NST/CST Part 1A Mathematics Past Exam Questions

This document last updated: 13 August 2018.

These pages provide comments on past exam papers for the Natural Sciences/Computer Sciences Part 1A Mathematics Course. Where there is no comment for a question, it probably means I've not yet had a go at it. An asterisk (*) denotes particularly good questions to try.

NB: the question numbering for Section B of the 2008 papers differed between the paper actually sat in the examination hall and the version now online. The paper sat in the examination hall had questions starting at B1, whereas the version now online had questions numbered as below.

2006 Papers 1 and 2, 2008 Paper 1, 2010 Paper 1 and 2016 Paper 1 were hard overall. Don't start there when revising! 2012 Paper 2 was relatively easy.

Comments on papers prior to 2001 are here

2001I1Vectors: geometricOK
2001I3Matrices: eigenvalues/vectorsOK
2001I4Misc: integration(c) may mislead you as to which end to start from
2001I5Misc: integration, stationary valuesOK
2001I6Vector surface integralsOK
2001I7Matrices, simultaneous eqnsOK
2001I8Vector surface integralsOK
2001I9Taylor seriesOK
2001I11Complex numbersOK
2001II1ODEs: 1st orderVery hard, and (c) is vindictive
2001II2LeibnitzOK but hard and rather dull
2001II3MiscSeems to me to need 2nd year material
2001II4ODEs: 2nd orderOK
2001II5Line integrals, conservative fieldsOK
2001II7Matrices: determinants, simultaneous eqnsOK
2001II8Multiple integralsOK
2001II9Differentials: transformationsLooks like (a) still holds for (b), which is bad style, but otherwise OK
2001II10Differentials: thermodynamicsOK
2001II11Probability/statisticsOK (rather easy)
2001II12Series, limitsOK (pretty easy)
2002I1Vectors: geometricOK
2002I3Misc: stationary values, eqns of linesOK
2002I4LagrangeOK though not easy to solve the simultaneous eqns
2002I5Multiple integralsOK
2002I6MatricesOK (rather easy)
2002I7ODEs: 1st orderOK though a bit long and not easy
2002I9Line integrals, conservative fieldsOK
2002I10PDEs: diffusionOK though last part unclear
2002I11Complex numbersOK
2002I12PDEsOK but not well-designed
2002II1Vectors: geometricCurve in (b) looks to be a confusing mess
2002II2Differentiation of integralsOK
2002II3Multiple integralsOK
2002II4Probability/statisticsOK (potentially quite tricky)
2002II5Matrices, simultaneous eqnsOK
2002II6Vector surface integralsOK (not easy)
2002II7ODEs: 2nd orderOK
2002II8Matrices: orthogonal, eigenvalues/vectorsOK
2002II9Taylor seriesOK
2002II10Series, limitsOK (not easy)
2002II11Differentials: transformationsOK*
2002II12Stationary valuesOK*
2003I1Vectors: eqnsOK*
2003I3Stationary values/Curve sketching/IntegrationOK
2003I6PDEs: heat conductionOK* (not long)
2003I7Line integrals, conservative fieldsOK
2003I8Taylor seriesOK*
2003I9Hyperbolic functionsOK (fairly easy)
2003I10Misc: series, integrationOK though (b) is slightly unnerving
2003I11ODEs: 1st orderOK (pretty quick)
2003I12Multiple integralsNot easy
2003II1Vectors: geometricOK (fairly easy)
2003II3Matrices: eigenvalues/vectorsOK
2003II4Vector surface integralsOK* except for the ambiguity regarding the word "evaluate"
2003II5Probabilitiy/Probability distributionsOK
2003II6Matrices, simultaneous equationsOK (quite quick)
2003II7Vector surface integralsUnusual - intellectually interesting but a bit odd
2003II8Matrices, suffix notationOK*
2003II9Differentials: thermodynamicsOK (easier than many of this type)
2003II10Partial differentiation: transformationsOK
2003II11ODEs: 2nd orderOK
2003II12Lagrange/transformation of axesQuite hard for the unfamilar
2004I1Vectors: eqns, geometricOK
2004I3Vector surface integralsOK
2004I4IntegrationOK (fairly easy)
2004I5Probability distributionsOK (slightly unusual)
2004I7Matrices, determinantsOK if you know about epsilon(ijk)
2004I8Line integrals, conservative fieldsOK (easy)
2004I9Complex numbers, hyperbolic functionsOK (fairly easy after (a))
