NST/CST Part 1A Mathematics Past Exam Questions

This document last updated: 17 November 2011.

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 and 2010 Paper 1 were hard overall. Don't start there when revising!

Comments on papers prior to 2001 are here

Year	P	Qu	Topic	Comments
2001	I	1	Vectors: geometric	OK
2001	I	2	Fourier	OK
2001	I	3	Matrices: eigenvalues/vectors	OK
2001	I	4	Misc: integration	Unclear
2001	I	5	Misc: integration, stationary values	OK
2001	I	6	Vector surface integrals	OK
2001	I	7	Matrices, simultaneous eqns	OK
2001	I	8	Vector surface integrals	OK
2001	I	9	Taylor series	OK
2001	I	10	Matrices	OK
2001	I	11	Complex numbers	OK
2001	I	12	Probability/statistics	OK
2001	II	1	ODEs: 1st order	Very hard, and (c) is vindictive
2001	II	2	Leibnitz	OK but hard and rather dull
2001	II	3	Misc	Seems to me to need 2nd year material
2001	II	4	ODEs: 2nd order	OK
2001	II	5	Line integrals, conservative fields	OK
2001	II	6	Lagrange	OK
2001	II	7	Matrices: determinants, simultaneous eqns	OK
2001	II	8	Multiple integrals	OK
2001	II	9	Differentials: transformations	Looks like (a) still holds for (b), which is bad style, but otherwise OK
2001	II	10	Differentials: thermodynamics	OK
2001	II	11	Probability/statistics	OK (rather easy)
2001	II	12	Series, limits	OK (pretty easy)
2002	I	1	Vectors: geometric	OK
2002	I	2	Fourier	OK
2002	I	3	Misc: stationary values, eqns of lines	OK
2002	I	4	Lagrange	OK though not easy to solve the simultaneous eqns
2002	I	5	Multiple integrals	OK
2002	I	6	Matrices	OK (rather easy)
2002	I	7	ODEs: 1st order	OK though a bit long and not easy
2002	I	8	Matrices	OK
2002	I	9	Line integrals, conservative fields	OK
2002	I	10	PDEs: diffusion	OK though last part unclear
2002	I	11	Complex numbers	OK
2002	I	12	PDEs	OK but not well-designed
2002	II	1	Vectors: geometric	OK though not especially good
2002	II	2	Differentiation of integrals	OK
2002	II	3	Multiple integrals	OK
2002	II	4	Probability/statistics	OK (potentially quite tricky)
2002	II	5	Matrices, simultaneous eqns	OK
2002	II	6	Vector surface integrals	OK (not easy)
2002	II	7	ODEs: 2nd order	OK
2002	II	8	Matrices: orthogonal, eigenvalues/vectors	OK
2002	II	9	Taylor series	OK
2002	II	10	Series, limits	OK (not easy)
2002	II	11	Differentials: transformations	OK*
2002	II	12	Stationary values	OK*
2003	I	1	Vectors: eqns	OK*
2003	I	2	Limits	OK
2003	I	3	Stationary values/Curve sketching/Integration	OK
2003	I	4	Integration	OK
2003	I	5	Matrices	OK
2003	I	6	PDEs: heat conduction	OK* (not long)
2003	I	7	Line integrals, conservative fields	OK
2003	I	8	Taylor series	OK*
2003	I	9	Hyperbolic functions	OK (fairly easy)
2003	I	10	Misc: series, integration	OK though (b) is slightly unnerving
2003	I	11	ODEs: 1st order	OK (pretty quick)
2003	I	12	Multiple integrals	Not easy
2003	II	1	Vectors: geometric	OK (fairly easy)
2003	II	2	Fourier	OK
2003	II	3	Matrices: eigenvalues/vectors	OK
2003	II	4	Vector surface integrals	OK* except for the ambiguity regarding the word "evaluate"
2003	II	5	Probabilitiy/Probability distributions	OK
2003	II	6	Matrices, simultaneous equations	OK (quite quick)
2003	II	7	Vector surface integrals	?
2003	II	8	Matrices, suffix notation	OK*
