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An eigenvector is going to be some element from the domain vector space of the linear map (which are generally called "vectors", even if they may be matrices, polynomials, functions, whatever). So in this case, an eigenvector is a matrix, since the domain vector space is a space of matrices (so...

Re: ATAR Notes vs Bored of Studies How badly would one need to misbehave on that site vs. on BOS in order to get an IP Ban?

Re: ATAR Notes vs Bored of Studies Yeah that was the main reason for my guess.

Re: ATAR Notes vs Bored of Studies I have a feeling this thread may get deleted or locked.

$\noindent Thanks for taking the trouble to write these up!$ $\noindent Really minor thing, but for $16$ (c) (iv), you used an initial condition of $D(0) = 1$. But the question as written only ever defined $D(n)$ for $n\geq 1$, giving $D(1) = 0$ and $D(2) = 1$ as the initial conditions. (Of...

Final answer is 0.

They should, pretty sure many others in the state would've used pencil too. But best to use pen from now on I think.

Your R is still the parabola. Did you mention the concave down parabola part?

$\noindent Note we don't need a GP to find $p_n$. To find $p_{n}$, it's just $1$ minus the probability the game \emph{doesn't} end in the first $n-1$ rolls. The game fails to end in the first $n-1$ rolls with probability $q^{n-1}$ (since this happens if and only if the first $n-1$ rolls are all...

$\noindent Let $X$ be the roll number the game ends on ($X$ can take on values $1,2,3,\ldots$). Let $p = \frac{1}{8}$ and $q = 1-p$. Then $\mathbb{P}\left(X = k\right) = q^{k-1}p$, $k=1,2,3,\ldots$. Let $p_{n}$ be the probability that the game ends \textbf{before} the $n$-th roll, for...

I don't know but it seems to just be a misinterpretation of the wording/requirement of the question (although the first part of the Q. should've made it clear what the interpretation is). So you should at least get some marks (guessing). Maybe lose 1? You could ask Carrotsticks for his opinion.

The reason is we're not allowed to link that page or type its name on BOS.

What's your definition of n and p()? In the Q., we're saying the game ended at or before roll#(n-1). So if n = 10, you'd need to have gone up to roll #9.

What do you mean by the first comma placement? That makes it sound like "before the nth roll is more than 3/4", as in somehow the nth roll can be more than three-quarters, which is nonsensical of course. Basically it's before like strictly before. I.e. find the smallest value of n such that...

What method did you use (sorry if you said this above already, I haven't followed this thread too closely)? If you did the parabola method I mentioned before, it's self-evident really. Plus the question's wording makes it sounds like they let you assume it (if I recall the question correctly)...

Remember, the constant terms disappear when we differentiate.

Yeah it's definitely strictly before the n-th roll, e.g. looking at the first part of that question, they asked for before the 4th roll and went up to roll #3.

For Question 2, option (A) doesn't satisfy p'(1) = 0. Remember, to be a multiple root, we also need p'(1) = 0. For Question 7, it's based on the following result (should be in at least one of the textbooks, maybe Arnold and Arnold?): $\noindent The graphs of $xy = \frac{k}{2}$ and $x^2 -y^2 =...

Well -i is cis(-pi/2), so it rotates by pi/2 clockwise.

Yeah period of tan(3x) is pi/3.