Mark Scheme Pure Mathematics Year 1 (AS) Unit Test 8 .

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Mark schemePure Mathematics Year 1 (AS) Unit Test 8: Exponentials and LogarithmsMarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b6thM11.1bMakes an attempt to solve the expressions by division. For1example, b3 (or equivalent) seen.8M11.1b12A11.1bA11.1bQ1aSchemeSubstitutes (2, 400) into the equation. 400 ab 2Substitutes (5, 50) into the equation.Solves for b. b 0.5 or b 50 ab5Solves for a. a 1600Set up, use andcritiqueexponentialmodels of growthand decay.(5)1bDivides by ‘1600’ and takes logs of both sides.M1ft1.1bx 1 k log log 2 1600 5thUnderstand anduse the three lawsof logarithms. 1 1 Uses the third law of logarithms to write log x log 2 2 xor log 2 x log 2 anywhere in solution.B12.1 1 Uses the law(s) of logarithms to write log log 2 2 anywhere in solution.B12.1 1600 log k *Uses above to obtain x log 2A1*2.1x(4)(9 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 8: Exponentials and LogarithmsMarksAOsPearsonProgression Stepand ProgressdescriptorUses appropriate law of logarithms to writelog11 2 x 1 x 4 1M13.1a5thInverse log11 (or 11 to the) both sides. 2 x 1 x 4 11M11.1bSolve simplelogarithmicequations usingthe laws of logs.M11.1bM11.1bA11.1bB13.2Q2SchemeDerives a 3 term quadratic equation.2 x 2 7 x 15 0Correctly factorises 2 x 3 x 5 0 or uses appropriatetechnique to solve their quadratic.Solves to find x 32Understands that x 5 stating that this solution wouldrequire taking the log of a negative number, which is notpossible.(6 marks)Notes Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemeQ3aPure Mathematics Year 1 (AS) Unit Test 8: Exponentials and LogarithmsMarksAOsPearsonProgression Stepand ProgressdescriptorGraph has correct shape anddoes not touch x-axis.M13.1a3rdThe point (0, 1) is given orlabelled.A1SchemeFigure 13.1aSketch the graphof y ax (fora 1)(2)3biii 1 Translation 1 unit right (or positive x direction) or by 0 B1 0 Translation 5 units up (or positive y direction) or by 5 B12.2a5th2.2aTransform thegraphs ofexponentialfunctions usingtranslations andstretches.(2)(4 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 8: Exponentials and LogarithmsQ4Scheme Correctly factorises. 8x 1 2 8x 1 16 0MarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b5thSolve exponentialequations usinglogarithms.(or for example, y 2 y 16 0 )States that 8 x 1 2 , 8 x 1 16 (or y 2, y 16).Makes an attempt to solve either equation (e.g. uses laws ofindices. For example,38 2 or183 2 or 8 34A11.1bM12.2a 16 or48 3 16 (or correctly takes logs of both sides).Solves to find x 4o.e. or awrt 1.333A11.1bSolves to find x 7o.e. or awrt 2.333A11.1b(5)(5 marks)Notes42nd M mark can be implied by either x 1 14or x 1 33 Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemeQ5aPure Mathematics Year 1 (AS) Unit Test 8: Exponentials and LogarithmsMarksAOsPearsonProgression Stepand ProgressdescriptorAttempt to find intersectionwith x-axis. For example,log9 x a 0M11.1b4thSolving log9 x a 0 tofind x a 1, so coordinatesof x-intercept are ( a 1, 0)oeA11.1bSubstituting x 0 to derivey log9 x a , socoordinates of y-intercept are 0,log9 x a B13.1aAsymptote shown at x astated or shown on graph.B13.1aIncreasing log graph shownwith asymptotic behaviour andsingle x-intercept.M13.1aFully correct graph withcorrect asymptote, all pointslabelled and correct shape.A12.2aSchemeFigure 2Sketch the graphy log(x).(6)5blog9 x a 2log9 x a seen.M12.15thThe graph of y log9 x a is a stretch, parallel to the y-A12.2aUnderstand anduse the three lawsof logarithms.22axis, scale factor 2, of the graph of y log9 x a .(2)(8 marks)Notes5aAward all 5 points for a fully correct graph with asymptote and all points labelled, even if all working is notpresent Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

Mark schemePure Mathematics Year 1 (AS) Unit Test 8: Exponentials and LogarithmsMarksAOsPearsonProgression Stepand ProgressdescriptorMakes an attempt to subsitute 7 into the equation, for example,P 100e0.4 7 seen.M11.1b4th1644 or 1640 only (do not accept non-integeric final answer).A13.4Understand theproperties offunctions of theform ax.2.2a4thQ6aScheme(2)6bIt is the initial bacteria population.B1Understand theproperties offunctions of theform ax.(1)6cStates that 100e0.4t 1000000 or that e0.4t 10000Solves to find t ln 10000 M13.4M11.1bA13.50.424 (hours) cao (do not accept e.g. 24.0).6thSet up, use andcritiqueexponentialmodels of growthand decay.(3)(6 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 8: Exponentials and LogarithmsQ7aSchemeUses the equation of a straight line in the formMarksAOsPearsonProgression Stepand ProgressdescriptorM11.1b6th1.1bSet up, use andcritiqueexponentialmodels of growthand decay.1.1b6thM11.1bSet up, use andcritiqueexponentialmodels of growthand decay.A11.1bA11.1blog 4 V mt c or log 4 V k m(t t0 ) o.e.Makes correct substitution. log 4 V 1t log 4 40000 o.e.10A1(2)7bEither correctly rearranges their equation by exponentiation M11t log4 4000010For example, V 4or takes the log of both sidestof the equation V ab . For example, log 4 V log 4 (abt ) .Completes rearrangement so that both equations are in directlycomparable form V log 4 V 140000 4 10 t t and V ab or 1t log 4 40000 and log 4 V log 4 a t log 4 b .10States that a 40 000States that b 4 110(4)7ca is the initial value of the car o.e.B12.2ab is the annual proportional decrease in the value of the caro.e. (allow if explained in figures using their b. For example,(since b is 0.87) the car loses 13% of its value each year.)B12.2a(2) Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.6thSet up, use andcritiqueexponentialmodels of growthand decay.

Mark scheme7dPure Mathematics Year 1 (AS) Unit Test 8: Exponentials and LogarithmsSubstitutes 7 into their formula from part b. Correct answer is 15 157, accept awrt 15 000B1ft3.44thUnderstand theproperties offunctions of theform ax.(1)7eUses 10000 abt with their values of a and b or writes1log 4 10000 t log 4 40000 (could be inequality).10M1Solves to find t 10 years.A1ft3.45thSolve exponentialequations usinglogarithms.1.1b(2)7fAcceptable answers include.B13.5b6thSet up, use andcritiqueexponentialmodels of growthand decay.The model is not necessarily valid for larger values of t.Value of the car is not necessarily just related to age.Mileage (or other factors) will affect the value of the car.(1)(12 marks)Notes7b2nd M mark can be implied by correct values of a and b.7cAccept answers that are the equivalent mathematically. For example, for b. the value of the car in 87% of the valuethe previous year. Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free.

8 2 8 16 0xx 11 (or for example, yy 2 16 0 ) M1 1.1b 5th Solve exponential equations using logarithms. States that 82x 1, 8 16x 1 (or y 2, y 16). A1 1.1b Makes an attempt to solve either equation (e.g. uses laws of indices. For example, 382 or 1 823 or 4 3 8 16 or 4 8 163 (or correctly takes logs of both sides). M1 2.2a Solves to find 4 3 x

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