This document last updated: 14 August 2015. Please note that this document is not being updated, because the changes that the Economics Faculty are proposing to make to Paper 6 for 2015-16 mean that questions from 2015 and before are unlikely to be relevant. You can find my page of comments on questions since the new course began here.
These pages provide comments on past exam papers for the Economics Tripos Part IIA Paper 6 (Mathematics and Statistics for Economists). Where there is no comment for a question, it probably means I've not yet had a go at it yet. A star (*) means the question is good in some way - interesting or particularly useful for practice.
General note for 2014/15: the syllabus has changed slightly over the years, so you may not have covered all the topics needed to answer all questions from past papers. In particular, you have not appear to have been taught:
This may hinder you slightly on some past exam questions, including:
| Year | Sec | Qu | Topic | Comments |
| 1999 | A | 1 | Linear Algebra (diagonalisation) | OK but quick |
| 1999 | A | 2 | Linear Algebra (securities, vector spaces) | OK |
| 1999 | A | 3 | Linear Algebra (difference eqns) | OK but relatively long |
| 1999 | A | 4 | Statistics (distribution functions) | OK, except you can't have a "density" in a discrete situation |
| 1999 | A | 5 | Statistics (estimators) | OK, though done as an examples sheet question |
| 1999 | A | 6 | Statistics (Bayes) | OK but quick |
| 1999 | A | 7 | Calculus (optimisation) | OK |
| 1999 | A | 8 | Calculus (equilibrium points & stability) | OK |
| 1999 | A | 9 | Calculus (Taylor Series) | OK but rather dull |
| 1999 | B | 1 | Linear Algebra (linear programming) | No longer on syllabus |
| 1999 | B | 2 | Linear Algebra (vector spaces) | OK, though unclear how much detail is needed in (a) |
| 1999 | B | 3 | Statistics (estimators) | OK but rather long and not easy |
| 1999 | B | 4 | Statistics (MGFs) | OK |
| 1999 | B | 5 | Calculus (optimisation) | OK |
| 1999 | B | 6 | Calculus (difference eqns) | OK |
| 2000 | A | 1 | Linear Algebra (vector spaces) | OK |
| 2000 | A | 2 | Linear Algebra (vector spaces) | OK except I think "orthogonal components" no longer in syllabus |
| 2000 | A | 3 | Linear Algebra (linear programming) | No longer on syllabus |
| 2000 | A | 4 | Statistics (MGFs) | OK (useful) |
| 2000 | A | 5 | Statistics (estimators) | OK |
| 2000 | A | 6 | Statistics (hypothesis testing) | (a) and (b) OK; (c) no longer in syllabus |
| 2000 | A | 7 | Calculus (differential eqns) | OK but quick |
| 2000 | A | 8 | Calculus (homogeneous functions) | OK but easy and dull |
| 2000 | A | 9 | Calculus (optimisation) | OK |
| 2000 | B | 1 | Linear Algebra (definiteness) | OK |
| 2000 | B | 2 | Linear Algebra (input/output models) | OK (not easy but interesting) |
| 2000 | B | 3 | Statistics (estimators) | (a) and (b) OK; (c) is confusing |
| 2000 | B | 4 | Statistics (hypothesis testing) | OK, except (a) and (d) no longer in syllabus |
| 2000 | B | 5 | Calculus (equilibrium points & stability) | OK though (d) is unclear |
| 2000 | B | 6 | Calculus (optimisation) | Long and unclear |
| 2001 | A | 1 | Linear Algebra (vector spaces) | OK |
| 2001 | A | 2 | Linear Algebra (eigenvalues/vectors) | (a) and (b) OK; (c) is unclear |
