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Re: MX2 Integration Marathon The answer is: I_n=\int_0^{\pi} \frac{\sin(nx)}{\sin x} \ dx = \begin{cases} 0 , \ \ $for even$ \ \ n \\ \pi, \ \ $for odd$ \ \ n \end{cases} This is because, I_{n} = I_{n-2} (using sum to product formula) So if n is even, it becomes I(0), and if n is odd, it...

Re: HSC 2013 4U Marathon If its in geometric progression, then for integers a and k, it must be in the form: a, ak, ak^2 Essentially, we are finding how many integer solutions there are for: ak^2 \leq 100 Where (a,k) are positive integers with a \leq 100 So lets sketch the graph: y^2...

Re: HSC 2013 4U Marathon a_{2m-1} + a_{2m-2} = \binom{m-1}{m-1} + \left(\binom{m}{m-1} + \binom{m}{m-2} \right) + \left( \binom{m+1}{m-2} + \binom{m+1}{m-3} \right) + \dots + \left(\binom{2m-2}{0} + \binom{2m-2}{1} \right ) + \binom{2m-1}{0} $Use the identity$ \binom{n}{r} =...

Re: MX2 Integration Marathon \int_0^{\pi} \frac{\sin(nx)}{\sin x} \ dx

Re: HSC 2013 4U Marathon Oh ok As for calculation of E(x) $Finding the sum$ \sum_{k=0}^r k \binom{r}{k} \binom{m-r}{n-k} k Consider the identity: r(1+x)^{r-1} (1+x)^{m-r} = r (1+x)^{m-1} Equate co-efficient of x^{n-1} on both sides, where for the first binomial, take the...

Re: HSC 2013 4U Marathon Ahh ok, I'm not too sure about how to interpret 'average' though.. If I did: P(k) = \frac{C(r,k) \times C(m-r,n-k)}{C(m,n)} And took the average probability: A = \frac{1}{n} \sum_{k=1}^r P(k) And then, if: P(m) < A < P(m+1) (so its bounded by two consecutive...

Re: HSC 2013 4U Marathon Here is an attempt: $i)$ \ \ \ \frac{C(r,k) \times C(m-r,n-k)}{C(m,n)} $ii) \ \ $let$ \ \ T_{k} = \binom{r}{k} \binom{m-r}{n-k} \frac{T_{k+1}}{T_{k}} > 1 \Rightarrow \ \ \frac{nr-m+r+n-1}{m+2} > k $Therefore the$ \ \ k \ \ $that maximizes probability, thus the...

Re: HSC 2013 4U Marathon Ohhh I see what you mean now, nice work. I squared both sides and cancelled out the a_k^2 and b_k^2 terms, and then by grouping the terms proving \sqrt{(a_k^2 + b_k^2)(a_m^2 + b_m^2)} \geq a_k a_m + b_k b_m Which is done by squaring both sides and expanding, to...

Re: HSC 2013 4U Marathon I don't think so

Re: HSC 2013 4U Marathon $Prove for real$ \ \ \{ a_k \} \ \ $and$ \ \ \{b_k \} \sum_{k=1}^{n} \sqrt{a_k^2+ b_k^2} \geq \sqrt{ \left(\sum_{k=1}^n a_k \right)^2 + \left(\sum_{k=1}^n b_k \right )^2}

Re: HSC 2013 4U Marathon Ah yep ok that is quite similar to what I did.

Re: HSC 2013 2U Marathon $In an economy, in the short run the relationship between inflation and unemployment (both measured in percentages) can be modeled by$ \frac{di}{du} = -k(i-c)^2 \ \ \ $where$ \ \ u \ $is unemployment$ \ \ i \ $is inflation $and$ \ \ k \ \ $and$ \ \ c \ \ $are...

Re: MX2 Integration Marathon :P Replace the roots with alpha and beta if you want, and anyway, I don't really want a solution I know people can do partial fractions. The aim of the question was more getting people to recognize the different cases

I'm going to try and revive this as HSC approaches: Why is deflation worse than inflation?

Re: HSC 2013 4U Marathon How did you get (2r-1)/(r^3-r), that is the correct answer though, I split the 1/k(k+2) to 1/2 ( 1/k - 1/(k+2)), then manipulated and grouped certain terms to telescope twice.

Re: MX2 Integration Marathon That is partially correct (even with the constant)

Re: HSC 2013 4U Marathon Yep, well done --- $i)$ \ \ \sum_{k=1}^{\infty} \frac{1}{k(k+2)} \left(1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{k+2} \right ) $ii)$ \ \ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{mn^2 + 2mn + m^2n}

Re: MX2 Integration Marathon u = \sqrt{x+1} 2u du = dx \int \frac{2u(u^2+1)}{u (u^4 + u^2 + 1)} \ du 2\int \frac{1 + \frac{1}{u^2}}{u^2 + 1 + \frac{1}{u^2} } \ du t = u - \frac{1}{u} . . . and so on ========= \int \frac{dx}{ax^2 + bx + c}

Re: HSC 2013 2U Marathon $A particle is moving with a displacement$ \ \ x \ $respect to the origin, that moves such that$ x= a\cos(n t) \ \ \ $for time$ \ \ t $i) Show that$ \ \ \ddot{x} = -n^2 x $ii) Find the displacement, velocity and acceleration when$ \ \ t = \frac{1}{2} a

Re: HSC 2013 2U Marathon My bad I just tried to make a weird substitution quadratic