2004I10Complex numbers, hyperbolic functionsOK (though a strange combination of parts)
2004I11ODEs: 1st orderOK
2004I12Multiple integralsOK
2004II1Vectors: geometricOK (though not clear which way they intend for (b) to be done)
2004II3MatricesOK* (not easy)
2004II4Misc: integration, stationary valuesNot easy and not fun
2004II5Probabilitiy/Probability distributionsOK
2004II6PDEsOK* (not easy but interesting)
2004II7Vector surface integralsOK
2004II8Taylor SeriesOK ((a) is dull and a bit tedious)
2004II9Partial differentiation, Taylor SeriesOK but a bit dull
2004II10Integration, SeriesDoable but tedious
2004II11ODEs: 2nd orderOK
2005I1Taylor SeriesHarder and less well-designed than usual
2005I2MatricesHard if you really have to use suffix notation througout
2005I4Probability distributionsOK but wording vague in (b(i))
2005I5Vectors: geometricOK
2005I6Vector surface integralsOK (hard but illustrative)
2005I7Misc: integration, approximations(c) is interesting; rest is tedious
2005I8Multiple integralsOK ((c) is interesting and not easy)
2005I9ODEs: 1st orderOK*
2005I10ODEs: 2nd orderOK but tedious
2005I11Misc: integration, stationary valuesOK*
2005I12FourierOK (fairly quick)
2005II1Line integrals, conservative fields, Vectors: geometricOK
2005II2Vector surface and volume integralsOK once you understand it's just a thick spherical shell
2005II5Complex numbersOK*
2005II6Vectors: geometricOK
2005II7Differentials: thermodynamicsOK
2005II8Differentials: exactOK (not easy but doable)
2005II9Matrices: eigenvalues/vectorsOK
2005II10Integration, SeriesOK (not easy but doable)
2005II12PDEs: diffusion equationOK (seems very quick)
2006I1MatricesPart (b) is very hard in that the obvious approach doesn't work
2006I2Matrices: eigenvalues/vectorsOK though a bit long
2006I3Complex numbers(a) and (b) are OK but rather tedious; (c) is hard until you spot the method
2006I4Stoke's TheoremOK
2006I5Vectors: geometricOK
2006I6Vectors: algebraic, geometricOK but dull
2006I7ProbabilityOK (fairly quick)
2006I8LagrangeOK (very quick if you understand (b))
2006I9Taylor SeriesTedious
2006I10ODEs: 2nd orderNot without merit, but rather lost in the tedium
2006I12PDEs: LaplaceOK (mischievous but interesting)
2006II1Misc: Differentiation, Taylor, IntegrationOK (not easy)
2006II2Matrices: suffix notationPretty hard, as well as scary
2006II3Line integrals, conservative fieldsOK*
2006II4Vector surface integralsStrange and a bit confusing
2006II5Volume integralsStrange: easy but confusing - best avoided
2006II6Misc: integration, approximationsOK
2006II7Differentials: exactOK* (Quick if you can spot the shortcuts)
2006II8ProbabilityOK (quick)
2006II9ODEs(b) is very hard until you see it and (d) is a plausible solution for (b), so not terribly satisfactory overall
2006II10SeriesQuite hard
2006II11PDEs: transformationsA standard method, but very tedious here
2006II12Stationary valuesVery tedious
2007IB1Vectors: algebraicOK
2007IB2Complex numbersOK*
2007IB3Taylor Series(a) OK* (b) Tedious
2007IB5Exact differentials, ODEs: 1st orderOK
2007IB6Stationary valuesOK*
2007IB7Multiple integralsOK*
2007IB8MatricesOK (indicative marks for (a) subparts unindicative!)
2007IB9Series and integrationOK (last parts of (a) are quite hard)
2007IB10Differentiation of integralsHorrid
2007IIB1Misc: Cartesian and polar coordinatesOK (unnervingly unusual)
2007IIB2InegrationOK* - though quick
2007IIB3ProbabilityOK if you know enough stats
2007IIB4ODEs: 2nd order(a) too simple (b) sketch seems hard for just 4 marks
2007IIB5Differentials: thermodynamicsOK - straightforward
2007IIB6Vector surface and line integrals(b) needs one to assume the coordinates are spherical polars and is rather tedious
2007IIB8FourierOK* though time consuming
2007IIB9LagrangeOK - rather quick though diagram is fiddly
2007IIB10PDEsOK - straightforward
2008IB11Vectors: geometricOK but confusing
2008IB12Complex numbersQuite hard; only 2 marks for (b)iii!