2003	II	9	Differentials: thermodynamics	OK (easier than many of this type)
2003	II	10	Partial differentiation: transformations	OK
2003	II	11	ODEs: 2nd order	OK
2003	II	12	Lagrange/transformation of axes	Quite hard for the unfamilar
2004	I	1	Vectors: eqns, geometric	OK
2004	I	2	Series	OK
2004	I	3	Vector surface integrals	OK
2004	I	4	Integration	OK (fairly easy)
2004	I	5	Probability distributions	OK (slightly unusual)
2004	I	6	Matrices	OK
2004	I	7	Matrices, determinants	OK if you know about epsilon(ijk)
2004	I	8	Line integrals, conservative fields	OK (easy)
2004	I	9	Complex numbers, hyperbolic functions	OK (fairly easy after (a))
2004	I	10	Complex numbers, hyperbolic functions	OK (though a strange combination of parts)
2004	I	11	ODEs: 1st order	OK
2004	I	12	Multiple integrals	OK
2004	II	1	Vectors: geometric	OK (though not clear which way they intend for (b) to be done)
2004	II	2	Fourier	OK
2004	II	3	Matrices	OK* (not easy)
2004	II	4	Misc: integration, stationary values	Not easy and not fun
2004	II	5	Probabilitiy/Probability distributions	OK
2004	II	6	PDEs	OK* (not easy but interesting)
2004	II	7	Vector surface integrals	OK
2004	II	8	Taylor Series	OK ((a) is dull and a bit tedious)
2004	II	9	Partial differentiation, Taylor Series	OK but a bit dull
2004	II	10	Integration, Series	Doable but tedious
2004	II	11	ODEs: 2nd order	OK
2004	II	12	Lagrange	OK
2005	I	1	Taylor Series	Harder and less well-designed than usual
2005	I	2	Matrices	Hard if you really have to use suffix notation througout
2005	I	3	Probability	OK
2005	I	4	Probability distributions	OK but wording vague in (b(i))
2005	I	5	Vectors: geometric	OK
2005	I	6	Vector surface integrals	OK (hard but illustrative)
2005	I	7	Misc: integration, approximations	(c) is interesting; rest is tedious
2005	I	8	Multiple integrals	OK ((c) is interesting and not easy)
2005	I	9	ODEs: 1st order	OK*
2005	I	10	ODEs: 2nd order	OK but tedious
2005	I	11	Misc: integration, stationary values	OK*
2005	I	12	Fourier	OK (fairly quick)
2005	II	1	Line integrals, conservative fields, Vectors: geometric	OK
2005	II	2	Vector surface and volume integrals	OK once you understand it's just a thick spherical shell
2005	II	3	Matrices	OK
2005	II	4	Lagrange	OK
2005	II	5	Complex numbers	OK*
2005	II	6	Vectors: geometric	OK
2005	II	7	Differentials: thermodynamics	OK
2005	II	8	Differentials: exact	OK (not easy but doable)
2005	II	9	Matrices: eigenvalues/vectors	OK
2005	II	10	Integration, Series	OK (not easy but doable)
2005	II	11	Integration	OK*
2005	II	12	PDEs: diffusion equation	OK (seems very quick)
2006	I	1	Matrices	Part (b) is very hard in that the obvious approach doesn't work
2006	I	2	Matrices: eigenvalues/vectors	OK though a bit long
2006	I	3	Complex numbers	(a) and (b) are OK but rather tedious; (c) is hard until you spot the method
2006	I	4	Stoke's Theorem	OK
2006	I	5	Vectors: geometric	OK
2006	I	6	Vectors: algebraic, geometric	OK but dull
2006	I	7	Probability	OK (fairly quick)
2006	I	8	Lagrange	OK (very quick if you understand (b))
2006	I	9	Taylor Series	Tedious
2006	I	10	ODEs: 2nd order	Not without merit, but rather lost in the tedium
2006	I	11	Fourier	OK*
