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The actual integral is: \int \sqrt{\tan{x}} + \sqrt{\cot{x}} \ dx = \sqrt{2}\bigg[\tan^{-1}{\left(\sqrt{2\tan{x}} + 1\right) + \tan^{-1}{\left(\sqrt{2\tan{x}} - 1\right)\bigg] + C \qquad \text{for some constant $C$}

There is a problem with the answer here. The function f(x) = \sqrt{\tan{x}} + \sqrt{\cot{x}} is positive throughout its domain. Yet, using the result presented: \begin{align*} \int_{\frac{7\pi}{6}}^{\frac{5\pi}{4}} \sqrt{\tan{x}} + \sqrt{\cot{x}}\ \!dx &=...

There are lots of places that sell papers and plenty of schools who make an exam by taking parts from multiple sources... plus many schools write the exam themselves. Swapping just a few questions can change the difficulty of an exam by quite a bit. Also, Trials are meant to suit the cohort...

Flying like a bird is easy... all you have to do is throw yourself at the ground, and then miss! ... wirth acknowledgement to Douglas Addams

Watch out with these sorts of questions, with an unknown as a boundary of integration as there can be multiple answers. In fact, if this question did not implicitly require a > 0, there would be two answers. If this same diagram were given (but in a non-MCQ) along with the definite integral...

No. Suppose they start together (i.e. n = 0), then Din wins and crosses the finish line 6.39 seconds before Bo-Katan. So long as n < 6.39 s, Din crosses the line first.

Of course I'm not in High School, it's 11 o'clock at night... and it's a Sunday... what kind of a weirdo would I have to be to be in a high school now?

Ah, Wednesday... hopefully a day for a good sleep in...

This question is a modified form of a question from the old Fitzpatrick book, which also shows up (with a different variation) in Cambridge... and the original Fitzpatrick question had no structure provided to guide you.

The direction vectors given are not unit vectors as they each have a magnitude that isn't 1. The vector 3i + 4j has magnitude 5, for example. If you multiplied the given direction vector by 50, you would get a vector for the force that has magnitude 250 N. So, to get a magnitude = 50 N vector...

I see what has happened, I misread your post as saying that the limit of \frac{1}{f(x)} had to be non-zero. Whoops!

But here, both of those limits are zero, and the horizontal asymptote is y=0 - as it must be for for \frac{1}{f(x)} where f(x) is a concave up parabola.

Continuity correction is needed (formally) when approximating a discrete distribution (like a binomial) with a continuous distribution. Mathematically, it should always be included when making such an approximation, and leads to a more accurate result. However, HSCially, solutions will be...

Yes A' is another notation for not A or A with a bar over it

It depends on how you view rigour and technicalities in proof. Either approach will be fine in the HSC, so long as you do specify that k is odd or even (as appropriate) in stating the induction hypothesis when taking the first approach. The second approach is mathematically preferable because...

If A and B are mutually exclusive (i.e. non-overlapping, as shown in the illustration posted above, then A' (everything not in A) intersects with B for everything in B and nothing else. Thus, P(A' AND B) = P(B)

As we all know, English teachers are noted for their proficiency in Sciences and Mathematics. After all, Shakespeare wrote so eloquently of the unparalleled beauty of quartic polynomials with complex coefficients in his famous Sonnet 18 (Shall I compare thee to a summer's day? ...)

Aren't we all students? Learning new things every day, as students of life? ... Though, I do believe that there is not much about HSC chemistry left for me to learn.

From the data, the pH of the equivalence point is around 8... so long as the volume is consistent with the graph drawn and a sensible pH, whatever concentration is calculated is reasonable. I see no reason that you should have been penalised... though the examiners / markers might have a...

This is not correct. Just as we reserve the term "aqueous" for situations where the solvent is water, we reserve the term "liquid" for a pure substance or for the solvent. In an esterification process where the reaction mixture contains several different liquid substances that are miscible and...