Mathematics Marathon HSC 09 (4 Viewers)

[ForrestGump]

Okay time for a new question:

If a sector har an arc length of 3pi and a radius of 4units,
a) what is the angle (exact)
and b) calculate the area of the sector.

 

[s2Vicki]

Okay time for a new question:

If a sector har an arc length of 3pi and a radius of 4units,
a) what is the angle (exact)
and b) calculate the area of the sector.

Answer:
a) arc length=radius * theta
3pi = 4 theta
theta = 3pi / 4

b) A=0.5 * (r^2) * theta
= 0.5 * (4^2) * (3pi/4)
= 24pi/4
= 6pi units sq.

 

[s2Vicki]

Question:

Express (4x + 1)/[(x^2)-x-2] in the form [a/(x-2)] + [b/(x+1)].

 

[ForrestGump]

Answer:
a) arc length=radius * theta
3pi = 4 theta
theta = 3pi / 4

b) A=0.5 * (r^2) * theta
= 0.5 * (4^2) * (3pi/4)
= 24pi/4
= 6pi units sq.

Tick!

 

[s2Vicki]

Tick!

can you answer my question?
or give me another question. i'm bored.

 

[xFusion]

Question:

Express (4x + 1)/[(x^2)-x-2] in the form [a/(x-2)] + [b/(x+1)].

a(x+1) +b(x-2)=4x + 1
a +b=4 --->a=4-b
a-2b=1
4 -3b=1 ---> b=1, a=3
hence [3/(x-2) + 1/(x+1)]

Question: The probability of a win is 27%. What is the probability that Bobbie will win 3 or more games out of 5.

 

[ForrestGump]

im still working on it
and heres another one:

Solve the pair of simultaneous equations: x-2y = 8 and 2x + y = 1

 

[00iCon]

im still working on it
and heres another one:

Solve the pair of simultaneous equations: x-2y = 8 and 2x + y = 1

multiplying (2) by 2
4x+2y=2
adding together
5x=10
x=2
2*2+y=1
y=-3
check
2-(-6)=8
YAY!

 

[ninetypercent]

we need another question..

 

[xFusion]

a(x+1) +b(x-2)=4x + 1
a +b=4 --->a=4-b
a-2b=1
4 -3b=1 ---> b=1, a=3
hence [3/(x-2) + 1/(x+1)]

Question: The probability of a win is 27%. What is the probability that Bobbie will win 3 or more games out of 5.

answer my question?

 

[untouchablecuz]

a(x+1) +b(x-2)=4x + 1
a +b=4 --->a=4-b
a-2b=1
4 -3b=1 ---> b=1, a=3
hence [3/(x-2) + 1/(x+1)]

Question: The probability of a win is 27%. What is the probability that Bobbie will win 3 or more games out of 5.

P(3/5 or more)=1-P(2/5 or 1/5 or 0/5)

P(2/5 or 1/5 or 0/5)=(0.27)²(0.73)³(⁵C₂)+(0.27)¹(0.73)⁴(⁵C₁)+(0.73)⁵=0.8742~0.87

.'. P(3/5 or more)=1-0.87=0.13

edited due to xFusion

 

[xFusion]

P(3 or more games out of 5)=1-P(2 games out of 5 won or 1 game out of 5 won)

P(2 games out of 5 won or 1 game out of 5 won)=(0.27)²(0.73)³(⁵C₂)+(0.27)¹(0.73)⁴(⁵C₁)=0.66697~0.7

.'. P(3 or more games out of 5)=1-0.7=0.3

correct?

You forgot no wins

 

[ForrestGump]

Question: The probability of a win is 27%. What is the probability that Bobbie will win 3 or more games out of 5.

Lol i tried drawing a massive tree diagram but when it went over an A4 page i gave up hahaha

 

[00iCon]

P(3/5 or more)=1-P(2/5 or 1/5)

P(2/5 or 1/5 or 0/5)=(0.27)²(0.73)³(⁵C₂)+(0.27)¹(0.73)⁴(⁵C₁)+(0.73)⁵=0.8742~0.87

.'. P(3 or more games out of 5)=1-0.87=0.13

correct?

Crap i did the 5c3, 5c4 and 5c5, but then my iexplore.exe failed and i i got a syntax error on my calc.

 

[00iCon]

question 1)a)

find x and y
3 marks
EDIT: BTW Tech ext1 trial '02

 

[untouchablecuz]

question 1)a)

find x and y
3 marks
EDIT: BTW Tech ext1 trial '02

assuming [x, y] are integral

(x+2y)+2√(x-y)=(11)+√(20)

(x+2y)+√4(x-y)=(11)+√(20)

comparing the rational and irrational parts

x+2y=11 → x=11-2y [1]

4(x-y)=20 → x-y=5 [2]

sub [1] in [2]

11-2y-y=5 → y=2

sub in [1] → x=7

find k if the roots of the equation x²-kx+1=0 differ by 6

 

[xFusion]

find k if the roots of the equation x²-kx+1=0 differ by 6

let roots be @, @ +6
2@ +6=k -------->@=(k-6)/2 -----1
@(@ +6)=1----------2
sub 1 in 2
(k-6)/2[(k-6)/2 +6)=1
k^2=40
k=√40

Question: Show that y=e^x can be written as x=1/2 lny

 

[00iCon]

assuming [x, y] are integral

(x+2y)+2√(x-y)=(11)+√(20)

(x+2y)+√4(x-y)=(11)+√(20)

comparing the rational and irrational parts

x+2y=11 → x=11-2y [1]

4(x-y)=20 → x-y=5 [2]

sub [1] in [2]

11-2y-y=5 → y=2

sub in [1] → x=7

find k if the roots of the equation x²-kx+1=0 differ by 6

XD stupid techers, question did not assume anything. I graphed this on wolframalpha.com and it's a function. so any set of coordinates on the function would satisfy the question.

 

[ninetypercent]

find k if the roots of the equation x2-kx+1=0 differ by 6

Let the roots be a + 6 and a

a(a+6) = 1
a^2 + 6a = 1
2a + 6 = k

(k - 6)/2 = a
[(k-6)/2]^2 + [6 x (k-6)/2] = 1
(k^2 - 12k + 36)/4 + 3k - 18 = 1
k^2 - 12k + 36 + 12k - 72 = 4
k^2 - 40 = 0

k = +/- root 40

damn got beaten

 

[00iCon]

Question: Show that y=e^x can be written as x=1/2 lny

ummm. it cant?
x=ln y
rite guys?