2u Mathematics Marathon v1.0 (1 Viewer)

[DeanM]

Q73 answer:

Q74: A army base has 2 defence guns. The first one has a success rate of 0.8 of destroying incoming enemy aircraft and the second has a success rate of 0.9.
a) Find the probability of an enemy aircraft passing through both guns without being shot down.
b) Find the probability of either the first or second gun shooting down the first 100 enemy aircraft that attack the army base.

Spoiler

P(gun 1 letting one in ) = 1/10
P(gun 2 letting one in ) = 1/5

a) P(both fail) = 1/10 X 1/5
=1/50
b)not to sure... ( not even sure about part a)
but ill give it ago..
is it 49/50 ? OR is it

 

[icycloud]

Q74: A army base has 2 defence guns. The first one has a success rate of 0.8 of destroying incoming enemy aircraft and the second has a success rate of 0.9.
a) Find the probability of an enemy aircraft passing through both guns without being shot down.
b) Find the probability of either the first or second gun shooting down the first 100 enemy aircraft that attack the army base.

Spoiler

(a) P(1st fails) = 0.2
P(2nd fails) = 0.1
P(both fails) = 0.2 * 0.1 = 0.02

(b) I'm taking this to mean the probability of all 100 aircraft being shot down? Then, it would be:

P(an aircraft gets taken down) = 1 - 0.02 (from part a)
= 0.98

Thus, P(all 100 aircrafts gets taken down) = (0.98)¹⁰⁰
= 0.1326 (4dp) #

Please tell me if that is not what you meant by part (b) of the question...

Question 75:
A random number generator can generate numbers from 1 to 1024 which are multiples of 3 and 7. What is the probability that the same two numbers are generated in a row, assuming that the generator is fair?

 

[DeanM]

solution to some question... (bit late now)

Spoiler

10000(1 + r/100)¹⁰=35478
divide both sides by 10000
so we have :
(1 + r/100)¹⁰=3.5478
take the 10th root
so we now have:
1 + r/100 = 1.350
therefore r = 13.5

Question 76
solve the pair of simultanous equations
x + y = 2
2x - y = 7

 

[yabby]

Q76

Spoiler

x=2-y ->sub into equation (2x-y=7)
2(2-y)-y=7
4-2y-y=7
-3y=3
y=-1 ->sub into other question
x+(-1)=2
X=3

Q77 integrate x+1/x

*BONUS* IDIOT QUESTION sinx/n

 

[lozabella]

Spoiler

1/2x^2 + lnx + c

new question.....
state the domain and range of
y= 2(square root of 25-x^2)

 

[yabby]

Spoiler

1/2x^2 + lnx + c

new question.....
state the domain and range of
y= 2(square root of 25-x^2)

can i see ur working on that...cos my answer is different

int x+1/x
= int x/x + 1/x
= int 1 + 1/x
= x + lnx + c

 

[Riviet]

(a) P(1st fails) = 0.2
Please tell me if that is not what you meant by part (b) of the question...

Yes, your interpretation is correct, i meant every single one of those 100 being blown up literally lol.
State the domain and range of
y= 2(square root of 25-x^2).
Answer:

Spoiler

Domain: 25-x²>=0
(5+x)(5-x)>=0
-5<=x<=5
Range: 0<=y<=10

Q78. Find the equation with roots -2 and 1.

 

[Riviet]

int x+1/x
= int x/x + 1/x
= int 1 + 1/x
= x + lnx + c

Answer Q77 (is it that hard?):

Spoiler

How the hell did you get from x to x/x??? Integrating x is as easy as it gets...
You simply add 1 to the index and divide by the new index.
int x= x¹⁺¹/(1+1)
=x²/2

.: int(x+1/x)=x²/2+log_ex+C

P.S Good luck for monday.

 

[yabby]

Q78. Find the equation with roots -2 and 1.[/QUOTE]

Answer:

Spoiler

Work backwards
(x+2)(x-1)
=x^2+x-2

Q79 (From 2001 HSC paper)
x=t-2/t+2

Show x=1-(4/t+2)
Hence find velocity and acceleration in terms of t

 

[yabby]

Answer Q77 (is it that hard?):

Spoiler

How the hell did you get from x to x/x??? Integrating x is as easy as it gets...
You simply add 1 to the index and divide by the new index.
int x= x¹⁺¹/(1+1)
=x²/2

.: int(x+1/x)=x²/2+log_ex+C

P.S Good luck for monday.

Oh shit my bad, i get what u guys were on about....the question was meant to be
Int (x+1)/x <----my bad!

 

[Riviet]

Q77 [corrected] Find the integral of (x+1)/x
Answer:

Spoiler

int (x+1)/x= int(x/x) + int(1/x)
=int(1)+int(1/x)
=x+lnx+C

 

[yabby]

Q77 [corrected] Find the integral of (x+1)/x
Answer:

Spoiler

int (x+1)/x= int(x/x) + int(1/x)
=int(1)+int(1/x)
=x+lnx

YEP! thats the question i was thinking off, not use to putting around brackets..
and dont forget the +c

 

[word.]

Spoiler

x = (t - 2)/(t + 2) = (t + 2)/(t + 2) - 4/(t + 2) = 1 - 4/(t + 2)
v = dx/dt = 4/(t + 2)²
a = dv/dt = -8/(t + 2)³

Question 80
Find the range of values of k such that the following simultaneous equations have two solutions:
y = x + k
2x² + y² = 6

 

[skyrockets1530]

Question 80
Find the range of values of k such that the following simultaneous equations have two solutions:
y = x + k
2x² + y² = 6

Spoiler

sub y in
2x² + (x + k)² = 6
2x² + x² + 2xk + k² = 6
3x² + 2xk + k² - 6 = 0
discriminant must be > 0
a = 3, b = 2k, c = k² - 6
(2k)² - 4(3)(k²-6)
let = 0 for now
4k² - 12k² + 72 = 0
-8k² + 72 = 0
k² - 9 = 0
k = +3 and -3
test on number plane
-3 < k < 3

Question 81
the area under the curve y = 1 / sqrt(x) for 1 <= x <= e² is rotated about the x-axis , find the exact volume of the solid of revolution

 

[icycloud]

Question 81
the area under the curve y = 1 / sqrt(x) for 1 <= x <= e^2 is rotated about the x-axis , find the exact volume of the solid of revolution

Spoiler

y = 1 / Sqrt(x)
V = pi * ∫(1/sqrt(x))^2 dx
bounds: 1 ----> e^2

V = pi * ∫ 1/x dx
= pi * [ln(x)] bounds 1--->e^2
= pi * [ln(e^2) - ln(1)]
= pi * [2]
= 2pi units³#

Question 82
Prove that sec²x [cos⁴x + 2cos³x sin x + sin²x - sin⁴x] = (cos x + sin x)²

 

[skyrockets1530]

thats some prove question!

 

[yabby]

Spoiler

y = 1 / Sqrt(x)
V = pi * ∫(1/sqrt(x))^2 dx
bounds: 1 ----> e^2

V = pi * ∫ 1/x dx
= pi * [ln(x)] bounds 1--->e^2
= pi * [ln(e^2) - ln(1)]
= pi * [2]
= 2pi units³#

Question 82
Prove that sec²x [cos⁴x + 2cos³x sin x + sin²x - sin⁴x] = (cos x + sin x)²

Spoiler

LHS=sec^2x(cos^4x + 2cos^3x.sinx + sin^2 - sin^4x)
= cos ^2x + 2cosxsinx +tan^2x + tan^2x sin ^2x
= cos ^2x + 2cosxsinx +tan^2x(1- sin^2x)
= cos ^2x + 2cosxsinx +tan^2x(cos^2x)
= cos ^2x + 2cosxsinx + sin^2x
factorise
=RHS!

Im here all night, hit me!
Q83
use the Simpson rule with 3 function values to find an approximation

Int (between 4 and 2) Ln(x-1)dx

 

[DeanM]

too tired to to even attempt that one.. someone give it ago....

 

[DeanM]

cant resist the simpsons...

Spoiler

sub x=2, 3, 4 into ln(x-1) to get the following...
1/3 { 0 + (4x0.6931) + 1.0986 }
~1.2904

someone else can post a question.. im off to bed.. for real this time..

 

[icycloud]

Alright then DeanM, here goes:
Question 84
Find the equation of the tangent to the curve y = e^xln(x) at x = 2.