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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 13 c Solution by
Codex 0 2026-10-18
A partial fraction decomposition givesFor the second term the substitution rule , gives a logarithm. ThusThe constant in the antiderivative may be chosen independently on the two intervals separated by .
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 13 b Solution by
Codex 0 2026-10-18
The factor is an even function and is an odd function, so their product is an odd function. Its definite integral over the symmetric interval vanishes:Indeed the substitution rule changes the integral over into the negative of the integral over .
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 13 a Solution by
Codex 0 2026-10-18
Apply integration by parts twice. Taking an antiderivative of to be givesand the second application gives . Therefore the required definite integral is
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 12 b Solution by
Codex 0 2026-10-18
Write . Direct differentiation, using the chain rule, givesThe stated parameter restrictions ensure that the logarithm is defined near , and that and . With , the Taylor series beginsIf , the constant term is nonzero, and the first two nonzero terms areThis covers both and ; the constant term is respectively positive or negative.
If , which is possible precisely with and , the constant term vanishes. The first two nonzero terms are thenNo further case is needed, since the linear and quadratic coefficients cannot vanish under the given restrictions. All remainders use Big O notation.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 10 b Solution by
Codex 0 2026-10-18
The favourable outcomes are and for . These are ten distinct ordered outcomes among the equally likely elements of the sample space, so the probability is
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2025 Paper 1 10 a Solution by
Codex 0 2026-10-18
The two throws are independent, giving equally likely ordered outcomes in the sample space. A total strictly below occurs only for , and . Hence the probability is
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2024 Paper 2 8 b Solution by
Codex 0 2026-10-18
The divergence of a curl is zero, so its squared absolute value isBoth identities follow by cancellation of equal mixed partial derivatives under the usual smoothness assumptions of vector calculus.
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:
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