Okay to be clear, this is a practice paper that follows the Pearson Edexcel Syllabus, and was designed for first assessment Summer 2018.
As someone who has taken the British style A-levels, I can attest that these questions are more difficult than what students will normally face in the exams.
I have chosen 3 questions today to go through with you. We have one on geometry, one of proof and one on calculus.
Hopefully that should be a broad enough range to scratch your maths itch. If you want more, please let me know as I can continue this series of looking at maths exam questions from around the world.
Now as usual, I recommend you pause the article and give each of these questions a go. And when you are ready, keep reading for the solution.
Question 1: Geometry
For this geometry question, the key is to label the radius of the circle. Then as we can see, we can construct a right-angled triangle MOB. In turn, we use the Pythagoras theorem to find the value of the radius.
I really like these sort of questions because they really feel like a ‘puzzle’. You know, piecing the whole picture together bit by bit.
Here’s another way to get to the answer.
Instead of using the Pythagoras theorem, we construct two similar triangles NMB and BME. Look at the diagram and mentally rotate the green triangle by 90 degrees and put it next to the yellow triangle, and you will see that they are similar.
Once we find the length of MN, which is 1/2. Then we can work out the radius.
Isn’t that so amazing?
Question 2: Proof
Back in school, I remember learning how to prove by contradiction that the square root of 2 is irrational. I must say trying to prove log10(5) is irrational will require much more ingenuity and thinking.
Let’s take a look.
Here the proof begins by assuming that log10(5) is rational. This means it can be written as a/b, which eventually gives 5b=10a. But 5^b is always odd, while 10^a is always even, creating a contradiction. Therefore, log10(5) must be irrational.
Remember kiddos, irrational here doesn’t mean unreasonable. It means the number cannot be expressed as a ratio of two integers.
Question 3: Calculus
For the final question we look at in this post, we are given equation of the curve y = ln(1 + cosx).
Our job is to manipulate the equation so that it reaches the desired form, which looks like a differential equation to me.
So we differentiate the given equation twice with respect to x. That way, we get dy/dx and the second derivative.
After that we exponentiate the original equation to get rid of the natural log.
And that’s our answer. How amazing 🙂
But wait, don’t go just yet. The bonus challenge for you is to solve this differential equation.
Let me know your answer in the comments below, or better yet submit to our Math Games publication on Medium 🙂