2006	I	12	PDEs: Laplace	OK (mischievous but interesting)
2006	II	1	Misc: Differentiation, Taylor, Integration	OK (not easy)
2006	II	2	Matrices: suffix notation	Pretty hard, as well as scary
2006	II	3	Line integrals, conservative fields	OK*
2006	II	4	Vector surface integrals	Strange and a bit confusing
2006	II	5	Volume integrals	Strange: easy but confusing - best avoided
2006	II	6	Misc: integration, approximations	OK
2006	II	7	Differentials: exact	OK* (Quick if you can spot the shortcuts)
2006	II	8	Probability	OK (quick)
2006	II	9	ODEs	(b) is very hard until you see it and (d) is a plausible solution for (b), so not terribly satisfactory overall
2006	II	10	Series	Quite hard
2006	II	11	PDEs: transformations	A standard method, but very tedious here
2006	II	12	Stationary values	Very tedious
2007	I	B1	Vectors: algebraic	OK
2007	I	B2	Complex numbers	OK*
2007	I	B3	Taylor Series	(a) OK* (b) Tedious
2007	I	B4	Probability	OK*
2007	I	B5	Exact differentials, ODEs: 1st order	OK
2007	I	B6	Stationary values	OK*
2007	I	B7	Multiple integrals	OK*
2007	I	B8	Matrices	OK (indicative marks for (a) subparts unindicative!)
2007	I	B9	Series and integration	OK (last parts of (a) are quite hard)
2007	I	B10	Differentiation of integrals	Horrid
2007	II	B1	Misc: Cartesian and polar coordinates	OK (unnervingly unusual)
2007	II	B2	Inegration	OK* - though quick
2007	II	B3	Probability	OK if you know enough stats
2007	II	B4	ODEs: 2nd order	(a) too simple (b) sketch seems hard for just 4 marks
2007	II	B5	Differentials: thermodynamics	OK - straightforward
2007	II	B6	Vector surface and line integrals	(b) needs one to assume the coordinates are spherical polars and is rather tedious
2007	II	B7	Matrices	OK
2007	II	B8	Fourier	OK* though time consuming
2007	II	B9	Lagrange	OK - rather quick though diagram is fiddly
2007	II	B10	PDEs	OK - straightforward
2008	I	B11	Vectors: geometric	OK but confusing
2008	I	B12	Complex numbers	Quite hard; only 2 marks for (b)iii!
2008	I	B13	Taylor Series	OK but (c) is tedious
2008	I	B14	Multiple integrals	To 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...
2008	I	B15	Probability	I gave up on (b)
2008	I	B16	ODEs: 1st order	Long and difficult
2008	I	B17	Stationary values; grad	OK
2008	I	B18	Matrices	Straightforward if dull
2008	I	B19	Limits and series	OK
2008	I	B20	Leibnitz; Schwartz	Bookwork or hard
2008	II	B10	Vectors: geometric, algebraic	OK but long and not easy
2008	II	B11	Probability	OK but be very careful how you interpret part (a)
2008	II	B12	Integration	I can now do this, but it is very difficult
2008	II	B13	ODEs: 2nd order	OK but dull
2008	II	B14	PDEs	OK
2008	II	B15	Line integrals, conservative fields	OK - bit fiddly
2008	II	B16	Matrices	OK but long and a bit tedious
2008	II	B17	Fourier	OK
2008	II	B18	Vector surface integrals	OK (looks scary but doable)
2008	II	B19	PDEs	(a) long but OK (b) OK if you realise they are using Sigma to denote a function of x, with Sigma0 being a constant
2009	I	B11	Misc: differentiation, mainly	Seems tedious and error-prone
2009	I	B12	Probability	Fairly easy
2009	I	B13	Matrices: eigenvalues/vectors	Tedious and you need to know about diagonalisation