| 2001 | A | 3 | Linear Algebra (definiteness) | OK if you ignore their statement that one of the eigenvalues is 2 |
| 2001 | A | 4 | Calculus (stationary values) | OK |
| 2001 | A | 5 | Calculus (difference eqns) | OK |
| 2001 | A | 6 | Calculus (equilibrium points & stability) | OK |
| 2001 | A | 7 | Statistics (MGFs) | OK* |
| 2001 | A | 8 | Statistics (estimators) | OK |
| 2001 | A | 9 | Statistics (estimators) | OK (bit odd in (a)) |
| 2001 | B | 1 | Linear Algebra (input/output models, vector spaces) | Unclear |
| 2001 | B | 2 | Linear Algebra (difference eqns) | OK |
| 2001 | B | 3 | Calculus (homogeneous functions) | OK but hard |
| 2001 | B | 4 | Calculus (homogeneous functions) | Algebra tiresome on (a)(iii) and (b) is unclear |
| 2001 | B | 5 | Statistics (moments) | Useful though quite quick and easy for Section B |
| 2001 | B | 6 | Statistics (hypothesis testing) | OK, except (a) no longer in syllabus |
| 2002 | A | 1 | Linear Algebra (definiteness) | OK* |
| 2002 | A | 2 | Linear Algebra (Misc) | OK |
| 2002 | A | 3 | Linear Algebra (Misc) | Some find this non-obvious |
| 2002 | A | 4 | Statistics (probabilities) | OK if one assumes one has to prove (iii) from the others! |
| 2002 | A | 5 | Statistics (Chebyshev) | OK though the last bit is tiresome in the detail, and I think the question should have saved you from even considering what happens if k<-1 |
| 2002 | A | 6 | Statistics (probabilities) | OK |
| 2002 | A | 7 | Calculus (differential/difference eqns) | OK |
| 2002 | A | 8 | Calculus (stationary values) | OK |
| 2002 | A | 9 | Calculus (equilibrium points & stability) | OK |
| 2002 | B | 1 | Linear Algebra (input/output models) | OK |
| 2002 | B | 2 | Linear Algebra (input/output models/difference eqns) | OK |
| 2002 | B | 3 | Statistics (estimators) | OK |
| 2002 | B | 4 | Statistics (estimators) | OK |
| 2002 | B | 5 | Calculus (Misc) | OK |
| 2002 | B | 6 | Calculus (optimisation) | Mostly OK but confusing as to what they want in places |
| 2003 | A | 1 | Linear Algebra (simultaneous eqns and vector spaces) | OK |
| 2003 | A | 2 | Linear Algebra (Misc) | OK |
| 2003 | A | 3 | Linear Algebra (Misc) | OK |
| 2003 | A | 4 | Calculus (stationary values) | Dull and (a) is not entirely clear |
| 2003 | A | 5 | Calculus (differential eqns) | OK |
| 2003 | A | 6 | Calculus (homogeneous functions/Lagrange) | OK |
| 2003 | A | 7 | Statistics (discrete probability distributions) | OK |
| 2003 | A | 8 | Statistics (estimators/hypothesis testing) | OK but I don't see the point |
| 2003 | A | 9 | Statistics (estimators) | Unclear |
| 2003 | B | 1 | Linear Algebra (difference eqns) | OK |
| 2003 | B | 2 | Linear Algebra (Misc) | OK (not easy) |
| 2003 | B | 3 | Calculus (equilibrium points & stability) | OK (nowhere near as bad as it looks at first) |
| 2003 | B | 4 | Calculus (Euler/stationary values) | (a) and (b) OK; (c) seems to me to have an error |
| 2003 | B | 5 | Statistics (estimation) | OK |
| 2003 | B | 6 | Statistics (joint distributions) | OK |
| 2004 | A | 1 | Linear Algebra (ISLM and Cramer's Rule) | OK |
| 2004 | A | 2 | Linear Algebra (determinants and inverses) | OK (easy) |
| 2004 | A | 3 | Linear Algebra (Vector spaces) | OK* |
| 2004 | A | 4 | Calculus (limits) | OK (easy & quick) |
