This second session of the course is about using vectors and algebra to solve problems about lines, intersection points of lines and intersection points of lines with curves. We'll kick off with a quick review of finding gradients and equations of lines and then move on to some geometric problems involving straight lines. We'll then link our algebraic knowledge to investigate how and where lines cross curves, and introduce the idea of the discriminant of a quadratic. We'll use this to help find the equation of a tangent to a curve. Again, each sub-section below should take around 40-60 minutes to work through.
Some of the tasks below come from the excellent Underground Maths site.
2. Problem-solving with straight lines
The purpose of this section is two-fold: to gain some fluency in working with straight lines and their equations; and to begin gaining some experience of open-ended problem solving. Start with the problems below:
3. Review questions
Here are some review questions. You will need to recall (re-learn?) some of the properties of shapes such as parallelograms, kites, etc to help with solving them.
Here's a set of Underground Maths problems which discusses approaches to questions like number 7.
4. Intersecting lines and curves
In completing some of the questions in the first few sections, you will, I hope, have recalled that in order to find the intersection point of two lines you must solve a pair of simultaneous equations. The same is true of finding where a line intersects a curve. Watch the video below, and then complete these questions. Then have a go at the problem below:
In completing the questions, I hope you noticed that sometimes there were two solutions, sometimes just one (corresponding to a repeated linear factor) and sometimes there was no solution. How can we tell which of the three cases we have?
5. Finding tangents and turning points (without differentiation)
If you studied the IGCSE course you will have seen the technique of differentiation to find a gradient formula for a curve (don't worry if you haven't studied this - you'll meet it in detail next year!) Watch the videos below, about finding the equation of a tangent to a curve without relying on finding a gradient formula.