Mark Scheme Pure Mathematics Year 1 (AS) Unit Test 7 .

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Mark schemePure Mathematics Year 1 (AS) Unit Test 7: IntegrationQScheme1Makes an attempt to expand 5 3 x 5 3 x . Must be 4 MarksAOsPearsonProgression Stepand ProgressdescriptorM13.15thA11.1bB11.1bIntegrate morecomplicatedfunctions such asthose requiringsimplification orrearrangement.M11.1bA11.1bterms (or 3 if x terms collected).Fully correct expansion 25 30 x 9 x or125 30 x 2 9x1Writes x as x 2 (or subsequently correctly integrates thisterm)1 Makes an attempt to find (25 30 x 2 9 x)dx . Raising xpower by 1 at least once would constitute an attempt.3Fully correct integration. 25 x 20 x 2 9 2x C o.e.2(5 marks)NotesAward all 5 marks for a fully correct final answer, even if some working is missing. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemeQ2Pure Mathematics Year 1 (AS) Unit Test 7: IntegrationSchemeUses laws of indices correcty at least once anywhere insolution131 1(e.g. x 2 or x x 2 or x x x 2 seen or implied).xMakes an attempt at integrating h '( x)3 15 x 2 40 x 12MarksAOsPearsonProgression Stepand bA12.2aFind the equationof a curve giventhe gradientfunction and apoint on thecurve.Raising at least one x power by 1 would constitute an attempt.51Fully correct integration. 6 x 2 80 x 2 (no need for C here).Makes an attenpt to substitute (4, 19) into the integratedexpression. For example,519 6 4 21 80 4 2 C is seen.Finds the correct value of C. C 13States fully correct final answer h( x) any equivalent form.56x 2 80 x 13 or(6 marks)NotesAward all 6 marks for a fully correct final answer, even if some working is missing. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemePure Mathematics Year 1 (AS) Unit Test 7: IntegrationQ3aSchemeMakes an attempt to find 10 6 x dxMarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b5thFind definiteintegralsanalytically.Raising x powers by 1 would constitute an attempt.A12.2a 10 a 3 a orM1ft1.1bRearranges to a 3-term quadratic equation (with 0).9a 2 10a 1 0M1ft1.1bCorrectly factorises the LHS: (9a – 1)(a – 1) 0 or uses avalid method for solving a quadratic equation (can be impliedby correct answers).M1ft1.1bA11.1b2aShows a fully correct integral with limits. 10 x 3x 2 1aMakes an attempt to substitute the limits into their expression. For example, 10 2a 3 2a 22 20a 12a 10a 3a is seen.22States the two fully correct answers a 1or a 19For the first solution accept awrt 0.111(6)3bFigure 1Straight line slopingdownwards with positive xand y intercepts. Ignoreportions of graph outside0 x 2M1Fully correct sketch withpoints (0, 10), and ( 5 , 0)A13.11stAssumedKnowledge2.2a3labelled. Ignore portions ofgraph outside 0 x 2(2) Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark scheme3cPure Mathematics Year 1 (AS) Unit Test 7: IntegrationStatements to the effect thatthe (definite) integral will onlyequal the area (1) if thefunction is above the x-axis(between the limits)B12.1Find an areabelow the x-axisusing integration(including anappreciation ofthe meaning of anegative definiteintegral).ANDwhen a 1, 2a 2, so part ofthe area will be above the xaxis and part will be below thex-axis.Greater than 1.5thB12.2a(2)(10 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemeQPure Mathematics Year 1 (AS) Unit Test 7: IntegrationScheme4Writes t as1t2150t 2or 50 t asMarksAOsPearsonProgression Stepand ProgressdescriptorB12.2a5th(can be implied by correct integral).1Makes an attempt to find(50t 2 20t 2 t 3 )dt . Raising at20least one t power by 1 would constitute an attempt.M11.1bMakes a fully correct integration (ignore limits at this stage).M11.1bM1ft1.1bA11.1b s 31 100 2 20 3 t 4 t t 20 334 1Integrate morecomplicatedfunctions such asthose requiringsimplification orrearrangement.200Makes an attempt to substitute the limits into their integratedfunction. For example,331 10020 203 204 100 2 20 03 04 20 2 0 20 334 334 is seen. Award mark even if the 0 limit is not shown.States fully correct answer. s 816 cao.(5 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemePure Mathematics Year 1 (AS) Unit Test 7: IntegrationQ5aSchemeAttempts to take out x or –x. y x x 2 2 x 8 or y x x 2 2 x 8MarksAOsPearsonProgression Stepand ProgressdescriptorM13.14th Fully and correctly factorised cubic.M11.1bA11.1bFactorise cubicexpressions withmonomial factors.y x(4 x)(2 x) or y x( x 4)( x 2)Correct coordinates written. A( 2,0) and B(4, 0).(3)5b Makes an attempt to find ( x3 2 x 2 8 x)dxM13.1aRaising at least one x power by 1 would constitute an attempt.Fully correct integration seen.A11.1bM11.1bA11.1b 20stated or used as area here or later in solution (could be3implied by correct final answer).B13.2Makes an attempt to substitute limits into integrated functionto find the area between x 0 and x 4128 64 3 64 (0) M11.1bA11.1b0 x4 2 32 x 4 x (ignore limits at this stage) 4 3 2Makes an attempt to substitute limits into integrated functionto find the area between x 2 and x 0 0 4 16 16 3 Finds the correct answer. Finds the correct answer.2031283 Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.5thFind an areabelow the x-axisusing integration(including anappreciation ofthe meaning of anegative definiteintegral).

