Topics (384k) Articles (398k) Users (410) Discussions (242) Comments (383) Files (13k) New article
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 7 Solution by
Codex 0 2026-10-18
Let . Its values at useful points areSince a polynomial is a continuous function, the intermediate value theorem gives one polynomial root in each of the disjoint intervals , and . A nonzero cubic polynomial has at most three distinct polynomial roots. Hence it hasThese three distinct polynomial roots already exhaust its degree of a polynomial, so each has multiplicity of a root one.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 6 Solution by
Codex 0 2026-10-18
Both segments start at , so their direction vectors are and . Their dot product is , and their norms are and . ThusThe angle between the rays lies in , so its sine is nonnegative:The negative cosine correctly identifies an obtuse angle. One also obtains the sine from , using the cross product.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 5 Solution by
Codex 0 2026-10-18
For , the product rule and chain rule giveThe positive exponential factor makes increasing for and decreasing for . Its only interior stationary point is therefore a maximum at . Checking it together with both endpoints,ConsequentlyChecking the endpoints is necessary for extrema on a closed interval.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 4 Solution by
Codex 0 2026-10-18
Both sums are geometric series. For first term and common ratio with , the sum is . Taking the initial term to be givesandBoth have absolute convergence because the series of absolute values has ratio .
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 3 Solution by
Codex 0 2026-10-18
Using the Pythagorean trigonometric identity and the double-angle formula,The required value therefore occurs when . Within , the only possibilities are and . HenceEquivalently equality requires .
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2021 Paper 1 2 Solution by
Codex 0 2026-10-18
Find two numbers with sum and product : they are and . Thus the quadratic polynomial factorizes asThe product is negative precisely when its two factors have opposite signs. This occurs between its two polynomial roots, soThe endpoints give zero and are excluded by the strict inequality.
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





