Hard Questions for the New AS Maths!

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I have to apologise for something, I was so tired yesterday I didn't spot a much cleaner way of tackling my own extension question 2 that I should have told you about.

Just had a read of some AS spec things. Very interesting it seems they've added more proof stuff. In fact the AM-GM inequality proof for two terms was a two marker in some sample material provided by Edexcel.

Exam question (?)

Prove that $n^3+6n^2+8n$ is divisible by 3 for any positive integer $n$.

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This is harder than an AS exam question.

The circle C has equation $x^2+6x+y^2-2y-7=0$. The lines $l_1$ and $l_2$ are tangents to the circle and intersect at the point $(0,a)$ where $a>0$.

Prove that $a>1+2\sqrt{2}$.

This is harder than an AS exam question.

The circle C has equation $x^2+6x+y^2-2y-7=0$. The lines $l_1$ and $l_2$ are tangents to the circle and intersect at the point $(0,a)$ where $a>0$.

Prove that $a>1+2\sqrt{2}$.

Oh yeah, forgot to ask actually but is this really harder than an AS exam question? Unless I'm seriously missing something here.

My solution would be harder than an AS question but I haven't thought about other methods. What did your solution involve?

Um if you're not Notnek don't be cheeky and read this.

Not sure if normal distribution is still in AS level stats or not, here you go nevertheless:

Joe is buying two tires from two separate car manufacturers, iCar and FireWheelz.
iCar sells tires which have mean diameter of 30mm and variance of 2mm
FireWheelz sells tires which have mean diameter of 33mm and variance of 4mm

It can be assumed that the diameter of both tires follow a normal distribution.

(a) State the probability that one of the tires that Joe buys has diameter of 32mm
(b) Determine the probability that the tire from iCar has its diameter between 31mm and 32mm, given that the diameter of the other tire is between 0mm and 0.0001mm
(c) Determine the probability that the tire from FireWheelz is of diameter less than its mean, while the tire from iCar is of greater diameter than its mean
(d) Determine the probability that the tire from iCar has diameter between 31mm and 32mm while the tire from FireWheelz has diameter between 35mm and 40mm

This is (arguably) harder than an AS exam question.

The circle C has equation $x^2+6x+y^2-2y-7=0$. The lines $l_1$ and $l_2$ are tangents to the circle and intersect at the point $(0,a)$ where $a>0$.

Prove that $a>1+2\sqrt{2}$.