Yeah I would love to see something like this in the HSC this year. Without the hint to multiply by e^nt (this might be taking it too far though), I think something similar could be a really nice, challenging last question, much better than something like the geo question from last year.
It was basically just reverse quotient rule
Is this taught in the syllabus?
Yup what TheOnePheeph said.
Ahh, that's so simple I'm surprised I didn't notice it. Thanks for the explanation. Good to know that if I'm struggling on an integral with a power of 2 in the denominator I'll consider this.Yup what TheOnePheeph said.
Reverse quotient rule is a dodgy inspection method of getting the answer. I'll explain
So we have
Observe that the denominator is of the form (something)^2. The only way we can get this is when you differentiate something of the form u/v. As this will give you
Notice the denominator is of the form (something)^2.
We can therefore say the Anti derivative takes the form of
Now comes the dodgy part. You guess the numerator. Usually it's something simple and you should guess it as something that is similar to the integral. For example if the integral contained a bunch of sines and coses, you might guess sin(x) as the numerator.
If you test 1 and -1 it doesn't work but if you test x which is the second most simplest thing it checks out
And no this isn't taught in the syllabus. This is just intuition you develop over time or by a tutor or someone telling you
They generally make it pretty clear when you have to do this, but using the property:
Why not have a crossover of trig and exponential?They generally make it pretty clear when you have to do this, but using the property:
Usually when it is done it is for something like trigonometric integrals, where you substitute pi/2-x, and change a sinx to a cosx. When you add this to the original integral, I , it simplifies it a lot and allows for you to solve for 2I. On the extremely rare chance they didn't make it clear they wanted you to use this method, look for bounds involving pi/2 and 0, with trig integrals, as it will normally be what its used for. There's also sometimes substitutions of u=-x, usually when you have a complicated function involving a lot of exponentials. Look out for bounds in the form of +/-a for these types of questions, and functions that you could simplify by adding this new form on to the original.
haha...Observe that the denominator is of the form (something)^2. The only way we can get this is when you differentiate something of the form u/v. As this will give you
Not sure, but everyone says 1993 was super hard, and the average for that was like in the 40s out of 120, and imo 1995 is of similar difficulty to that, possibly a bit easier.
Ironically that year Anthony Henderson got 100 raw, the only one so far.
Yeah lol I heard about that. Tbh the paper isn't that bad. It gets hard really early on like with that circle question in 3, which would be why the average is so low, but q8 isn't difficult at all. I'm not sure why its hyped up so much, years like 2000 and 1989 are harder.
I would like it if they did, those non uniform circular motion qs are great
That's what I was thinking as well, although I don't think they will include it purely just for "old time's sake". I think they'll do some sort of probability question, and perhaps throw in a bit of sequences and series amongst it all. They may even also do something that is going to be assessed in the new syllabus. I've seen a lot of questions lately involving the Taylor series and how it allows you to express complex numbers in a different form. They may play on that, or possibly something similar.
The problem is that it’s pretty damn hard to come up with a good creative circular motion question that’s still confined within the HSC syllabus. It was a struggle for us coming up with good questions on this topic for the BoS trials (that hasn’t been done before) so I would imagine the writers of the HSC paper would have similar challenges.
If you mean the ones where you get the t- series representation of the trig finctions and manipulating them to show that:
I'm still not sure what the average or highest mark is, but I just sat the paper personally today under timed conditions (Don't have the solutions, but I was going off the exam report). If I was to guess, I would say the average was in the 50s, with the E4 cutoff being in the 70s somewhere (out of 120 that is). There weren't many difficult questions (Apart from that circle one) but it got trickier early on.
Yeh that circle geometry question was killer. If you read the markers comments, only one person in the state got full marks for all of the parts involved in the question. However I felt that question 8 was relatively easier compared to other years.