Extension II Mathematics Game (1 Viewer)

[gurmies]

Hey guys, just thought that the Extension II section is usually a tad empty, and with upcomming assessments, I thought it would be a good idea to reincarnate this thread/game. At this point i'd assume the majority of questions will be targeting complex numbers/polynomials/conics but feel free to post any harder 3 unit or anything for that matter =)

Rules are same as always, one question at a time, and person who answers a question should post up one of their own. Try to make the questions quite difficult and unusual otherwise there'd be no challenge.

I'll start things off with a strange locus question:

Prove that if z lies on the circle x² + y² = 1, the points representing Z = √[(1+z)/(1-z)] lie on an orthogonal line pair (Lines at 90° to eachother)

 

[azureus88]

Z = √[(1+z)/(1-z)]

Z^2 = (1-z)/(1+z)

(1-z)Z^2 = (1+z)

z(Z^2 - 1) = Z^2 - 1

z = (Z^2 - 1)/(Z^2 + 1)

|z| = |Z^2 - 1|/|Z^2 + 1| = 1 since x^2 + y^2 = 1

|Z^2 - 1| = |Z^2 + 1|

let Z = X+iY

(X^2 - Y^2 - 1)^2 + (2XY)^2 = (X^2 - Y^2 + 1)^2 + (2XY)^2

4(X^2 - Y^2) = 0

X^2 = Y^2 is locus which lies on an orthogonal line pair.

 

[gurmies]

Very nicely done. My method was different =) Post a question?

 

[azureus88]

[maths]$Let the points A_1, A_2,...A_n$ represent the nth roots of unity, w_1, w_2,...,w_n,$and suppose P represents any complex number z such that \left |z \right |=1[/maths]

[maths](i)$Prove that w_1+w_2+...+w_n=0$[/maths]

[maths](ii)$Show that PA^2_i=(z-w_i)(\overline{z}-\overline{w_i}) for i=1,2,...,n$[/maths]

[maths](iii)$Prove that \sum_{z=1}^{n}PA_i^2=2n [/maths]

 

[Templar]

Try using the LaTeX editor here, it's really annoying to go off and read each line separately.

And why doesn't the TeX mode support \begin{itemize}?

 

[gurmies]

(i)






(ii)

 

[Templar]

(iii) No clue yet...

You post question.

 

[gurmies]

The locus of the complex number is defined by the equation

Find the least value of

 

[Trebla]

The locus of the complex number is defined by the equation

Find the least value of

The locus basically looks like y = x + 1 for y > 0.
The shortest distance from the origin is the minimum value of |z|

min |z| = |x - y + 1| / √2
Sub the origin gives min |z| = 1/√2

 

[Trebla]

Question:

Consider a regular pentagon ABCDE with each side being 1 unit. Let the length of the diagonals be φ.

(i) Show that the length of the diagonals are given by φ = 2cos (π/5)

(ii) Show that φ is a solution of x² - x - 1 = 0 (use the pentagon, not manual calculation)

(iii) Hence deduce the exact value of 2cos (π/5)

 

[azureus88]

Question:

Consider a regular pentagon ABCDE with each side being 1 unit. Let the length of the diagonals be φ.

(i) Show that the length of the diagonals are given by φ = 2cos (π/5)

(ii) Show that φ is a solution of x² - x - 1 = 0 (use the pentagon, not manual calculation)

(iii) Hence deduce the exact value of 2cos (π/5)

stuck on part (ii). Hint plz?

 

[lolokay]

might help:
φ is a solution to x - 1 - 1/x = 0

 

[Trebla]

BIG hint: Consider triangle ADC...

For anyone who is interested, the number φ in this question is a special number known as the "golden ratio". It has significance in arts and nature. It's also the ratio that any two adjacent Fibonacci numbers converges to...

One of the many neat expressions of it is

 

[azureus88]

ok i got it, but its kinda hard to explain without a diagram:

In regular pentagon ABCDE, construct AD and AC and BD. BD and AC intersect at a point X.

(i) φ/2 = cos(π/5) [by drawing pendicular from side to diagonal]
φ = 2cos (π/5)

(ii) Let AD = AC = x

angle(AED)= 540/5=108
angle(ADE)=angle(ADB)=angle(BDC)=108/3=36 (since equal chords subtend equal angles at circumference)
angle(BDE)=angle(ADE)+angle(ADB)=72
so AE is parallel to BD and by similar argument, DE is parallel to AC.
so AEDX is a rhombus and AX=1

triangle(BCX) is similar to triangle(ADX) [equiangular]
therefore x/1 = 1/(x-1)

φ is a solution of x² - x - 1 = 0

(iii) φ = 2cos (π/5) = [1+sqrt(5)]/2 by quadratic formula

 

[gurmies]

^ Well done =) Post question?

 

[azureus88]

Prove by induction that a regular polygon with n sides (where n is even) has at least one side parallel to a diagonal.

 

[tommykins]

Prove by induction that a regular polygon with n sides (where n is even) has at least one side parallel to a diagonal.

doesn't work with a square ? :S

 

[azureus88]

doesn't work with a square ? :S

lol yeh, i mean n is even and is bigger than or equal to 6. actually the question sucks. someone else can ask another one.

 

[Trebla]

Suppose that p is the probability that event A occurs and q is the probability event A does not occur. Prove that:

 

[shaon0]

Suppose that p is the probability that event A occurs and q is the probability event A does not occur. Prove that:

n.p^(n-1)=A
not A=1-(n.p^(n-1))
= (p-p^(n))/p

stuff it i don't even know probability.