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  1. ivanradoszyce

    Vectors q

    Here is the solution written in Latex.
  2. ivanradoszyce

    Vectors q

    Yes, that's the way I did it.
  3. ivanradoszyce

    4u reduction formula help

    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...
  4. ivanradoszyce

    Application of Complex Numbers question

    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.
  5. ivanradoszyce

    Application of Complex Numbers question

    This may help. The PDF was written using Latex.
  6. ivanradoszyce

    Reduction q

    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...
  7. ivanradoszyce

    Need Projectile Motion questions

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

    Easy Integral Question

    \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...
  9. ivanradoszyce

    Trig

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

    Complex q

    Got it - thanks.
  11. ivanradoszyce

    Complex q

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

    Complex q

    Attached is a complete solution written in Latex
  13. ivanradoszyce

    Complex q

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

    Combinatorics at Stage 6

    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.
  15. ivanradoszyce

    Combinatorics at Stage 6

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

    Question

    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...
  17. ivanradoszyce

    Roots of a complex number question

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

    Roots of a complex number question

    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 +...
  19. ivanradoszyce

    de moivre's question help

    Apologies for my ignorance, but what is "kurt"?
  20. ivanradoszyce

    de moivre's question help

    Can you please indicate what text book this came from. BTW I have the solution in PDF form if you wish.
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