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Re: HSC 2016 4U Marathon - Advanced Level \\ $Is$ \ 2^n + 3^n $, for positive integers$ \ n $, ever a square number?$

Re: HSC 2016 4U Marathon - Advanced Level Yes this is what I got as well, I wasn't able to find a closed form either which is why I simply asked for an expression

Re: HSC 2016 4U Marathon \\ $Let there be polynomials of the form$ \\ p(x) = x^n - nx^{n-1} + \frac{n(n-1)}{2}x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0 \\ $the roots of which are$ \ \alpha_1, \alpha_2, \dots , \alpha_n \\\\ $i) Show that$ \ (\alpha_1 - 1)^2 + (\alpha_2 - 1)^2 + \dots +...

Some exercises in formal logic: \\ $Prove formally the following:$ \\ $1.$ \ p \Rightarrow (q \wedge r) \vdash (p \Rightarrow q) \wedge (p \Rightarrow r) \\ $2.$ \ \sim p \Rightarrow p \vdash p \\ $3.$ \ (p \vee q) \Leftrightarrow p \vdash q \Rightarrow p For a list of the symbols used...

Re: HSC 2016 4U Marathon - Advanced Level \\ $Prove that there are no positive integers$ \ a,b \ $satisfying$ \ \frac{a}{b} + \frac{b}{a} = 4

Re: MX2 2016 Integration Marathon \\ $Evaluate$ \ \frac{\int_0^1 (1-x^n)^m \ dx}{\int_0^1 (1-x^n)^{m+1} \ dx} \ $for positive integers$ \ m,n

Re: MX2 2016 Integration Marathon Yes well done, alternatively x= \tan u and then v = \frac{\pi}{2} - u , adding up the integrals in u and v to get what you have.

Re: HSC 2016 4U Marathon - Advanced Level \\ x \ $is a real number randomly chosen from the interval$ \ 0 < x < 1 \\ $Let$ \ f(x) = \left\{\frac{1}{x} \right\} \ $where$ \ \{A\} \ $is the fractional part of$ \ A. \\ $Examples of this function include,$ \ \{\pi \} = \pi - 3 \ $or$ \ \{1.5 \}...

Re: HSC 2016 4U Marathon \\ z^3 = -\overline{w}^7 \ $(from (1))$ \\ z^5 w^11 = - z^2 w^7 \overline{w}^7 w^4 1 = - z^2 w^4 |w|^{14} \\ \Rightarrow |z|^2 |w|^{18} = 1, \ (3) \\ $(from (2))$ |z|^5 |w|^{11} = 1, \ (4) \\ (3) - (4): \ |z|^2 |w|^{11}(|w|^7 - |z|^3) = 0 \\ |z| \neq 0 , |w|...

Re: HSC 2016 4U Marathon \\ $Let$ \ P(x_1,y_1) \ $and$ \ Q(x_2, y_2) \ $be points on ellipse$ \ b^2x^2 + a^2y^2 = a^2b^2 \\ $Let the tangents at$ \ P \ $and$ \ Q \ $meet at$ \ R(x_0, y_0) \\ $Immediately the equation of$ \ PQ \ $is$ \ x \frac{y_2 - y_1}{x_1y_2 - x_2 y_1} - y \frac{x_2 -...

Re: HSC 2016 4U Marathon \\ $By the very definition of differentiability$ \ f \ $is differentiable at$ \ a \ $if and only if the limit$ \ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \ $exists$ \\ f'(a) \ $is also simply the name we give to this limit, and since$ \ f'(a) \ $is independent of$ \ h...

Re: MX2 2016 Integration Marathon \\ \lim_{a \to \infty} \int_0^a \frac{\text{d}x}{(1+x^2)(1+x^3)}