| 2004 | A | 5 | Calculus (partial differentiation, Taylor Series) | OK though wording is confusing |
| 2004 | A | 6 | Calculus (coupled differential equations) | OK (though long via the matrix method) |
| 2004 | A | 7 | Statistics (probabilities) | OK (not easy) |
| 2004 | A | 8 | Statistics (probability distributions, Chebyshev) | OK |
| 2004 | A | 9 | Statistics (Bayes with distributions) | OK (though may not be your cup of tea) |
| 2004 | B | 1 | Linear Algebra (vector spaces) | OK but not easy |
| 2004 | B | 2 | Linear Algebra (Markov) | OK (relatively easy) |
| 2004 | B | 3 | Calculus (stationary values) | OK |
| 2004 | B | 4 | Calculus (concavity, homogeneity) | Unreasonably hard |
| 2004 | B | 5 | Statistics (estimators) | OK* |
| 2004 | B | 6 | Statistics (estimators) | OK though (d) should say Bernoulli, and is not easy; quite long overall |
| 2005 | A | 1 | Linear Algebra (Vector spaces) | OK |
| 2005 | A | 2 | Linear Algebra (Vector spaces) | OK other than lack of definition for x2 |
| 2005 | A | 3 | Linear Algebra (Misc) | OK (not easy) |
| 2005 | A | 4 | Calculus (homogeneous functions) | Design of question confusing |
| 2005 | A | 5 | Calculus (hessians, optimisation) | OK* |
| 2005 | A | 6 | Calculus (differential equations) | (c) is absurdly tedious |
| 2005 | A | 7 | Statistics (expectations) | Question confusing |
| 2005 | A | 8 | Statistics (expectations, independence) | OK |
| 2005 | A | 9 | Statistics (probabilities, Bayes) | OK (relatively easy) |
| 2005 | B | 1 | Linear Algebra (misc, eigenvalues/vectors) | (a) is bookwork but hard if do not know, rest is easy |
| 2005 | B | 2 | Linear Algebra (vector spaces) | (a) is hard, (b) is interesting, (c) is bookwork |
| 2005 | B | 3 | Calculus (distributions, l'Hopital) | OK* ((c) is hard) |
| 2005 | B | 4 | Calculus (optimisation, Envelope Theorem) | OK, though (b) is bookwork |
| 2005 | B | 5 | Statistics (estimators) | Seems easy but not clear what they want |
| 2005 | B | 6 | Statistics (moment generating functions, hypothesis testing) | Seems easy but not clear what they want |
| 2006 | A | 1 | Statistics (estimators) | OK |
| 2006 | A | 2 | Statistics (estimators) | OK |
| 2006 | A | 3 | Statistics (estimators) | (a) Requires a principle not taught (b) is badly-worded and identical to a question in the previous year |
| 2006 | A | 4 | Calculus (misc) | OK* |
| 2006 | A | 5 | Calculus (Mean Value Theorem) | Great idea for a question, messed up by MVT |
| 2006 | A | 6 | Calculus (coupled differential equations) | OK |
| 2006 | A | 7 | Linear Algebra (Markov) | OK |
| 2006 | A | 8 | Linear Algebra (misc) | OK (bit simple) |
| 2006 | A | 9 | Linear Algebra (portfolios) | OK |
| 2006 | B | 10 | Statistics (estimators) | For b read beta; OK but identical to an examples sheet question |
| 2006 | B | 11 | Statistics (Bayes) | OK except that (a) is ambiguous and (c) is ambiguous in whether you need to prove result for mean of Beta distribution |
| 2006 | B | 12 | Calculus (stationary values) | OK* though ignore (a) initially |
| 2006 | B | 13 | Calculus (homogeneous functions/Lagrange) | Identical to 2003 A6 :-( |
| 2006 | B | 14 | Linear Algebra (input/output models) | OK if you can be bothered |
| 2006 | B | 15 | Linear Algebra (vector spaces) | OK except (d) is not clear |
| 2007 | A | 1 | Linear Algebra (IS-LM) | OK except we don't have enough info on coefficients for (b) |