2008IB13Taylor SeriesOK but (c) is tedious
2008IB14Multiple integralsTo do with multiple integrals (as the question implies) is very hard unless you get the right order of integration variables, and still not easy even then. There is a very quick geometrical argument...
2008IB15ProbabilityI gave up on (b)
2008IB16ODEs: 1st orderLong and difficult
2008IB17Stationary values; gradOK
2008IB18MatricesStraightforward if dull
2008IB19Limits and seriesOK
2008IB20Leibnitz; SchwartzBookwork or hard
2008IIB10Vectors: geometric, algebraicOK but long and not easy
2008IIB11ProbabilityOK but be *very* careful how you interpret part (a)
2008IIB12IntegrationI can now do this, but it is very difficult
2008IIB13ODEs: 2nd orderOK but dull
2008IIB15Line integrals, conservative fieldsOK - bit fiddly
2008IIB16MatricesOK but long and a bit tedious
2008IIB18Vector surface integralsOK (looks scary but doable)
2008IIB19PDEs(a) long but OK (b) OK if you realise they are using Sigma to denote a function of x, with Sigma0 being a constant
2009IB11Misc: differentiation, mainlySeems tedious and error-prone
2009IB12ProbabilityFairly easy
2009IB13Matrices: eigenvalues/vectorsTedious and you need to know about diagonalisation
2009IB14Complex numbersFairly quick
2009IB15Vectors: geometric, algebraic(a)(i) unclear, rest OK
2009IB17ODEs: 1st order(a)(ii) very hard unless you know how; rest OK
2009IB18Partial differentiationVery tedious - I gave up on (c)
2009IB20Series; limitsOK* ((b)(iii) is hard)
2009IIB11Misc: geometryOK
2009IIB12ODEs: 2nd orderTedious and error-prone
2009IIB13Stationary values(a) is very tedious and error-prone; (b) is interesting but needs (a)
2009IIB14Matrices: equations, determinantsOK*
2009IIB15FourierRather tedious unless there's a trick for (b)
2009IIB16Div/Grad/CurlRather tedious unless there are shortcuts
2009IIB18Multiple integralsOK
2009IIB19Vector surface integralsOK
2009IIB20Misc: Leibnitz, Binomial expansionOK but badly structured ((a) presumably still holds for (b), and (b) for (c))
2010IB11Taylor SeriesOK; relatively quick
2010IB12ProbabilityHard unless you spot the right approaches for (e) and (f)
2010IB13Matrices: eigenvalues/vectorsUtterly tedious
2010IB14Complex numbersOK but repetitious
2010IB15Vectors: algebraic, geometricHard
2010IB16IntegrationOK though (d) needs a Jacobian and (c) is quite hard
2010IB17ODEs: 1st, 2nd orderOK; bit fiddly
2010IB18Partial differentiationOK; not easy
2010IB19Misc: partial differentiation, PDEsOK though quite quick; (c) is interesting
2010IB20Limits and seriesOK; interesting though quick if you know how
2010IIB11Misc: geometryOK
2010IIB12Misc: vectors, ODEsOK
2010IIB13Stationary valuesOK except the sketch is unpleasant
2010IIB14MatricesOK; interesting though quick if you know how
2010IIB16Line integrals, conservative fieldsOK
2010IIB17ProbabilityOK; quite quick; (c) is interesting
2010IIB18Multiple integralsOK
2010IIB20Misc: LeibnitzOK
2011IB11Taylor SeriesOK; very quick
2011IB12ProbabilityToo long
2011IB13Matrices: eigenvalues/vectorsOK but fiddly
2011IB14Fourier, ODEsOK, though wording in (e) could be clearer
2011IB15Vectors: algebraic, geometricOK; fairly quick
2011IB16ODEs: 1st, 2nd orderOK
2011IB17Complex numbersOK; very quick
2011IB18Differentials: exactOK
2011IB19Vector surface integralsOK*
2011IB20Functions, continuity, differentiability(a) is quite hard, and horrendous to sketch f1 and f2 without a computer; (b) is interesting
2011IIB11Line integrals, vector surface integralsTechnically doable but crazy
2011IIB12Spherical trigonometry(b) is fiddly and tiresome; (c) the l^2+m^2+n^2=1 bit is best ignored, I think; (d) very quick
2011IIB13Misc: hyperbolic functions, grad, stationary valuesOK; quite quick
