IB Questionbank Test



HL Week 4a Revision - Integration1a. [4 marks] Markscheme? ? ?(M1)(A1)? ? ?A1? ? ?A1[4 marks] 1b. [2 marks] Markscheme?? ?(M1)= 12? ? ?A1[2 marks] 2a. [2 marks] Markscheme????M1A1Note: M1 is for use of the chain rule.[2 marks] 2b. [7 marks] Markschemeattempt at integration by parts? ? ?M1? ? ?(A1)? ? ??A1using integration by substitution or inspection? ? ? (M1)? ? ??A1Note: Award A1 for??or equivalent.Note: Condone lack of limits to this point.attempt to substitute limits into their integral? ? ?M1? ? ?A1[7 marks] 3a. [2 marks] MarkschemeMETHOD 1? ? ?M1A1? ? ?AG[2 marks]?METHOD 2? ? ?M1? ? ?A1? ? ?AG[2 marks]? 3b. [5 marks] MarkschemeMETHOD 1? ? ?M1? ? ?M1? ? ?M1A1? ? ?A1Note: For the final A mark,??must be expressed in the form?.[5 marks]?METHOD 2? ? ?M1? ? ?M1? ? ?M1? ? ?A1? ? ?A1Note: For the final?A?mark,??must be expressed in the form?.[5 marks]? 3c. [5 marks] Markschemethe area of R is?? ? ?M1? ? ?A1? ? ?A1? ? ?M1? ? ?A1Note: Only follow through from part (b) if??is in the form?[5 marks] 4a. [2 marks] Markschemeattempt at product rule? ? ? M1?? ???A1[2 marks] 4b. [1 mark] Markscheme?? ???A1[1 mark] 4c. [4 marks] MarkschemeMETHOD 1Attempt to add? and?? ? ? (M1)?? ?A1 (or equivalent)? ? ? A1Note: Condone absence of limits.?? ?A1?METHOD 2 OR?? ? ?M1A1? ? ?A1? ??A1[4 marks] 5a. [4 marks] Markscheme?(accept? or equivalent)? ? ? ?A1substitution, leading to an integrand in terms of?? ? ?M1?or equivalent? ? ? A1=?2 arctan?? ? ?A1[4 marks]? 5b. [3 marks] Markscheme? =?arctan?3 ? arctan?1? ? ?A1tan(arctan?3?? arctan?1) =?? ? ? (M1)tan(arctan?3?? arctan?1) =?arctan?3?? arctan?1 = arctan?? ? ?A1[3 marks] 6a. [2 marks] Markscheme? ? ?A1? ? ?A1[2 marks] 6b. [5 marks] Markscheme? ? ??M1M1Note: Award M1 for dividing by? to get?, M1 for separating the? and 1.? ? ?(M1)A1A1Note: Award (M1)A1 for integrating?, A1?for the other two terms.[5 marks] 7a. [7 marks] Markschemedifferentiating implicitly:? ? ? ?M1? ? ?A1A1Note: Award A1 for each side.if??then either? or?? ? ? ?M1A1?two solutions for?? ? ? R1?not possible (as 0?≠ 5)? ? ?R1hence exactly two points? ? ? AGNote: For a solution that only refers to the graph giving two solutions at???and?no solutions for? award R1 only.[7 marks] 7b. [5 marks] Markschemeat (2, 1)??? ? ?M1? ? ?(A1)gradient of normal is 2? ? ? ?M11 =?4?+ c? ? ? ?(M1)equation of normal is?? ? ?A1[5 marks] 7c. [3 marks] Markschemesubstituting? ? ? (M1) or?? ? ? ?(A1)? ? ? A1[3 marks] 7d. [7 marks] Markschemerecognition of two volumes? ? ? (M1)volume?? ? ??M1A1A1Note: Award M1 for attempt to use?,?A1 for limits, A1 for??Condone omission of at this stage.volume 2EITHER? ? ?(M1)(A1)OR? ? ?(M1)(A1)THENtotal volume?= 19.9? ? ? A1[7 marks] 8a. [2 marks] Markschemeattempt to solve? for?t or equivalent? ? ?(M1)t1?= 0.441(s)? ? ?A1[2 marks] 8b. [2 marks] Markscheme? ????M1A1Note: Award M1 for attempting to differentiate using the product rule.[2 marks] 8c. [1 mark] Markscheme?(ms?2)?? ? ?A1[1 mark] 9. [5 marks] Markschemeattempt at integration by parts ? ? M1 ? ? A1 ? ? (A1)?Note: ? ? Condone absence of limits (or incorrect limits) and missing factor of 10 up to this point.? ? ? (M1) ? ? A1[5 marks] 10a. [5 marks] Markschemeattempt to use quotient rule or product rule ? ? M1 ? ? A1A1?Note: ? ? Award A1 for or equivalent and A1 for or equivalent.?setting ? ? M1 or equivalent ? ? A1 ? ? AG[5 marks] 10b. [2 marks] Markscheme ? ? A1A1?Note: ? ? Award A1 for and A1 for . Accept .?[2 marks] 10c. [3 marks] Markschemeconcave up curve over correct domain with one minimum point above the -axis. ? ? A1approaches asymptotically ? ? A1approaches asymptotically ? ? A1?Note: ? ? For the final A1 an asymptote must be seen, and must be seen on the -axis or in an equation.?[3 marks] 10d. [4 marks] Markscheme ? ? (A1)attempt to solve for ? ? (M1) ? ? A1 ? ? A1[4 marks] 10e. [3 marks] Markscheme ? ? (M1)(A1)?Note: ? ? M1 is for an integral of the correct squared function (with or without limits and/or ).? ? ? A1[3 marks] 11. [7 marks] MarkschemeEITHER ? ? M1A1 ? ? M1A1OR ? ? M1A1 ? ? M1A1THEN ? ? (M1) ? ? A1 ? ? M1?Note: ? ? This M1 may be seen anywhere, including a sketch of an appropriate triangle.?so ? ? AG[7 marks] 12a. [1 mark] Markscheme???? A1[1 mark] 12b. [1 mark] Markscheme???? A1[1 mark] 12c. [5 marks] MarkschemeA1 for the shapeA1 for the equation A1 for asymptotes and A1 for coordinates A1 -intercept [5 marks] 12d. [1 mark] Markscheme???? M1???? AG[1 mark] 12e. [4 marks] Markscheme???? A1???? M1???? M1A1[4 marks] 12f. [2 marks] Markschemesymmetry about the -axis???? M1correct shape???? A1?Note:???? Allow FT from part (b).?[2 marks] 12g. [3 marks] Markscheme???? (M1)(A1)???? A1?Note:???? Do not award FT from part (e).?[3 marks] 13. [5 marks] Markschemeattempt at integration by parts with and ???? M1?? ?A1A1?Note:???? Award A1 for and A1 for .?solving by substitution with or inspection???? (M1)?? A1[5 marks] 14a. [5 marks] Markscheme???? M1A1???? M1???? A1A1?Note:???? Award A0A0 if answers are given in degrees.?[5 marks] 14b. [2 marks] Markscheme???? A1A1[2 marks] 15a. [4 marks] Markschemelet ???? (A1)???? M1?Note:???? The method mark is for an attempt to substitute for both and .? (or equivalent)???? A1when and when ? ? M1?? ?AG[4 marks] 15b. [3 marks] Markscheme???M1???? A1???? A1[3 marks] 16a. [2 marks] MarkschemeA1 for correct shapeA1 for correct and intercepts and minimum point[2 marks] 16b. [4 marks] MarkschemeA1 for correct shapeA1 for correct vertical asymptotesA1 for correct implied horizontal asymptoteA1 for correct maximum point[??? marks] 16c. [2 marks] MarkschemeA1 for reflecting negative branch from (ii) in the -axisA1 for correctly labelled minimum point[2 marks] 16d. [5 marks] MarkschemeEITHERattempt at integration by parts???? (M1)???? A1A1???? A1???? A1ORattempt at integration by parts???? (M1)???? A1A1???? A1???? A1[5 marks] 16e. [4 marks] Markscheme ? ? M1A1A1?Note:???? Method mark is for differentiating the product. Award A1 for each correct term.?both parts of the expression are positive hence is positive???? R1and therefore is an increasing function (for )???? AG[4 marks] 17a. [2 marks] Markscheme???? (M1)???? A1[2 marks] 17b. [2 marks] Markscheme?? ?