| 2007 | A | 2 | Linear Algebra (simultaneous eqns) | OK (quite interesting but a bit messy) |
| 2007 | A | 3 | Linear Algebra (eigenvalues and Markov) | (a) depends on your definition of eigenvalues/vectors (b) OK |
| 2007 | A | 4 | Calculus (misc) | Doable but tiresomely theoretical |
| 2007 | A | 5 | Calculus (misc) | OK but not easy |
| 2007 | A | 6 | Calculus (coupled and uncoupled differential equations) | OK* |
| 2007 | A | 7 | Statistics (probability density functions) | OK* |
| 2007 | A | 8 | Statistics (moment generating functions) | OK |
| 2007 | A | 9 | Statistics (estimators) | OK |
| 2007 | B | 10 | Linear Algebra (matrices) | OK |
| 2007 | B | 11 | Linear Algebra (vector spaces) | (a) regurgitate your lecture handout (b) OK* |
| 2007 | B | 12 | Calculus (Lagrange) | Pointlessly messy - I gave up |
| 2007 | B | 13 | Calculus (misc) | (b) is gory and tiresome |
| 2007 | B | 14 | Statistics (estimators) | Doable though a bit odd; I assume (b) means possible *unbiased* estimator |
| 2007 | B | 15 | Statistics (maximum likelihood estimators) | Doable but a bit odd; I assume alpha0 is still zero in (d) |
| 2008 | A | 1 | Linear Algebra (determinants) | OK (quick) |
| 2008 | A | 2 | Linear Algebra (linear dependency) | OK ((b) not easy) |
| 2008 | A | 3 | Linear Algebra (in/definite matrices) | OK |
| 2008 | A | 4 | Calculus (continuity/differentiability) | (c) is interesting; rest OK but dull |
| 2008 | A | 5 | Calculus (misc) | Avoid (unusual, ambiguous and confusing) |
| 2008 | A | 6 | Calculus (misc) | Avoid (unusual, confusing in the sheer detail, and seems incomplete) |
| 2008 | A | 7 | Statistics (probability set theory) | OK (quick) |
| 2008 | A | 8 | Statistics (probability distribution functions) | OK if you've seen it before, otherwise hard |
| 2008 | A | 9 | Statistics (hypothesis testing) | Similar to a supervision question, and not very clear |
| 2008 | B | 10 | Linear Algebra (vector spaces: securities and portfolios) | Challenging conceptually and horrendous in the detail |
| 2008 | B | 11 | Linear Algebra (coupled difference equations) | Infamous in that r1=2, not 0.2, quite long and error prone |
| 2008 | B | 12 | Calculus (misc) | Avoid (unusual, long, confusing, unclear) |
| 2008 | B | 13 | Calculus (optimisation) | OK-ish |
| 2008 | B | 14 | Statistics (prior/posterior probability distributions) | OK but very simple |
| 2008 | B | 15 | Statistics (maximum likelihood estimators) | OK, though I think their hint in (b) is invalid |
| 2009 | A | 1 | Linear Algebra (determinants) | OK (quick) |
| 2009 | A | 2 | Linear Algebra (simultaneous eqns) | OK (quick but useful) |
| 2009 | A | 3 | Linear Algebra (vector spaces) | OK (quick and easy) |
| 2009 | A | 4 | Statistics (expectations) | Seems to need strict convexity rather than just convexity? |
| 2009 | A | 5 | Statistics (moment generating functions) | OK (good) |
| 2009 | A | 6 | Statistics (Bayes) | Repeat of 2006 B11 |
| 2009 | A | 7 | Statistics (regression) | Hard! |
| 2009 | A | 8 | Calculus (functions) | OK |
| 2009 | A | 9 | Calculus (difference equations) | OK |
| 2009 | A | 10 | Calculus (differentiation) | Near-identical to a supervision question |
| 2009 | B | 11 | Linear Algebra (ISLM) | (a) is 1st year material, (b) is tiresome |
