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Re: HSC 2017 Maths (Advanced) Marathon \begin{align*}\text{Let }f(x)&=-x^3\\ f^{\prime}(x)&=-3x^2\end{align*} \text{For all }x\in\mathbb{R},x\neq0\text{, }x^2>0\text{, so }-3x^2<0 \therefore\text{As }f^{\prime}(x)<0\text{ for all }x\text{ in the given domain, }x^3\text{ is monotonic decreasing.}

Re: HSC 4U Integration Marathon 2017 \begin{align*}\text{So }&\int_{-1}^{1}{\sin{(\pi|x|)}\sin^{-1}{\sqrt{|x|}}\text{d}x}\\ =&2\int_{0}^{1}{\sin{(\pi x)}\sin^{-1}{\sqrt{x}}\text{d}x}\\ \text{Let }&\pi x=\cos^{-1}{u}\\ &\text{d}x=\frac{-\text{d}u}{\pi\sqrt{1-u^2}}\\ &x=0\implies u=1\\...

\text{The graph of }\cos{\theta}\text{ is simply that of }\sin{\theta}\text{ shifted }\frac{\pi}{2}\text{ units to the left.} \begin{align*}\text{So as }\sin{(\theta+a)}\text{ shifts the graph of }\sin{\theta}\text{, }a\text{ units to the left...

Such as a function defined as f(x)=(-1)^{\lceil{x}\rceil}|x|?

A function with no two points x_1 and x_2 such that f(x_1)=f(x_2). An example would be: f(x)=\begin{cases}(x+1)^3&x\leq -1\\ 0&-1\leq x\leq 1\\ (x-1)^3&1\leq x\end{cases} Which fits the definition for monotonic increasing, but isn't strictly monotonic increasing. (From Wikipedia) A function can...

For a function to be invertable, it must be defined such that it is one-to-one. If a function is monotone, then every x value will result in a single value of f(x), however, the function must be strictly monotonic, else there could exist two values x_1 and x_2 such that f(x_1)=f(x_2). So if the...

Re: HSC 2017 Maths (Advanced) Marathon Thanks for clearing that up; none of my teachers were able to tell me definitively.

Re: HSC 2017 Maths (Advanced) Marathon \begin{align*}\text{Let }h(x)&=f(g(x))\\ h^{\prime}(x)&=f^{\prime}(g(x))g^{\prime}(x)\\ \text{So }&f^{\prime}(x)>0,\forall x\in\mathbb{R}\\ &g^{\prime}(x)>0,\forall x\in\mathbb{R}\\ \therefore&h^{\prime}(x)>0,\forall x\in\mathbb{R}\\ \text{So...

Re: HSC 4U Integration Marathon 2017 I didn't actually do 4U, so there's probably a method to do it properly, but: \begin{align*}\text{So }&\int_{-1}^{1}{\sin{(\pi |x|)}\sin^{-1}{\sqrt{|x|}}\text{d}x}\\ =&2\int_{0}^{1}{\sin{(\pi x)}\sin^{-1}{\sqrt{x}}\text{d}x}\\ =&\frac{2}{\pi}\big[\cos{(\pi...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread It's asking what values of a, b, and c are required to change the degree of the polynomial. For example, if you had a polynomial ax^3+bx^2+cx+d, and you wanted a polynomial of degree 1, a and b would both have to be zero, so that...

You don't, the answer is x < 2, 2 < x < 4.

I know that all of the teachers for maths I've had in the past have said that the markers for the HSC don't care whether you include the roots of the derivative function. Although, in this case, because the graph meets the axis, being a stationary point, I imagine they'd want x < 2, 2 < x < 4.

\begin{align*}\text{So we're given that }\sqrt{a}(b-a)+\sqrt{c}(c-b)&>\frac{c^2-a^2}{2\sqrt{b}}\\ \frac{a^2}{2\sqrt{b}}-\sqrt{a}(a-b)&>\frac{c^2}{2\sqrt{b}}-\sqrt{c}(c-b)\end{align*} \begin{align*}\text{So let }f(x)&=\frac{x^2}{2\sqrt{b}}-\sqrt{x}(x-b)\\...

\begin{align*}\text{So }\left(1-x\right)^n\left(1+\frac{1}{x}\right)^n=\sum_{r=0}^{n}{{{n}\choose{r}}\left(-1\right)^rx^r}\sum_{k=0}^{n}{{{n}\choose{k}}x^{-k}}\end{align*} \begin{align*}&\text{Now notice that if we let }x=1\text{ we get an expression with each term in the form...

This is probably what the question wants, then: \begin{align*}\text{3. }&\text{So by the definition of the derivative, }f^{\prime}(x)=\lim_{h \to 0}{\frac{\sin^{-1}{(x+h)}-\sin^{-1}{x}}{h}}\\ &\text{Let }y=\sin^{-1}{(x+h)}-\sin^{-1}{x}\\ &\sin{y}=\sin{(\sin^{-1}{(x+h)}-\sin^{-1}{x})}\\...

Re: HSC 2017 Maths (Advanced) Marathon If I'm interpreting your question correctly: \text{So we're looking for three numbers }T_1\text{, }T_2\text{, and }T_3\text{ such that }8,T_1,T_2,T_3,\frac{1}{32}\text{ is a geometric series.} \text{We can express }8\text{ as }2^3\text{ and...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread If N is to the left of U, the answers must all be in the form ____N____U____, where the underscores represent places the remaining four letters can go. So: Consider _N_U_, where we are trying to insert M. There are 3 places for...

A slightly more "rigorous" proof (not really): \text{Take the general form of the parabola }x^2=4ay\text{, and the points }P(2ap,ap^2)\text{ and }A(x_0,y_0)\text{.} \text{For a parabola with a vertex at }(h,k)\text{, just shift }A\text{ to an equivalent point.} \begin{align*}\text{So }x^2&=4ay\\...

\begin{align*}\text{1. }&\text{We have }\lim_{a \to 0}{\frac{\sin^{-1}{a}}{a}}\\ &\text{Let }a=\sin{u}\\ &\text{So }\sin^{-1}{a}=\sin^{-1}{(\sin{u})}=u\\ &\therefore\lim_{a \to 0}{\frac{\sin^{-1}{a}}{a}}=\lim_{a \to 0}{\frac{u}{\sin{u}}}\\ &\text{Now as }a \to 0\text{, }\sin^{-1}{a} \to 0\text{...