sorry can someone help me with these questions please. with the volume question I'm confused because they did not specify which axis its rotated by so how am I supposed to know.
thank you
Actually that is a fair call with the volumes question. That question is in theory not doable because they haven't specified the line about which the rotation is taken with respect to.
As this is a reasonably huge dump of strenuous questions I (more or less because I'll get weary
) won't go into complete depth with them. However it is worth mentioning that for the 3 variable AM-GM inequality you've mentioned, in the current HSC there would be hints that help you build up to that result, as opposed to being shoved straight into the deep end.
This is one way of proving the AM-GM inequality for 3 variables given that you've already proven it for 4 variables, and to prove it for 4 variables you can work through my trial lecture handout solutions.
The angular velocity \( \omega\) is defined by \( \omega = \frac{\d\theta}{dt} \), i.e. the rate at which the angle the particle makes, at the origin in the positive \(x\)-axis, changes with respect to time. That derivation was examined in the 1981 paper (and from memory it should've been stated in my 4U notes book), but it essentially relies on you starting with \( x = r\cos \theta\), \(y = r\sin \theta\) and then using implicit differentiation to obtain results like \( \dot{x} = \frac{d\theta}{dt} \times -r\sin \theta = -r\omega \sin \theta\), etc.
That locus essentially describes the arc, with endpoints at \(z_1\) and \(z_2\), going in an anticlockwise direction from \(z_1\) to \(z_2\) and not including the points \(z_1\) and \(z_2\) themselves.