HSC Revision Thread 2006 (1 Viewer)

[bboyelement]

help me with this question please

 

[Riviet]

bboyelement: I know you need the previous bit of the question as this continues on, it's not that hard actually, if you use one or two of the previous results in that question.

 

[3li]

actualli u dont need previous parts, i just used a vector, pythag identity, and roots of unity (DMT)

 

[vafa]

Solution:
View attachment 13980

View attachment 13981

View attachment 13982

 

[vafa]

Eventually one of my attach look nice!

 

[Riviet]

Eventually one of my attach look nice!

Yeah, especially the first one! So much clearer, now I'm actually going to try reading.

 

[Trebla]

Help with these questions please (preferably before tonight). Hope you can read my writing.....Thanks.

 

[shimmerz_777]

help me with this question please

yea that question doesnt make sense to me either, i had a look at the answers posted, but even with the first bit its still confusing, i keep ending up with1 as the answer. like bar l1wl is how i see that, which just = 1? what am i missing or not getting. ill check over the answer again

 

[shimmerz_777]

Help with these questions please (preferably before tonight). Hope you can read my writing.....Thanks.

1

let ST = x, PS=b RS=a (althought typo says e in answer)

by alternate angles on parallel lines,
BST=STP=@ (@ = pheta)

tan@= b/x
x=bcot@

volume of water V= ((b^2)cot@)/2

when it is turned, V=ah
((b^2)cot@)/2 = ah
((b^2)cot@)/2a = h, as required.

 

[shimmerz_777]

7i)
consider triangles MPV, WPM and VWP
PV=PW=s (radii of same circle)
hence VWP is iscololese
(similiar proof for OVW reveals that OVPW is a kite)
diagonals of a kite intersect at right angles, if that isnt a general proof u can use angle sum of triangles to prove that PMV=90

ii)OT^2 = OU^2 + TU^2 (pythagaras in OPV)
PT^2= TU^2 +PU^2 (pythagaras in PTV)
hence OT^2 - PT^2 = OU^2 - PU^2

OM^2= r^2 - MW^2 pythagaras.....
PM^2 = s^2- MW^2 pythagaras....
OM^2 - PM^2 = r^2 - S^2

iii) when the point T is moved onto the line WV, the point U lies on point M
hence M=U, substituting M for U
OU^2 – PU^2 = OM^2 – PM^2 = r^2 – s^2
Hence now
OT^2 – PT^2 = r^2 – s^2

havent got the last question though...

 

[3li]

yea that question doesnt make sense to me either, i had a look at the answers posted, but even with the first bit its still confusing, i keep ending up with1 as the answer. like bar l1wl is how i see that, which just = 1? what am i missing or not getting. ill check over the answer again

k my methods pretty fast
draw vectors of unity 5 with mod 1
use tip to tail then ull find that vector from P1 - P0 is w-1
so to find mod of w-1 let w = cis2pie/5 cos its roots of unity remember each root rotated by cis 2pie/5

then mod is square root of{ (cos2pie/5-1)^2 + sin^2 2pie/5}
use pythag, double angle and ull get the answer

similar for part ii u can draw more vectors, find more mod and times em together ....
its an alternative if u forgot cos rule like me

 

[3li]

7i)
consider triangles MPV, WPM and VWP
PV=PW=s (radii of same circle)
hence VWP is iscololese
(similiar proof for OVW reveals that OVPW is a kite)
diagonals of a kite intersect at right angles, if that isnt a general proof u can use angle sum of triangles to prove that PMV=90

ii)OT^2 = OU^2 + TU^2 (pythagaras in OPV)
PT^2= TU^2 +PU^2 (pythagaras in PTV)
hence OT^2 - PT^2 = OU^2 - PU^2

OM^2= r^2 - MW^2 pythagaras.....
PM^2 = s^2- MW^2 pythagaras....
OM^2 - PM^2 = r^2 - S^2

iii) when the point T is moved onto the line WV, the point U lies on point M
hence M=U, substituting M for U
OU^2 – PU^2 = OM^2 – PM^2 = r^2 – s^2
Hence now
OT^2 – PT^2 = r^2 – s^2

havent got the last question though...

i did the last 1 a while back
simultaneously solve for two of the 3 lines then realise that the third line satisfies the equation. so all three pass thru it

 

[gamecw]

p( x)=3x^4-11x^3+14x^2-11x+3
solve p( x) over complex field + factorise P(x ) over real
do this one plz!