(M1)???? A1[2 marks] 17c. [5 marks] Markscheme???? (M1)???? A1?? ?(M1)(A1)???? A1[5 marks] 18a. [4 marks] MarkschemeMETHOD 1???? (M1)???? (A1)???M1A1?Note:???? Award M1 for an attempt to find the difference between two functions, A1 for all correct.?METHOD 2when ???? A1?? ?M1A1?Note:???? Award M1 for an attempt to find the inverse function.????? A1METHOD 3?? ?M1A1A1A1?Note:???? Award M1 for considering the area below the -axis and above the -axis and A1 for each correct integral.?[4 marks] 18b. [2 marks] Markscheme???? A2[2 marks] 19a. [6 marks] MarkschemeMETHOD 1???? (M1)(A1)?Note:???? Award M1A1 for finding using any alternative method.?hence gradient of normal ???? (M1)hence gradient of normal at is ???? (A1)hence equation of normal is ???? (M1)A1METHOD 2???? (M1)???? (A1)?Note:???? Award M1A1 for finding using any alternative method.?hence gradient of normal ???? (M1)hence gradient of normal at is ???? (A1)hence equation of normal is ???? (M1)A1[6 marks] 19b. [3 marks] MarkschemeUse of ?? ?(M1)(A1)?Note:???? Condone absence of limits or incorrect limits for M mark.Do not condone absence of or multiples of .????? A1[3 marks] 20a. [2 marks] Markscheme ? ?M1A1[2 marks] 20b. [2 marks] Markscheme ? ?M1A1 ? ?AG[2 marks] 20c. [2 marks] Markscheme ? ?R1 ? ?R1hence maximum at ?? ? AG[2 marks] 20d. [2 marks] Markscheme ? ?M1 ? ?A1?Note: Award M1A0 if extra zeros are seen.?[2 marks] 20e. [3 marks] Markschemecorrect shape and correct domain ? ? A1max at , point of inflexion at ?? ? A1zeros at and ?? ? A1?Note: Penalize incorrect domain with first A mark; allow FT from (d) on extra points of inflexion.?[3 marks] 20f. [6 marks] MarkschemeEITHER ? ?M1A1 ? ?A1OR ? ?M1A1 ? ?A1THEN ? ?M1A1 ? ?A1[6 marks] 20g. [3 marks] Markscheme ? ?(A1)? (A1) ? ?A1[3 marks] 20h. [2 marks] Markscheme ? ?A1the graph is approximated by a straight line ? ? R1[2 marks] 21a. [2 marks] Markschemeattempting to solve either or for ? ? (M1) (or equivalent eg?) ? ? A1?Note: Accept or equivalent eg .?[2 marks] 21b. [5 marks] Markschemeconsidering ?? ? (M1) ? ?A1considering one of ?or ?? ? M1 ? ?A1 ? ?A1?Note: Award A0A0 for ?and ?stated without any justification.?[5 marks] 21c. [3 marks] Markscheme ? ?M1A1A1 ? ?AG[3 marks] 21d. [4 marks] Markscheme is (strictly) decreasing ? ? R1?Note: Award R1 for a statement such as and so the graph of has no turning points.?one branch is above the upper horizontal asymptote and the other branch is below the lower horizontal asymptote ? ? R1 has an inverse ? ? AG ? ?A2?Note: Award A2 if the domain of the inverse is seen in either part (d) or in part (e).?[4 marks] 21e. [4 marks] Markscheme ? ?M1?Note: Award M1 for interchanging and (can be done at a later stage).? ? ?M1 ? ?A1 ? ?A1[4 marks] 21f. [4 marks] Markschemeuse of ?? ? (M1) ? ?