their correctly substituted volume of the cuboid.?= 165?cm3? ?(164.548…)? ? ??(A1)(ft) (C3)Note: The answer is 165?cm3; the units are required. Follow through from part (a)(i).?[3 marks] 15c. [2 marks] 1?cm3 of glass has a mass of 2.56?grams.Calculate the mass, in grams, of the paperweight.Markschemetheir 164.548… × 2.56? ? ? (M1)Note: Award (M1) for multiplying their part (a)(ii) by 2.56.?= 421?(g)? ?(421.244…(g))? ? ? (A1)(ft) (C2)Note: Follow through from part (a)(ii).?[2 marks]Printed for City Honors ? International Baccalaureate Organization 2019 International Baccalaureate? - Baccalauréat International? - Bachillerato Internacional? ................
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