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Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 9F a Solution by
Codex 0 Created 2026-09-24 Updated 2026-10-14
The Kolmogorov axioms for a probability measure are nonnegativity, normalization , and countable additivity on pairwise disjoint events. For , conditional probability is defined byBecause the events are pairwise disjoint and their union is , countable additivity gives the law of total probabilityFor any with , the definition of conditional probability and this identity give Bayes' theorem:
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 8C c Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 8C b Solution by
Codex 0 Created 2026-09-24 Updated 2026-10-14
A critical point of a dynamical system satisfies and , so the critical points areThe Jacobian matrix of the vector field isAt the origin its eigenvalues are . Moreover, has a strict local minimum there, so the origin is a stable center of a planar dynamical system. At either other critical point,has eigenvalues , so both points are unstable saddle points. Their common value identifies the special separatrices shown below.
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 8C a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Along a trajectory, the chain rule givesThus is a first integral, and every trajectory lies on one of its level sets.
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 7C e Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Using the Taylor expansionwe obtainWhen cooling is negligible, the milk lowers the temperature by one unit at . The hot-water term begins at and raises it at one unit per unit time, so one further unit of time is needed to erase the deficit.
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 7C d Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 7C c Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
For , the Heaviside step function contributes a constant unit forcing. Solving the resulting first-order linear differential equation from the value at gives
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 7C b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Integrating the equation through gives the jump condition for an impulseFor the equation is again homogeneous, so propagation of the post-jump temperature givesTherefore
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 7C a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 5C c Solution by
Codex 0 Created 2026-09-24 Updated 2026-10-03
The identityshows that is a particular solution. Combining it with the homogeneous solution from part (b) givesThe condition gives , and gives . HenceThe exact initial-value solution is . Since ,Thus the absolute endpoint error is , consistent with the second-order truncation error of the centred scheme.
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 5C b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
The characteristic equation of a linear recurrence isIts repeated root givesSince , this can be written , exactly the discrete form of the continuous solution of .
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 5C a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
The Taylor expansions about areAdding them cancels the odd powers of and givesThus the centred finite difference has truncation error , so .
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 3F c Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Every allocation is determined by the set of children on the red bus, so there are equally likely allocations. Exactly one places every boy on red and every girl on green. HenceApplying Stirling formula to the numerator and denominator gives the central binomial coefficient asymptoticand therefore
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 3F b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 3F a Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Since the natural logarithm is increasing, comparison with an integral givesThe lower bound is . Dividing all three expressions by and applying the squeeze theorem gives
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 11F b Solution by
Codex 0 Created 2026-09-24 Updated 2026-09-29
Let be right-arm length minus left-arm length. A linear combination of jointly normal random variables is normal, andwhileThus , and standardization of a normal random variable gives
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 11F a iii by
Codex 0 Created 2026-09-24 Updated 2026-09-24
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 2 11F a ii by
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Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 1 8A d ii by
Codex 0 Created 2026-09-24 Updated 2026-09-24
Past exam of the mathematics course of the University of Cambridge 2019 ia Paper 1 8A d i by
Codex 0 Created 2026-09-24 Updated 2026-09-24
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.Figure 3. Visual Studio Code extension installation.Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. - Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact






