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Hi all, I believe Conics was a topic in Extension 2 a while ago. I was just wondering if anyone had any old PDFs of Extension 2 text books that covered this topic. It more just form my personal interest that anything else. I wold really like to know how it was covered. Thanks

As a matter of interest, where does this question come from?

Here is the solution written in Latex.

Yes, that's the way I did it.

This looks very similar to the following problem: \begin{align*} \text{Given} \\ I_n &= \int \frac{x^n}{1 + x^2} dx \\ \text{show} \\ I_n &= \dfrac{x^{n - 1}}{n - 1} - I_{n - 2} \\ \text{From this expression $I_{n-2} = \int \frac{x^{n-2}}{1 + x^2} dx$ is required. }\\\\ \text{We can get...

Here is part a) and b) written in Latex. I'll post part c) soon. For c) use Vieta's formula. Attached are full solutions + a file about Vieta's formula for cubics.

This may help. The PDF was written using Latex.

The solution in Latex: \begin{align*} \text{Consider $I_n = \int_0^{\sqrt{3}}(3 - x^2)^n dx$} \\[5mm] \text{Using integration by parts } \\[5mm] u &= (3 - x^2)^n \hspace{60pt} v' = 1 \\[5mm] u' &= n(3 - x^2)^{n-1} \cdot -2x \hspace{20pt} v = x \\[5mm] I_n &= uv' - \int...

Here are a few from my favourite Year 12 Applied Mathematics book - with answers

\int \frac{1}{4 + \sqrt{x} } \, dx Let u = x^{1/2} \Rightarrow u^2 = x \begin{align*} u^2 &= x \\ \text{Differentiate both sides with respect to $x$} \\ \frac{d}{dx} (u^2) &= \frac{d}{dx}x \,\, \text{Apply chain rule to LHS} \\ \frac{d}{du} (u^2) \frac{du}{dx } &= 1 \\ 2u...

Part (c) and (d) involves Vieta's Formula for cubics. This PDF may help is solving such problems. Created in Latex.

Got it - thanks.

Thank-you very much Lith_30. I'll edit those domain restrictions to page 2.

Attached is a complete solution written in Latex

Can you please tell me where you got this question. I like it a lot!

Can you please expand this answer. Which mathematics subject is it taught? Is it an optional area of study or compulsory? I would appreciate a little more information.

Hi all, Excuse my ignorance since I do not live in NSW. Are permutations and combinations taught at stage 6 level? Thanks in advance.

For part a), write z^6 - 1 = 0 as \left[z^2\right]^3 - 1^3 = 0 a difference of 2 cubes. Recall that a^3 - b^3 = (a - b)(a^2 + ab + b^2), therefore \left[z^2\right]^3 - 1^3 = (z^2 - 1)(z^4 + z^2 + 1) The roots where z \in Q are z = \pm1 . b) Consider z^6 - 1 = 0 as z^6 = 1 In...

Thanks.... I was sort of on the right track, but your solution has clarified my confusion. Cheers.

Hi all, This question comes from Cambridge Chapter 3(C) Q15. a) Show that every root of the equation is imaginary for (1+z)^{2n} + (1-z)^{2n} =0 b) Let the roots be represented by the points P_1, P_2, P_3, ... ,P_{2n} in the Argand diagram, and let O be the origin. Show that: OP^2_1 +...