This document last updated: 23 March 2012.
These pages provide comments on past exam papers for the Economics Tripos Part IIA Paper 6 (Mathematics for Economists and Statisticians). 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 2011/12: 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 (vector spaces) | OK |
| 1999 | A | 3 | Linear Algebra (portfolios) | 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) | Theoretical proofs - I skipped it |
| 1999 | B | 3 | Statistics (estimators) | OK but rather long and not easy |
| 1999 | B | 4 | Statistics (MGFs) | OK |
| 1999 | B | 5 | Calculus (optimisation) | |
| 1999 | B | 6 | Calculus (difference eqns) | |
| 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 but quick and easy |
| 2000 | A | 5 | Statistics (estimators) | OK |
| 2000 | A | 6 | Statistics (hypothesis testing) | |
| 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) | |
| 2000 | B | 3 | Statistics (estimators) | (a) and (b) OK; (c) is confusing |
| 2000 | B | 4 | Statistics (hypothesis testing) | |
| 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) | |
| 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) | Largely OK though not entirely clear |
| 2001 | B | 5 | Statistics (moments) | OK though quite quick and easy |
| 2001 | B | 6 | Statistics (hypothesis testing) | |
| 2002 | A | 1 | Linear Algebra (definiteness) | OK* |
| 2002 | A | 2 | Linear Algebra (Misc) | OK |
| 2002 | A | 3 | Linear Algebra (Misc) | Ridiculously easy |
| 2002 | A | 4 | Statistics (probabilities) | Theoretical proofs - I skipped it |
| 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 though I forgot to do (c) |
| 2002 | B | 5 | Calculus (Misc) | OK |
| 2002 | B | 6 | Calculus (optimisation) | Mostly OK but some aspects confusing |
| 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) | 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 when you know but best avoided |
| 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) | Yuk |
| 2004 | B | 5 | Statistics (estimators) | OK* |
| 2004 | B | 6 | Statistics (estimators) | OK (I think (d) is Bernoulli) |
| 2005 | A | 1 | Linear Algebra (Vector spaces) | OK |
| 2005 | A | 2 | Linear Algebra (Vector spaces) | Question wrong in (a) |
| 2005 | A | 3 | Linear Algebra (Misc) | Hard |
| 2005 | A | 4 | Calculus (homogeneous functions) | Design of question confusing |
| 2005 | A | 5 | Calculus (hessians, optimisation) | Question wrong in (b) |
| 2005 | A | 6 | Calculus (differential equations) | (c) is absurdly tedious |
| 2005 | A | 7 | Statistics (expectations) | Question confusing |
| 2005 | A | 8 | Statistics (expectations, independence) | Not sure about (c) |
| 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) | OK if we can be excused the tedium of the MVT |
| 2006 | A | 6 | Calculus (coupled differential equations) | OK |
| 2006 | A | 7 | Linear Algebra (Markov) | OK |
| 2006 | A | 8 | Linear Algebra (misc) | Bit simple and (c) is unclear |
| 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) | Tiresomely theoretical |
| 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 |
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