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Re: HSC 2015 3U Marathon $ Note the integrand can be expressed $2^{\ln x}=e^{\ln x \ln2}=x^{\ln2}$ so our integral will be $\frac{x^{1+\ln2}}{1+\ln2}+C$ $

Re: HSC 2015 4U Marathon $ Using properties of tangents, discriminants and product of roots, we get $x^2+y^2=25$ i.e. the locus describes a circle centred at $(0,0)$, with radius 5 units. $

Re: HSC 2015 4U Marathon NEXT QUESTION $ Solve $3x^2-2x-2\le|3x|$. $

Re: HSC 2015 4U Marathon Modulus is $17\sqrt2$ , doesn't matter though since the question at hand is only asking about the properties of arguments.

Take the scholarship and do Commerce/Arts, work your ass off in first-year to get 80+ WAM average (very doable with enough effort) and keep this is as a safety net in case you want to transfer into Comm/Law or Arts/Law, that way you have nothing to lose. Still, it is easier said than done, if...

Pick the one you think you will enjoy the most, that way you will enjoy what you study (which is important) and won't have any regrets about what you are doing :) I am going to be biased and say go with UNSW (despite me not knowing what Arts is like), since they are very flexible when it comes...

Re: HSC 2015 4U Marathon Yeah, better you try this at home with a clear mind, draw a diagram and try to see what the question actually means, I am not sure if your method of equating gradients ?? is actually correct, since the question at hand refers to angles other than that from the positive...

Re: HSC 2015 4U Marathon NEW QUESTION (EASIER): $ If $|z+2|+|z-2|=8$ find the equation of the locus of the point $P$ representing $z$ on an Argand diagram. $

Re: HSC 2015 4U Marathon NEW QUESTION (HARDER): $ Prove that the tangent at a point $P$ on the ellipse is equally inclined to the focal chords through $P$. $

Re: HSC 2015 4U Marathon Why $ $e^2$ $ ?

Re: HSC 2015 4U Marathon It is the standard definition of the ellipse you learn when you start Conics i.e. the ellipse is defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the...

Re: HSC 2015 4U Marathon $ Using the fact that $PS=ePM$ ($0<\,$e$\,<1$), we can use this to show that $PS+PS'=ePM+ePM'=e(\frac{2a}{e})=2a$ i.e. the set of points such that the sum of the distances to two fixed points (the foci) is constant. $

Boundary interval should be $0 \le\, $x$ \le2$ so you would integrate from x=0 to x=2 to find your volume.

Re: HSC 2015 4U Marathon $ Simplified expression is $\frac{\cos\frac{x}{2}-\cos(n+\frac{1}{2})x}{2\sin\frac{x}{2}}$. \\ Substituting $x=\frac{\pi}{2015}$ with $n=2014$, and manipulating with our simplified expression, the desired result follows. $ EDIT: Not sure if the RHS value is correct...

Re: HSC 2015 4U Marathon First part is to 'show', then second part is to 'find' i.e. use the first part to do the second, so it is like a 'Hence, find...' sort of question.

Re: HSC 2015 4U Marathon .

Re: HSC 2015 4U Marathon $ Consider the polynomial $P(z)=a_0+a_1z+...+a_nz^n$, where $a_1, a_2,...,a_n$ are all real and $n$ is an odd integer. Suppose that $\alpha$ is a root of $P(z)$ so that $P(\alpha)=0$. \\ Then $P(\bar{\alpha})=\displaystyle\sum_{k=0}^{n}a_k...

Re: MX2 2015 Integration Marathon $ Express the denominator of the integrand as a product of two quadratic factors (by completing the square), then use partial fractions to get to the standard $\frac{f'(x)}{f(x)}$ form to get $\frac{1}{2\sqrt2}\ln\frac{x^2-\sqrt2x+1}{x^2+\sqrt2x+1}+C.$ $

Re: HSC 2015 4U Marathon Thanks, just fixed it, yeah I am also not sure if the area was referring to the whole circle or just that major segment of it described by the locus. Will just have to wait and see what the OP of the question says.

Re: HSC 2015 4U Marathon Bump