yr 11 revision (1 Viewer)

[budgie]

hello, i've got some year 11 revision questions to ask...

1. What does the graph of the polar equation r = a (1+cos@) where r is the distance to the origin of any point P on the curve and @ is the angle that OP makes with the positive side of the x-axis (counted anticlockwise) look like?? i don't understand what it's supposed to be.

2. A straight line AB of length 10 units is free to move with its ends A and B on the x-axis and y-axis respectively. Find the locus of a point P (x,y) on the line AB at a distance of 3 units from A.

3. Given a function f of a real variable which satisfies the function equation
f(x) + x f(1-x) = 1 + x^2
for all real x
determine f(x)
note; none of those x's mean 'times'.. they're all 'EX'-es...

4. if x and y are real numbers, find the minimum value of teh function
f(x,y) = 4x^2 + y^2 - 4x + 6y + 3

5. If p, q are the roots of the equation ax^2 - bx + c = 0, form the equation with the roots (p + 1/p), (q + 1/q)

i feel quite inadequate...
thanks to anyone who can help!

 

[Xayma]

Originally posted by budgie
hello, i've got some year 11 revision questions to ask...
2. A straight line AB of length 10 units is free to move with its ends A and B on the x-axis and y-axis respectively. Find the locus of a point P (x,y) on the line AB at a distance of 3 units from A.

This is what I got.

D²_PA=9=(x-A)²+y²

D²_PB=49=x²=(y-B)²

9=x²-2xA+A²+y²
49=x²+y²-2yB+B²
x²+y²=49+2yB-B<sup>2</SUP>
9=49-B²+2yB-2xA+A²
40=B²-A²+2xA-2yB
40=B(B-2y)-A(A-2x)

Bah thats alot of sups

 

[Affinity]

sounds more like someone preparing for olympiad and mucking around here.

anyway, the first graph is called a cardiod, it looks like a heart, why don't you plot it with some software or just search cardiod up the net

the second question.

say A is on the point (a,0) and B is on the point (0,b)
since the only constraint is a^2 + b^2 = 10^2

we can parameterize a and b to

a=10cos(t)
b=10sin(t)
for t in [0,2pi)

the point P has coordinate

(7/10)A + (3/10)B = (7cos(t),3sin(t))

and this can be seen to be an ellipse with cartesian equation:

x^2/49 +y^2/9 = 1

question 3

Substitue x = (1-y) into the equation

f(1 - y) + (1-y)f(y) =2 - 2y +y^2

change y to x

f(1-x) + (1-x)f(x) = 2 - 2x + x^2

multiply through by x

xf(1-x) +x(1-x)f(x) = 2x - 2x^2 + x^3

subtract the original equation from this

(-x^2 + x - 1)f(x) = x^3 -3x^2 +2x -1

notice -x^2 + x - 1 has a negative determinant, it cannot equal 0

so f(x) = (x^2 -3x^2 +2x -1)/(-x^2 + x -1)

question 4

you complete the squares here

f(x,y)=(2x-1)^2 +(y+3)^2 - 7

so the minimum value is -7 when x= (1/2) and y = -3

question 5.

well we have p+q = b/a (yes I noticed the negative sign)
and pq=c/a

(p+1/q)+(q+1/p)= p+q + (p+q)/pq = (b/a) + (b/a)/(c/a)

= b/a + b/c

(p+1/q)*(q+1/p) = pq + 1/pq + (p^2 +q^2)/pq

= c/a + a/c + [(b/a)^2 - 2(c/a)] / (c/a)

= c/a + a/c + b^2/ac - 2

so the equation is

x^2 -[(b/a) + (c/a)]x + (c/a + a/c + b^2/ac - 2] = 0

 

[Xayma]

stupid squashed circles... make me not have an elegant answer

 

[Affinity]

perhaps you might get more responses if you posted 'year 11' question in the preliminary forum

 

[Calculon]

Originally posted by Grey Council
whoa. You did olympiad affinity? Or did/do you do this stuff at uni?
and no wonder it looked so damn hard. meh, olympiad. if turtle was here, he'd have a field day.

PS, did you know that turtle taught himself the 3u and 4u course in year 9? wtf, how absolutely crazy is that?!?

Thats absolutely f***ing insane, how come he didn't beat keypad?

 

[iambored]

Originally posted by Calculon
Thats absolutely f***ing insane, how come he didn't beat keypad?

i'd like to know that as well!

 

[Calculon]

Hang on, i just searched board of studies website, turns out turtle topped it in 2002

 

[Rahul]

keypad in 2003
turtle in 2002
me in 2001

 

[Calculon]

Originally posted by Rahul
keypad in 2003
turtle in 2002
me in 2001

Hi Hugh Miller from Barker College Hornsby!

 

[Rahul]

hello! how do you do. would love to sit and chat, but i have to attend important matters. good-bye. [add accent for effect]

 

[Xayma]

Does that mean a member of the board has to top it this year as well? *looks around (go for the hat trick)

 

[freaking_out]

Originally posted by Rahul
keypad in 2003
turtle in 2002
me in 2001

tell us mr rahul...why can't i see your name on the board of studies website?

Originally posted by Affinity
perhaps you might get more responses if you posted 'year 11' question in the preliminary forum

yeah, way better chance to get them answered!

 

[Calculon]

Originally posted by Xayma
Does that mean a member of the board has to top it this year as well? *looks around (go for the hat trick)

You don't even do 4u Xayma, its gonna be me, even though I'm only coming 10th in the class, oh well theres always Bio

 

[Xayma]

Originally posted by Calculon
You don't even do 4u Xayma,

I know by *looking around* I was meaning *Looks for someone to do it,

 

[Calculon]

So who do you reckon is the biggest maths nerd of '04

 

[freaking_out]

Originally posted by Calculon
So who do you reckon is the biggest maths nerd of '04

abdoooo.

 

[Calculon]

i reckon grey council

 

[freaking_out]

Originally posted by Calculon
i reckon grey council

hehe, yeah u prolly saying that since u seem to have a grudge against abdooo.

 

[Xayma]

Stuff trying to find one on BOS you have to get the already super maths nerds onto BOS.