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Re: MATH1131 help thread How do you know it's outside? (It could just be laziness about brackets. It should be clear from the context which one was meant.) Was the question asking to differentiate (sin x)^x or something (if so, logs would be taken and then logarithmic differentiation used...

Re: MATH1131 help thread Judging by the next line, it's surely inside the log, i.e. log(sin(x)^x).

Re: MATH1131 help thread E.g. When x = 0, the value is negative. When x = 3, the value is positive. (In other words inspection as leehuan said above.)

Re: MATH1131 help thread It's not really scalar "product" (the term "scalar product" commonly refers to the dot product). Better to say scalar "multiple".

Re: MATH1131 help thread Since the two vectors are parallel, their components are in the same ratio, so 3:2 = s:1 = t:5. Since 3:2 = s:1 (i.e. 3/2 = s/1), we have s = 3/2. Since t:5 = 3:2 (t/5 = 3/2), we have t = 5*3/2 = 15/2.

$\noindent 2) I think you've mixed up the normal vector formula (or typoed, since you just put the $z$'s in the `denominators' instead of `numerators'). A normal to the graph of $z=f\left(x,y\right)$ at a given point $\vec{a}$ is $\vec{N} = \begin{bmatrix} - \frac{\partial f}{\partial x} \\...

No worries. :) $\noindent 1) a) A particular solution is $\vec{x}_p = (0,1,0,0) = \vec{e}_2$. In fact in general, for ANY $m\times n$ matrix $A$, we have $A \vec{e}_j = \text{ column } j\text{ of }A$, where $\vec{e}_j$ is the $j$th standard basis vector for $\mathbb{C}^n$ or $\mathbb{R}^n$, for...

Re: MATH1131 help thread $\noindent The answer will be a vector $\vec{v}$ that is of the form $\vec{v} = \alpha \vec{u}$, for some real constant $\alpha$ (so that they're parallel). We just need to find an appropriate $\alpha$. Since we want $\left \| \vec{v}\right \|=10$, we want $\left...

Re: First Year Uni Calculus Marathon What do you mean? Do you mean if a function is monotone on an interval, it has at most one zero on that interval? If so, that follows from the definition of monotone (if it's monotone, it can't take the same value at two different places).

$\noindent The \textit{definition} of events $A,B$ being independent is that $\mathbb{P}\left(A\cap B\right)=\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$. An intuitive `visualisation' for this is like if two coin tosses are independent, then indeed chance of two heads in a row is...

This is essentially referring to the definition of a function (no input has more than two outputs). For one-to-one, it's basically reverse: every y-value in the range has one and only one x-value in the domain. Or equivalently, we don't have two different x-values getting mapped to the same...

$\noindent Intuitively, the range is smaller if the projection angle is smaller (because you hit the ground pretty quickly, so not much time for the projectile to go forward much), increasing until you get to the angle with maximal range (famously known to be $45^\circ$), and then it just goes...

Re: MATH1131 help thread $\noindent Let $G$ have position vector $\vec{g}$. You should be able to see by drawing a diagram that for $OEFG$ to be a parallelogram (with the points in that order), we will have: $\overrightarrow{OG}=\overrightarrow{EF}$ (because opposite sides of a parallelogram...

$\noindent The range of a projectile is $d=\frac{u^2}{g}\sin 2\theta$, where $\theta$ is the angle of inclination, $u$ is the speed of projection, and we have assumed the projectile starts on ground level (and no air resistance of course). You can see this formula here:$...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Recall that a number is divisible by 3 if and only if the sum of its digits is a multiple of 3. So just find how many ways you can form numbers from that so that the digit sum is a multiple of 3.

Re: MATH1131 help thread $\noindent If we're translating under $\vec{t}$, it means we're moving everything in the direction and amount specified by the vector $\vec{t}$.$ $\noindent I.e. it is the mapping $\vec{x}\mapsto \vec{x} + \vec{t}$. So if we had a point before, we add $\vec{t}$ to...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Don't need to. Just combine the fractions in the given expression for T_n at the start of the Q.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread It's given.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Find T_3 first using the recursive formula. Then find T_4 by subbing in known or found values of T_2 and T_3, and rearranging. So T_2 = (1/2)*(T_1 + T_3), and we know T_2 and T_1, so we can solve for T_3. Then T_3 = (1/2)*(T_2...

Re: HSC 2016 MX2 Combinatorics Marathon $\noindent Let $n$ be a positive integer at least $2$. Consider permutations of the integers $1,2,\ldots,n$. Call a number $j\in \left\{1,2,\ldots n\right \}$ a \textit{local maximum} (plural \textit{maxima}) if it is greater than both its neighbours, or...