(A1)(A1)?Note: Award (A1) for the correct integrand and (A1) for the limits.? ? ?A1[4 marks] 22a. [1 mark] Markscheme ? ?A1[1 mark] 22b. [5 marks] Markscheme ? ?(A1) ? ?M1A1area or ?? ? A1 ? ?A1[5 marks] 22c. [7 marks] Markscheme(i) ? ? ?? ? (A1) ? ?A1(ii) ? ? use of integration by parts ? ? M1 ? ?A1A1 ? ?AG?Note: ? ? If the substitution is used A1A1 can be awarded for .?(iii) ? ? ?? ? (M1) ? ?A1[7 marks] 22d. [5 marks] Markscheme(d) ? ? volume ?? ? (A1)EITHER ? ?M1A1 ? ?M1ORusing parts ?? ? M1A1 ? ?M1THEN ? ?A1volume?[5 marks] 23a. [8 marks] Markschemeuse of ?? ? (M1)Note: ? ? Condone any or missing limits. ? ?(A1) ? ?A1 ? ?(M1) ? ?M1A1 ? ?(A1) ? ?A1Note: ? ? If the coefficient “” is absent, or eg, “” is used, only M marks are available.[8 marks] 23b. [4 marks] Markscheme(i) ? ? attempting to use with ?? ? M1 ? ?A1(ii) ? ? substituting into ?? ? (M1) ? ?A1Note: ? ? Do not allow FT marks for (b)(ii).[4 marks] 23c. [7 marks] Markscheme(i) ? ? ?? ? (M1) ? ?M1A1Note: ? ? Award M1 for attempting to find . ? ?A1(ii) ? ? ?? ? A1Note: ? ? Award A1 for ?from an incorrect .(iii) ? ? METHOD 1?is a minimum at and the container is widest at these values ? ? R1 is a maximum at ?and the container is narrowest at this value ? ? R1[7 marks] 24a. [1 mark] MarkschemeEITHERuse of a diagram and trig ratioseg,from diagram, ? ??R1ORuse of ? ??R1THEN ? ??AG[1 mark] 24b. [4 marks] Markscheme ? ??(A1)?Note: ? ? Limits (or absence of such) may be ignored at this stage.? ? ??(M1) ? ??(A1) ? ??A1[4 marks] 25a. [2 marks] Markscheme?? ? M1A1[2 marks] 25b. [2 marks] Markscheme?? ? M1A1Note: ? ? If simplified in part (a) award (M1)A1 for .Note: ? ? Award M1A1 for .[2 marks] 25c. [1 mark] Markscheme?? ? A1[1 mark] 26a. [9 marks] Markscheme(i) ? ? ?? ? (M1) ? ?(A1) ? ?A1 ? ?AG(ii) ? ? ?? ? A1 ? ?M1(when and when ) ? ?M1A1 ? ?(M1) ? ?A1[9 marks] 26b. [8 marks] Markscheme(i) ? ? ?? ? M1A1 ? ?M1 ? ?M1A1(ii) ? ? for two real solutions, we require ?? ? R1and we also require ?? ? R1 ? ?A1 ? ?AG[8 marks] 26c. [6 marks] MarkschemeMETHOD 1 ? ?M1A1 ? ?A1 ? ?AGMETHOD 2 ? ?M1A1 and ?? ? A1 ? ?AGMETHOD 3 ? ?M1A1 ? ?A1 ? ?AGMETHOD 4 ? ?M1A1 and gives ?? ? A1 ? ?AG(ii) ? ? METHOD 1 (or equivalent) ? ? M1A1 ? ?R1hence ?? ? AGNote: ? ? Award as above for use of either and or and .METHOD 2?(or equivalent) ? ? M1A1? ? R1hence? ? ??AGNote: ? ? Award as above for use of either and or and .METHOD 3 ? ?M1A1 ? ?R1hence ?? ? AG[6 marks] 27a. [2 marks] Markscheme ? ? M1 ? ? R1hence is odd ? ? AG[2 marks] 27b. [3 marks] Markscheme ? ? M1A1A1[3 marks] 27c. [3 marks] Markscheme ? ? A1?Note: ? ? This may be seen in part (b).?Note: ? ? Do not allow FT from part (b).? ? ? M1 ? ? A1[3 marks] 27d. [3 marks] Markscheme-coordinates of the Max Min Points are ? ? M1A1so range of is ? ? A1?Note: ? ? Allow FT from (c) if values of , within the domain, are used.