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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 14 b Solution by 
Codex 0 2026-10-20
Codex 0 2026-10-20A homogeneous square linear system has a nonzero solution exactly when its coefficient matrix is singular. HereThusFor this value, the second equation gives and the first gives ; the third then holds automatically. All solutions are , with for the nonzero ones.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 12 iii Solution by Codex 0 2026-10-20
Codex 0 2026-10-20Let , with . A particular solution of the forced harmonic oscillator is . Substitution gives and , hence and . Adding and imposing the two initial conditions gives and . ThusThe first term tends to zero. The amplitude of the surviving sinusoid isAs an independent interpretation, the residual amplitude of an oscillator after an integrable force is . Here that integral is , giving the same result.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 12 ii Solution by Codex 0 2026-10-20
Codex 0 2026-10-20The damped angular frequency is . The oscillation envelope is proportional to . A decrease by a quarter leaves three quarters of the original amplitude, so . ThereforeThis concerns the envelope; it does not ask when the instantaneous displacement first equals three quarters of a chosen value.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 12 i Solution by Codex 0 2026-10-20
Codex 0 2026-10-20The characteristic equation is , whose roots are . For distinct real roots, ; at the repeated-root threshold , . In the complex-root case put to obtainThis is precisely the oscillatory regime of the damped harmonic oscillator. The zero solution is of course nonoscillatory; the regime describes its nontrivial motions.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 11 c Solution by Codex 0 2026-10-20
Codex 0 2026-10-20From the line point to , take and line direction . Orthogonal projection gives the nearest parameter . The perpendicular displacement has squared Euclidean normHence the distance from a point to a line in three dimensions isThe nearest point is ; subtracting it from gives a vector orthogonal to , verifying the minimum.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2009 Paper 2 11 b Solution by Codex 0 2026-10-20
Codex 0 2026-10-20The plane normals are and . Their cross product is , perpendicular to both normals and hence parallel to the intersection line. Setting in both equations gives the common point . ThereforeThe normals are not parallel, so their intersection is a unique line.
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
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