2011IIB14MatricesOK, except (b) is equivalent to 2010 II 14(b)
2011IIB15ODEs: second orderTedious algebra though physically interesting
2011IIB17ProbabilityOK; quick
2011IIB18Multiple integralsOK; quick
2011IIB19PDEsHard but interesting
2011IIB20Misc: Leibnitz, ODEs, orthogonalityHard
2012IB11Simultaneous equations, vectorsOK; quite useful
2012IB12Complex numbersOK*; quite quick
2012IB13ODEs: 1st orderOK until (c) gets tedious
2012IB14Vectors: algebraic, geometricNot easy; you need to understand r x p=q
2012IB15Misc: coordinate transformations, integration(a) hard if you've not seen before (c) has nasty twist
2012IB16Stationary valuesToo easy
2012IB18Probability, distributionsOK; fairly easy
2012IB19Functions, continuity, differentiabilityOK
2012IIB11Matrices: eigenvalues/vectorsOK*; fairly easy
2012IIB12Misc: integration, recurrence relationsOK* except ignore their hint
2012IIB13ODEs: second orderOK; quick
2012IIB14Vectors, vector areasOK*
2012IIB15Vector fields, diferential equationsOK; quick if you know how
2012IIB16Exact differentials, ODEs: 1st order(a) is tedious; rest is very easy
2012IIB17Taylor SeriesOK; quite useful
2012IIB18ProbabilityOK; fairly easy but fiddly
2012IIB19Differentiation of integralsOK; (b) is interesting
2013IB11Simultaneous equations(a) tedious (b) OK (not easy)
2013IB12Complex numbers(a),(b),(c) good, (d) could be horrendous
2013IB13ODEs: 1st orderOK*
2013IB14Vectors(b) could be horrid unless you know how
2013IB15Multiple integrals(b) too hard for 4 marks (c) OK (not easy)
2013IB16Stationary values(a) tiresome (b) OK but mark distribution unfair
2013IB17FourierDull but OK
2013IB18ProbabilityVery easy and quick
2013IB19Limits and series(b) interesting, rest is dull bookwork
2013IB20Lagrange(a) easy (b) too long and somewhat ambiguous
2013IIB11DistributionsOK*; fairly easy
2013IIB12Matrices and suffix notation(a) useful (b)(vi) hard, rest OK
2013IIB13IntegrationOK* (not easy)
2013IIB14ODEs: 2nd order(a) tedious (b)(ii) hard
2013IIB15Equations of curves(a),(b),(c) OK but not easy, (d) horrendous unless you get lucky
2013IIB16Vector fieldsOK, albeit unusual in (a)(iv)
2013IIB17Exact differentialsStandard Maxwell stuff
2013IIB18Misc: partial differentiation, curve sketchingOK
2013IIB19Vector surface integralsOK
2014IB11Matrices: eigenvalues/vectorsOK except (d) unclear
2014IB12Complex numbersOK
2014IB13ODEs: 1st, 2nd orderOK
2014IB14Vectors: geometric(a) is not easy unless you know; (b) is ambiguous
2014IB15Multiple integralsOK though (a)(iii) and (b) hard if you don't see the tricks
2014IB16Stationary valuesOK
2014IB17FourierOK (fairly quick)
2014IB18Probability/statisticsRather easy
2014IB19Taylor series, series(d) is hard
2014IB20LagrangeTime-consuming and not easy
2014IIB12Matrices, simulataneous eqnsIrritating question
2014IIB13Integration(a)(ii) tedious (d) interesting
2014IIB14Differentials: exactOK
2014IIB15Taylor seriesOK (c) is tedious but adds variety
2014IIB16Vector surface integrals, line integrals, conservative fieldsBit tedious but otherwise fine
2014IIB17GradFar too quick
2014IIB18Misc: chain rule(e) is tedious and error-prone
2014IIB19PDEs: wave eqnHard
2014IIB20Integration, Schwarz(a) bookwork (b) ugly (c) interesting and not easy
2015IB11Complex numbersHard overall
2015IB12Differentials: exact, stationary valuesEasy and quick apart from the plot
2015IB13PDEsAlgebra tedious
2015IB14Misc: some integrationNot easy but interesting*
2015IB15Taylor seriesTedious but doable
2015IB16Probability/statisticsInteresting but too easy
2015IB17IntegrationOK, bit tedious; (a) is unusual
2015IB18Matrices: eigenvalues/vectorsNot without merit, but rather lost in the tedium