[3 marks] 27e. [3 marks] MarkschemeShape: The graph of an odd function, on the given domain, s-shaped,where the max(min) is the right(left) of ? ? A1-intercepts ? ? A1turning points ? ? A1[3 marks] 27f. [4 marks] Markscheme ? ? (M1)attempt at “backwards chain rule” or substitution ? ? M1?Note: ? ? Condone absence of limits for first two marks.? ? ? A1 ? ? A1[4 marks] 27g. [2 marks] Markscheme ? ? R1 ? ? R1so ? ? AG[2 marks]Total [20 marks] 28a. [2 marks] Markscheme ? ? A1A1[2 marks] 28b. [4 marks] Markscheme(i) ? ? attempting to find (graphically or analytically) the first ? ? (M1) ? ? A1(ii) ? ? attempting to find (graphically or analytically) the first ? ? (M1) ? ? A1[4 marks] 28c. [2 marks] Markschemedistance travelled (or equivalent) ? ? (M1) ? ? A1?Note: ? ? Award M1 for attempting to form a definite integral involving . To award the A1, correct limits leading to? must include the use of absolute value or a statement such as “distance must be positive”.In part (c), award A1FT for a candidate working in degree mode .[2 marks]Total [8 marks] 29a. [2 marks] Markscheme ? ? M1A1[2 marks] 29b. [3 marks] Markscheme ? ? M1A1 ? ? A1?Note: ? ? Allow integration by parts followed by trig identity.Award M1 for parts, A1 for trig identity, A1 final answer.[3 marks]Total [5 marks] 30a. [4 marks] Markscheme ? ? M1 ? ? M1 ? ? A1 ? ? A1?Note: ? ? and might be interchanged earlier.?Note: ? ? First M1 is for interchange of variables second M1 for manipulation?Note: ? ? Final answer must be a function of [4 marks] 30b. [2 marks] Markschemeequating coefficients and ? ? (M1) and ? ? A1?Note: ? ? Could also be done by division or substitution of values.[2 marks] 30c. [1 mark] Markscheme ? ? A1?Note: ? ? accept equivalent e.g. [1 mark]Total [7 marks] 31. [7 marks] Markscheme ? ? (A1)EITHERintegral is ? ? M1A1 ? ? M1A1?Note: ? ? Award M1 only if the integral has completely changed to one in .?Note: ? ? needed for final A1?ORintegral is ? ? M1A1?Note: Award M1 only if the integral has completely changed to one in .? ? ? M1A1?Note: ? ? In both solutions the two method marks are independent.?THEN ? ? (A1) ? ? A1Total [7 marks] 32. [6 marks] Markschemeany attempt at integration by parts ? ? M1 ? ? (A1) ? ? (A1) ? ? A1?Note: ? ? Condone absence of limits at this stage.? ? ? A1?Note: ? ? Condone absence of limits at this stage.? ? ? A1 ? ? AG[6 marks] 33. [8 marks] MarkschemeEITHER ? ? (M1) ? ? A1OR ? ? (M1) ? ? A1THEN ? ? (A1)?Note: ? ? This A1 is independent of the first two marks? ? ? M1A1?Note: ? ? Award M1 for attempting to obtain integral in terms of and ? ? ? A1 ? ? A1 ? ? A1[8 marks] 34. [4 marks] Markscheme ? ? (M1)(A1)(A1)?Note: ? ? Award M1 for integral involving the function given; A1 for correct limits; A1 for and ? ? ? A1[4 marks]Printed for International School of Monza ? International Baccalaureate Organization 2019 International Baccalaureate? - Baccalauréat International? - Bachillerato Internacional? ................
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