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Use the angle sum identity: \sin(\alpha + \beta) =\,\, ?

Suppose we have the integral \int \sqrt{1-x^2}\,dx We can make the substitution x = \sin \theta, which is undoubtedly valid because the range of the substitution -1 \leq \sin \theta \leq 1 is the same as the domain of the integrand -1 \leq x \leq 1 . But what about an integrand whose...

I do a lot of exercises, especially from textbooks. I try to familiarise myself with the course content. The SGS Notes are pretty good for this actually, I haven't had to use anything else. Doing questions on this forum, I think, has also improved my results. Personally I prefer not to use...

For Q3 you can consider the (isosceles) triangle AEC with side lengths 10 , 10 and \sqrt{18} . You can find the desired angle \angle AEC without the use of sine rule or cosine rule. Q4 is the same question as Q3 with the slanted height changed.

Re: HSC 2018 MX2 Integration Marathon Can someone give me a hint for this question? $Show that$\,\, \frac{\pi}{12} < \int_0^{\frac{\pi}{4}} \sec^{\frac 5 2} x\,dx - \frac{1}{3}\left(2^{\frac 5 4}\right) < \frac{\pi}{12}\left(2^{\frac 1 4}\right).

Yep it's 220. For the second question, the marker's answer sets up the rings as such: R_R_R_R_R_R_R_R_R Where the eight "_" characters represent possible spaces for a divider, of which there must be three. Thus the answer is simply ^8C_3 = 56.

Here's an easier question from one of my school's 4U Task 1 papers: In how many ways can nine identical rings be placed on the four fingers of the left hand, excluding the thumb, if: there are no restrictions? there must be at least one ring on each finger? there are no restrictions and...

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) Basically if you want to place n objects in k boxes you can model it using something like \star | \star \star \star | \star \star Where the stars are the objects and the bars represent the dividers between boxes. So in the...

Assuming the tigers and cages are identical and arranged in a straight line: If no cage is to be empty and each cage can fit a maximum of two tigers, then there must be either one or two tigers in each cage. Also, there must be exactly 6 tigers. The only way to make 6 from adding 1 or 2...

Are these four cages distinct or identical? Are the tigers distinct or identical?

Came bottom 5 in my class for prelim math HYs, rank 102/176. Pulled my shit together and came top 5 in my class in yearlies, overall 41/176.

Table A3 on this might be helpful: https://www.uac.edu.au/assets/documents/publications/2017-prelim-scaling-report.pdf

Good in Maths and SDD, don't have English but probably average-mediocre like usual. Pretty bad for Chemistry but IMO it wasn't entirely my fault... not the first time our school has sent out a shifty Chemistry exam.

There are \frac{^{12}P_6}{6} ways to take out six people from the group and arrange them around a table. Then, with the remaining six, there are \frac{^{6}P_4}{4} ways to take four and arrange them in a table. And there is only one way to arrange two people around a table. So the...

Suppose an acid (HCl), in a conical flask, is titrated with a base (NaOH). The equivalence point is reached and then more base is added. The substance in the conical flask is clearly basic, but is it correct or acceptable to refer to it as "the base"?

Fractional coefficients are generally only acceptable in combustion or half cell equations. So the second one.

My timetable goes 18-19-25-30-02-05, so I'm fine with that but I wish the exams could be a bit later in the day...

For sciences, the best thing to do is usually to make notes according to the dot points. Maths subjects aren't as heavy on content and memorisation so making notes on the dot points isn't as effective (but it is still useful to refer to them so you know what may be required of you). Doing...

With simultaneous equations and multiple unknowns, you need several pieces of information. Generally you need one equation for each unknown quantity. So if you have two unknowns you need two equations. And you can't "cheat" by rearranging the first equation and using that as the second...

From what I can see, the answer is no. I go to a top 5 selective school and I've never been under the impression that most, or even many people have rich parents. I can definitely name a few people who have a reputation for $$$ but I think most of us can be described as "well-off, but not...