2015IB19SeriesWay too hard, long and confusing
2015IB20Vector surface integrals, div, curlOK: quite interesting but not easy
2015IIB11Vectors: geometric, algebraicOK
2015IIB12Multiple integralsNot easy
2015IIB13Line integrals, conservative fieldsEasy and dull
2015IIB14Probability/statisticsDull and a bit confusing at the end
2015IIB15ODEs: 2nd orderOK*
2015IIB16ODEs: 1st order, Exact differentialsWay too long and hard
2015IIB17Matrices: eigenvalues/vectorsBit long, and confusing as to which matrix they mean in (c)
2015IIB18FourierToo long and tedious; I didn't bother to do (b) in full
2016IB11Complex numbersOK; (d) not easy
2016IB12Partial differentiation; stationary valuesOK except (b)(ii) too hard except by computer
2016IB13Line integrals, conservative fieldsToo long and tedious
2016IB14Matrices: eigenvalues/vectorsToo hard, long, unclear
2016IB15Taylor seriesToo long, boring and unpredictable
2016IB16Probability/statisticsToo long and confusing
2016IB18IntegrationBit too long and not easy
2016IB19Partial Differentiation: transformationsOK
2016IB20Line integrals; surface integralsOK
2016IIB11Vectors: geometric, algebraicNot easy but interesting*
2016IIB12Multiple integralsInteresting but too long and (e) is unpredictable
2016IIB13Curve sketching; miscHorrendous
2016IIB14ODEs: 1st orderWay too long
2016IIB15Matrices, simultaneous eqnsOK except I don't think (a) is within the syllabus
2016IIB16Probability/statisticsOK (fairly easy)
2016IIB18FourierOK but dull
2016IIB19Lagrange; limitsOK*
2016IIB20PDEsOK (fairly easy)
2017IB11Complex numbersOK*
2017IB12Partial differentiation; Exact differentialsA bit long and hard unless you did the supervision question
2017IB13ODEs: 1st orderHard
2017IB14Stationary valuesSomewhat long; last part strange
2017IB15Taylor SeriesOK
2017IB16Probability/statisticsOK, but note the hint
2017IB17IntegrationToo easy after first integral
2017IB18Matrices: eigenvalues/vectorsOK but too time-consuming; last part unclear
2017IB19Newton-Raphson; Series(d) seems ridiculous, the rest is potentially interesting but too long
2017IB20ODEs: 1st order; MiscOK*
2017IIB11Vectors: algebraic, geometricOK (not easy)
2017IIB12Multiple integralsOK, though (b) has a hard step
2017IIB13Line integrals, conservative fieldsOK but dull
2017IIB14Probability/statisticsOK (fairly easy)
2017IIB15ODEs: 1st order(a) too long; (b) OK
2017IIB16Surface integral, multiple integralsOK
2017IIB17ODEs with eigenvalues/vectorsStandard method but not in syllabus??
2017IIB20ODEs: 1st order, PDEsOK
2018IB11Complex numbersOK
2018IB12Multiple integralsBoth parts of (b) need a trick; otherwise OK
2018IB13ODEs: 1st order(c) is easy if you spot the trick and horrid otherwise
2018IB14Stationary valuesDanger of death by boredom
2018IB15Taylor SeriesOK
2018IB16Probability/statisticsOK, though not easy
2018IB17IntegrationOK if you see substitution for (a); nasty otherwise
2018IB18Matrices: eigenvalues/vectorsOK
2018IB19Continuity/differentiability; SeriesOK
2018IB20Misc: differentiation of integralsOK*
2018IIB11Vectors: algebraic, geometricToo long, quite hard and unsatisfactory; (c) is too long
2018IIB12Partial differentiationToo long and algebra is tedious
2018IIB13Line integrals, conservative fieldsOK
2018IIB14Probability/statisticsStrange question
2018IIB15ODEs: 2nd orderOK
2018IIB16Vector fieldsNot easy
2018IIB17Matrices, simulataneous eqnsOK though a bit unclear as to what proofs are required
2018IIB18FourierHard, especially as to what form to use for y_f_0 in (c)
2018IIB19Lagrange(b) is way too long and hard
2018IIB20PDEs(a)(ii) doesn't follow on from (a)(i), annoyingly

Ian Rudy (graphic containing email address for iar1)