2009	I	B14	Complex numbers	Fairly quick
2009	I	B15	Vectors: geometric, algebraic	(a)(i) unclear, rest OK
2009	I	B16	Integration	OK*
2009	I	B17	ODEs: 1st order	(a)(ii) very hard unless you know how; rest OK
2009	I	B18	Partial differentiation	Very tedious - I gave up on (c)
2009	I	B19	PDEs	OK
2009	I	B20	Series; limits	OK* ((b)(iii) is hard)
2009	II	B11	Misc: geometry	OK
2009	II	B12	ODEs: 2nd order	Tedious and error-prone
2009	II	B13	Stationary values	(a) is very tedious and error-prone; (b) is interesting but needs (a)
2009	II	B14	Matrices: equations, determinants	OK*
2009	II	B15	Fourier	Rather tedious unless there's a trick for (b)
2009	II	B16	Div/Grad/Curl	Rather tedious unless there are shortcuts
2009	II	B17	Probability/statistics	OK
2009	II	B18	Multiple integrals	OK
2009	II	B19	Vector surface integrals	OK
2009	II	B20	Misc: Leibnitz, Binomial expansion	OK but badly structured ((a) presumably still holds for (b), and (b) for (c))
2010	I	B11	Taylor Series	OK; relatively quick
2010	I	B12	Probability	Hard
2010	I	B13	Matrices: eigenvalues/vectors	Utterly tedious
2010	I	B14	Complex numbers	OK but repetitious
2010	I	B15	Vectors: algebraic, geometric	Hard
2010	I	B16	Integration	OK though (d) needs a Jacobian and (c) is quite hard
2010	I	B17	ODEs: 1st, 2nd order	OK; bit fiddly
2010	I	B18	Partial differentiation	OK; not easy
2010	I	B19	Misc: partial differentiation, PDEs	OK though quite quick; (c) is interesting
2010	I	B20	Limits and series	OK; interesting though quick if you know how
2010	II	B11	Misc: geometry	OK
2010	II	B12	Misc: vectors, ODEs	OK
2010	II	B13	Stationary values	OK except the sketch is unpleasant
2010	II	B14	Matrices	OK; interesting though quick if you know how
2010	II	B15	Fourier	OK
2010	II	B16	Line integrals, conservative fields	OK
2010	II	B17	Probability	OK; quite quick; (c) is interesting
2010	II	B18	Multiple integrals	OK
2010	II	B19	Lagrange	OK
2010	II	B20	Misc: Leibnitz	OK
2011	I	B11	Taylor Series	OK; very quick
2011	I	B12	Probability	Too long
2011	I	B13	Matrices: eigenvalues/vectors	OK but fiddly
2011	I	B14	Fourier, ODEs	OK, though wording in (e) could be clearer
2011	I	B15	Vectors: algebraic, geometric	OK; fairly quick
2011	I	B16	ODEs: 1st, 2nd order	OK
2011	I	B17	Complex numbers	OK; very quick
2011	I	B18	Differentials: exact	OK
2011	I	B19	Vector surface integrals	OK* though it rather overdoes one idea
2011	I	B20	Functions, continuity, differentiability	(a) is horrendous without a computer; (b) is interesting
2011	II	B11	Line integrals, vector surface integrals	Technically doable but crazy
2011	II	B12	Spherical trigonometry	(b) is fiddly and tiresome; (c) the l^2+m^2+n^2=1 bit is best ignored, I think; (d) very quick
2011	II	B13	Misc: hyperbolic functions, grad, stationary values	OK; quite quick
2011	II	B14	Matrices	OK, except (b) is equivalent to 2010 II 14(b)
2011	II	B15	ODEs: second order	Tedious algebra though physically interesting
2011	II	B16	Integration	OK*
2011	II	B17	Probability	OK; quick
2011	II	B18	Multiple integrals	OK; quick
2011	II	B19	PDEs	Hard but interesting
2011	II	B20	Misc: Leibnitz, ODEs, orthogonality	Hard

Ian Rudy (

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