 

[Mill]

divide through by x^2 since x != 0

then use some incredibly sexy complex results (let x = cis @)

 

[toadstooltown]

These ones are a bit weird in the way you have to approach them. You know it's one of them when the coëfficients 'rise and fall' but aren't a binomial expansion.
3x^4-11x³+14x²-11x+3
x²(3x²-11x+14-11/x+3/x²)
x²[3(x²+1/x²) -11(x+1/x) +14]
x²[3({x+1/x}²-2) -11(x+1/x) +14] - completing the square to make a quadratic
x²[3(x+1/x)²-11(x+1/x)+8]
x+1/x = [11±5]/6 - from quadratic formula
x+1/x = 8/3, 1
So,
x²(x+1/x-8/3)(x+1/x-1)
x²(3x+3/x-8)(x+1/x-1) - as we note the leading coëfficient
(3x²+3-8x)(x²+1-x) - multiplying each bracket by x
x = (4±sqrt7)/3, (1±isqrt3)/2 , x over C- from solving each quadratic
P(x)=3(x²-x+1)(x-4/3-sqrt7/3)(x-4/3+sqrt7/3) factorised over R. (NB sqrt7/3 = (sqrt7)/3)

If you sub the reals in at least you get 0, can't be bothered checking the Im answers. Let me know if I've made a mistake.

 

[gamecw]

These ones are a bit weird in the way you have to approach them. You know it's one of them when the coëfficients 'rise and fall' but aren't a binomial expansion.
3x^4-11x³+14x²-11x+3
x²(3x²-11x+14-11/x+3/x²)
x²[3(x²+1/x²) -11(x+1/x) +14]
x²[3({x+1/x}²-2) -11(x+1/x) +14] - completing the square to make a quadratic
x²[3(x+1/x)²-11(x+1/x)+8]
x+1/x = [11±5]/6 - from quadratic formula
x+1/x = 8/3, 1
So,
x²(x+1/x-8/3)(x+1/x-1)
x²(3x+3/x-8)(x+1/x-1) - as we note the leading coëfficient
(3x²+3-8x)(x²+1-x) - multiplying each bracket by x
x = (4±sqrt7)/3, (1±isqrt3)/2 , x over C- from solving each quadratic
P(x)=3(x²-x+1)(x-4/3-sqrt7/3)(x-4/3+sqrt7/3) factorised over R. (NB sqrt7/3 = (sqrt7)/3)

If you sub the reals in at least you get 0, can't be bothered checking the Im answers. Let me know if I've made a mistake.

thanks

 

[Trebla]

i did the last 1 a while back
simultaneously solve for two of the 3 lines then realise that the third line satisfies the equation. so all three pass thru it

I know that's what you have to do. I just don't know how to do it.

 

[toadstooltown]

Nah, first time I came across one of those I spent half an hour blindly guessing roots untill I finally gave up and asked my teacher. It was frustrating so wouldn't wanna leave someone else equally as frustrated right before the exam!

 

[Trebla]

Omfg!!!!!! If You Read The Attachment I Posted Earlier, You Would Realise That Question 2 On That Attachment Appeared In This Year's Paper In A Similar Form!!!!!!

 

[Riviet]

Omfg!!!!!! If You Read The Attachment I Posted Earlier, You Would Realise That Question 2 On That Attachment Appeared In This Year's Paper In A Similar Form!!!!!!

Well I guess that gives you a slight unfair advantage.

Closing the thread since the exam's over.