| 2009 | B | 12 | Linear Algebra (eigenvalues and eigenvectors) | Long and tedious but not difficult |
| 2009 | B | 13 | Statistics | Avoid (partly interesting but needs concepts not taught) |
| 2009 | B | 14 | Statistics (regression, maximum likelihood estimators) | Needs matrix notation and unclear in places what they want |
| 2009 | B | 15 | Calculus (optimisation) | Fairly easy |
| 2009 | B | 16 | Calculus (optimisation) | Confusing in places but OK if you can follow it |
| 2010 | A | 1 | Linear Algebra (vector spaces/linear dependency) | Interesting but (b) is quite time-consuming, and not easy overall |
| 2010 | A | 2 | Linear Algebra (simultaneous eqns) | OK* |
| 2010 | A | 3 | Linear Algebra (singular matrices) | OK, except (a) can be done in more than one way, hence potentially mucking up (b) |
| 2010 | A | 4 | Calculus (limits) | (a) useful (b) not easy if you've not seen it before |
| 2010 | A | 5 | Calculus (implicit function theorem) | OK (easy) |
| 2010 | A | 6 | Calculus (optimisation) | OK (easy if you understand the question) |
| 2010 | A | 7 | Statistics (regression) | Horrendous |
| 2010 | A | 8 | Statistics (probabilities) | Trivial |
| 2010 | A | 9 | Statistics (Chebyshev) | OK (quite quick) |
| 2010 | B | 10 | Linear Algebra (misc) | (a) OK though fiddly (b) quite hard (c) hard |
| 2010 | B | 11 | Linear Algebra (vector spaces/simultaneous eqns) | (a) confusing and quite hard (b) tedious |
| 2010 | B | 12 | Calculus (difference equations) | OK except (c) is a bit vague as to what they want |
| 2010 | B | 13 | Calculus (optimisation) | OK* |
| 2010 | B | 14 | Statistics (Bayes) | OK, in places a bit confusing or vague |
| 2010 | B | 15 | Statistics (maximum likelihood estimators) | Unfair to expect you to judge how to handle censored data, in my view |
| 2011 | A | 1 | Linear Algebra (traces and suffix notation) | OK (quick) |
| 2011 | A | 2 | Linear Algebra (vector spaces/linear dependency) | OK |
| 2011 | A | 3 | Linear Algebra (quadratic forms and definiteness) | OK (quick) |
| 2011 | A | 4 | Calculus (functions) | OK |
| 2011 | A | 5 | Calculus (limits) | OK* |
| 2011 | A | 6 | Calculus (differential equations) | OK (quick) |
| 2011 | A | 7 | Statistics (expectations, variances, Taylor Series) | OK once you know |
| 2011 | A | 8 | Statistics (Bayes) | OK |
| 2011 | A | 9 | Statistics (estimators) | OK |
| 2011 | B | 10 | Linear Algebra (simultaneous equations / linear mappings) | OK, though (a) is 2003A1, (b) looks scary but isn't so bad |
| 2011 | B | 11 | Linear Algebra (ISLM / convex functions / misc) | (b) is increasingly tough, rest OK |
| 2011 | B | 12 | Calculus (optimisation) | OK but too easy for Section B |
| 2011 | B | 13 | Calculus (Gini Index - unusual) | Hard if you've never encountered Gini Indexes |
| 2011 | B | 14 | Statistics (Chebyshev) | Bit confusing but OK |
| 2011 | B | 15 | Statistics (maximum likelihood estimators) | OK but too easy for Section B |
| 2012 | A | 1 | Linear Algebra (vector spaces) | OK |
| 2012 | A | 2 | Linear Algebra (eigenvalues and vectors) | Very quick but useful* |
| 2012 | A | 3 | Linear Algebra (orthogonal matrices, quadratic forms and definiteness) | Very easy |
| 2012 | A | 4 | Statistics (Bayes) | OK (quick) |
| 2012 | A | 5 | Statistics (Neyman-Pearson, UMP tests) | OK* though too long for Section A |