Mark schemeCorrectly adds the two areas.Pure Mathematics Year 1 (AS) Unit Test 7: IntegrationA1148o.e.32.2a(8)(11 marks)Notes5a Award method marks for substituting limits even if evaluation at x 0 is not seen.5b For the first integral, candidates may integrate –f(x) between 2 and 0 to obtain a positive answer directly. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemeQ6aPure Mathematics Year 1 (AS) Unit Test 7: IntegrationSchemeMarksAOsPearsonProgression Stepand ProgressdescriptorEquates the curve and the line. x 2 8 x 20 x 6M13.14thSimplifies and factorises. (x – 7)(x – 2) 0 (or uses other validmethod for solving a quadratic equation).M11.1bFinds the correct coordinates of A. A(2, 8).A11.1bInterpret solutionsto simultaneousequationsgraphically.Finds the correct coordinates of B. B(7, 13).A11.1b(4)6bMakes an attempt to find the area of the trapezium bounded byx 2, x 7, the x-axis and the line.For example,5 8 13 or2 72M13.1AssumedKnowledge( x 6)dx seen.A1Correct answer. Area 52.5 o.e.1st1.1b(2)6c3.15thM11.1bFind the areaunder a curveusing integration. 1 Correctly finds x3 4 x 2 20 x 3 2A11.1bMakes an attempt to substitute limits into the definite integral. 343 8 3 196 140 3 16 40 M11.1bA11.1b 72B1( x 2 8 x 20)dx .Makes an attempt to find the integral. Raising at least one xpower by 1 would constitute an attempt.7Correct answer seen. 95or 31.6 oe seen.3(5) Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark scheme6dPure Mathematics Year 1 (AS) Unit Test 7: Integration Understands the need to subtract the two areas. ( 52.5 31.6 )220.8 units seen (must be positive).M12.2a5thA12.2aFind the areaunder a curveusing integration.(2)(13 marks)Notes6a If A0A0, award A1 for full solution of quadratic equation (i.e. x 2, x 7). Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Simplifies and factorises. (x – 7)(x – 2) 0 (or uses other valid method for solving a quadratic equation). M1 1.1b Finds the correct coordinates of A. A(2, 8). A1 1.1b Finds the correct coordinates of B. B(7, 13). A1 1.1b (4) 6b Makes an attempt to find the area of the trapezium bounded by x 2, x 7, the x-axis and the line. For .

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