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This is just mindlessly plugging into formulas which should be avoided at all costs for this case. matchalolz does not have a good idea of how to do this fundamental question so giving formulas without explanation/proof is worthless for the purposes of learning which is top priority in this...

This may help, \ln(x) \leq x - 1.

You are correct. Easy verification by substitution then expansion and grouping of reals and imaginary components which I know you know, so why did you have doubts?

For the third question from the top, $Let $q(k) = p(k) - k^3.$ Then we wish to solve $q(k) = 0$, or to find the number of zeros $q(k)$ has. Sub in 100, $q(100) = -9900$. We can deduce the structure of $q$ by examining the prime factorization of -9900. \begin{align*}-9900 &= -3 \times 3 \times...

^LoLLL. I don't think Truongs accepts students halfway through year 12 but this was a few years ago, things may have changed.

What was the miscalculation?

Tell us what errors you made.

I believe there is another typo on the RHS. Expand via binomial theorem and use Pythagorean trig identities to get \begin{align*} (\sin x - \cos x)^3 &= \sin^3 x - 3 \sin^2 x \cos x + 3 \sin x \cos^2 x - \cos^3 x \\ &= \sin x (1 - \cos^2 x) - 3 \sin^2 x \cos x + 3 \sin x \cos^2 x - \cos x (1 -...

ah, I'd be careful of plagiarism if I were you.

Identity is false for all integer powers (of LHS).

You want to find angle FAC. Now which trig ratio should you use to find this angle? This is the same as choosing which two sides of triangle FAC to find. Give this decision a try.

Identify the aim (the last step). What angle do you want to find? Identify the penultimate (second-last) step. Identify the second-second-last step and so on, until you have an idea of what to do. This is called working backwards and allows you to open up paths to search for the answer. It...

This is normal and is good. Each paragraph of my first essay took at least 7 hours to complete and most of the time was spent thinking. By the way, I was not distracted during the 7+ hours. Just keep at it and the brain will adapt. Your thoughts will become more organised and mental clarity...

No way... Has to be a typo... The author should be behind bars for putting wrong information in a textbook! xd

Re: 3D Trig Help It depends on where exactly the observer is looking at. If the line of sight of the observer is parallel to the (flat) ground and directly towards the y-axis of the balloon, then SirKeynes' answer is right. Otherwise, if the line of sight of the observer is directly towards...

I have friends doing econometrics and they don't recommend it because it lacks rigour in mathematics. And if you want to really understand what's going on, you'll need the mathematics. Thus, you're better off doing stats.

Employers are probably aware that a CR at UNSW is at least a DN at UTS. It's also harder to score high for CS at UNSW because they use harmonic mean to calculate marks which means you have to do well in all assessments. _______ I don't think you should base your decision primarily on your...

Re: Probability @juantheron, your logic checks out but strange notation. Maybe the given answer is for three or four tests? Too lazy to do the computation.

Re: Probability P(1 machine then 1 machine) = 2/4 x 1/3, because there is a 2/4 chance of identifying the faulty machine with 1 test. The event "0 machines then 2 machines" is 3 tests.

1) Tried to prove, didn't seem right. Plugged in A = 40, B = 5 degrees and realised the identity was wrong. It's meant to be \cot (A+B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} 2) You are correct. Their answers are correct under the assumption that 0≤θ≤π