| 2012 | A | 6 | Statistics (maximum likelihood estimators) | OK* |
| 2012 | A | 7 | Calculus (functions, convexity) | OK* |
| 2012 | A | 8 | Calculus (functions, optimisation) | OK though hard if you aren't used to (b) |
| 2012 | A | 9 | Calculus (optimisation) | Hard going |
| 2012 | B | 10 | Linear Algebra (ISLM and Cramer's Rule) | OK but too easy for Section B |
| 2012 | B | 11 | Linear Algebra (matrices, determinants, induction) | OK* though too short for Section B |
| 2012 | B | 12 | Statistics (Chebyshev) | OK though too short for Section B |
| 2012 | B | 13 | Statistics (maximum likelihood estimators) | Long, confusing as to what they want in places |
| 2012 | B | 14 | Calculus (functions, Fundamental Theorem of Calculus) | OK though too short for Section B |
| 2012 | B | 15 | Calculus (Bidding functions, utility) | Odd but doable and quite short |
| 2013 | A | 1 | Linear Algebra (eigenvalues and vectors) | OK* |
| 2013 | A | 2 | Linear Algebra block matrices) | Pretty hard unless you spot it |
| 2013 | A | 3 | Linear Algebra (linear dependency) | OK (quick) |
| 2013 | A | 4 | Statistics (estimators) | OK* |
| 2013 | A | 5 | Statistics (matrix formulation of OLS) | Boring bookwork |
| 2013 | A | 6 | Statistics (moment generating functions, estimators) | Not ideal in that it needs Jensen's Inequality but unclear that one is allowed to assume it |
| 2013 | A | 7 | Calculus (functions, convexity, Taylor) | OK (very easy) |
| 2013 | A | 8 | Calculus (integration) | OK (very easy) |
| 2013 | A | 9 | Calculus (optimisation) | OK* |
| 2013 | B | 10 | Linear Algebra (input/output models) | OK, though a bit ambiguous |
| 2013 | B | 11 | Linear Algebra (determinant proofs) | Boring proof |
| 2013 | B | 12 | Statistics (maximum likelihood estimators) | Ridiculously long |
| 2013 | B | 13 | Statistics (Bayes) | OK |
| 2013 | B | 14 | Calculus (functions, differentiability) | OK* though (e) ambiguous |
| 2013 | B | 15 | Calculus (difference equations) | OK except needs complex numbers |
| 2014 | A | 1 | Linear Algebra (properties of matrices and determinants) | OK |
| 2014 | A | 2 | Linear Algebra (vector spaces) | OK (very easy) |
| 2014 | A | 3 | Linear Algebra (orthogonal matrices) | OK |
| 2014 | A | 4 | Statistics (moment generating functions) | OK |
| 2014 | A | 5 | Statistics (Bayes) | Not entirely clear what is wanted |
| 2014 | A | 6 | Statistics (Chebyshev) | OK |
| 2014 | A | 7 | Calculus (functions) | OK (not easy unless you spot the method) |
| 2014 | A | 8 | Calculus (quasiconcavity) | OK |
| 2014 | A | 9 | Calculus (optimisation) | Crazy: probably a mistake in question |
| 2014 | B | 10 | Linear Algebra (vector space bases) | OK except (c) is ambiguous as to "smallest" |
| 2014 | B | 11 | Linear Algebra (Markov) | OK unless you do (b) in fractions like I did |
| 2014 | B | 12 | Statistics (estimators) | Too long; (c) unclear on definition of Xi |
| 2014 | B | 13 | Statistics (maximum likelihood estimators) | Too long; (c) somewhat scary but OK |
| 2014 | B | 14 | Calculus (integration) | A bit odd, especially (c), where it is best to ignore the formula given |
| 2014 | B | 15 | Calculus (differential equations) | OK except for the phase plot |
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