2u Mathematics Marathon v1.0 (1 Viewer)

[word.]

Inspired by other boards because they worked well, I thought it'd be interesting to start a Mathematics Marathon here... although it might not work as well because only a few people come here.

Here's how it goes: You answer the previous question and post a new one.
You don't necessarily have to be able to answer your own questions although it'd be good so you can correct anyone that makes a mistake.

by the time the 2U hsc comes around we should have a large collection of questions that we can go over and just test ourselves...

Make sure to use spoiler tags when answering so other people can try the questions.

For example:
Question: What is 7 times 2?

Answer: [sp0iler]7 * 2 = 14[/sp0iler]
replacing the zero's with the letter o will output:

Spoiler

7 * 2 = 14

Also put which question number it is to avoid confusion. And try keep it 2U level. I also would imagine multiple people answering the same question... if they beat you to a question by a few minutes or seconds it'd be better if you delete your post or edit your post to answer their question instead.

Also try to show all working instead of just an answer.

Without further ado:

QUESTION 1
Two fair dice are thrown at the same time. What is the probability that the two numbers which appear uppermost on the dice are either both fives or both sixes?

 

[SaHbEeWaH]

Answer to Q1:

Spoiler

there is only 1 combination of getting two 5's.
and there is only 1 combination of getting two 6's.
there are a total of 6*6 = 36 combinations.
so the answer is 2/36 = 1/18

Question 2
If y = 2x²ln(5x) then find d²y/dx².

 

[AntiHyper]

Ans to Q2 (hope i'm right):

Spoiler

Product Rule: uv' +vu'
dy/dx = y'
= 2x²5/5x + ln(5x)4x = 10x²/5x + 4x*ln(5x)
= 2x + 4x*ln(5x) = 2x( 1 + 2ln(5x) )
d²y/dx² = y''
= 2x*10/5x + (1+2ln(5x))*2 = 20x/5x + 2 + 4ln(5x)
= 4x + 2 + 4ln(5x) = 2( 2x + 1 + 4ln(5x))

That was more of a typing excercise than maths with all the [ sup][ /sup]

 

[word.]

20x/5x = 4

anyway post up a new question

 

[switchblade87]

Well, I'll post one if no one else will.

Q3: A frog jumps 0.5m. It then jumps 0.1m and on each subsequent jump travels 0.2m of the previous distance. Find the distance through which the frog jumps.

 

[Riviet]

Here's the solution to Q3:

Spoiler

a=0.5, r=0.2
Sum (infinity)=a/(1-r)
=0.5/(1-0.2)
=0.5/0.8
=0.625m
.: The frog jumps a total distance of 0.625m

Q4: If the 11th term of an AP is 120 and the 4th term is 71, find the first term and common difference. Hence find the sum of the first 100 terms.

 

[Antwan23q]

or double?

 

[Riviet]

I think he means 1/5th of the previous jump because the 2nd jump (0.1m) is one fifth of the first jump (0.5m)

 

[Trev]

Riviet is correct.

 

[switchblade87]

Spoiler

t11 = a + 10d = 120 (1)
t4 = a + 3d = 71 (2)
(1) - (2): 49 = 7d
d = 7
At d = 7, (2): 71 = a + 21
a = 50
.: S100 = 50[100 + (99 * 7)
S100 = 39650

Hope thats right.

Q5. Find the gradient of the tangent to the parabola x² = 16y at the point (4,1).

 

[Antwan23q]

Spoiler

x²=16y tangent at (4,1)
x²/16=y
dy/dx = 2x/16
dy/dx = x/8
gradient at x=4
m=4/8
m=1/2

Integrate:
(e^x)dx
(e^x+1)

(thats divide)

 

[skyrockets1530]

Spoiler

ln(e^x + 1)

Q7
find the equation of the tangent to the graph y = 1 + sqrt(3)sinx + cosx at x = 5pi/6

 

[Riviet]

Correct me if i'm wrong ^^
Answer to Q7:

Spoiler

dy/dx=sqrt3cosx-sinx
y co-ordinate is (5+sqrt3)/2
m=-3/2-1/2
=-2
y-y1=m(x-x1)
y-(5+sqrt3)/2=-2(x-5pi/6)
y-(5+sqrt3)/2=-2x+5pi/3
6y-15-3sqrt3=-12x+10pi
12x+6y-10pi-15-3sqrt3=0

Q8 If a and b are the roots of the equation 3x²+4x-5=0, without finding the roots, find a³+b³

 

[word.]

I got a different answer to Q7
I think you made an error finding the y co-ordinate:

Spoiler

y = 1 + sqrt(3)sinx + cosx
dy/dx = Sqrt(3)cosx - sinx
at x = 5pi/6,
m = dy/dx = Sqrt(3)*(-Sqrt(3)/2) - 1/2 = -2

at x = 5pi/6, y = 1 + Sqrt(3)*1/2 + (-Sqrt(3)/2) = 1

by point-gradient formula:
y - 1 = -2(x - 5pi/6)
y - 1 = -2x + 5pi/3
so the req'd equation is 2x + y - (1 + 5pi/3) = 0

Answer to Question 8:

Spoiler

Note: (a + b)³ = a³ + 3a²b + 3ab² + b³
= a³ + 3ab(a + b) + b³
so a³ + b³ = (a + b)³ - 3ab(a + b)

sum of roots(a + b) = -b/a = -4/3
product of roots(ab) = c/a = -5/3

so a³ + b³ = (-4/3)³ - 3*(-4/3)(-5/3) = -244/27

Question 9

Use the trapezoidal rule with three function values to find an approximate value of the definite integral of (x+1)^-1 from 2 to 4.

 

[klaw]

Question 9

Spoiler

(1/2)[(1/3)+(1/5)+2(1/4)]
=31/60

Question 10
Prove x²+y²>=2xy, where x and y are integers

 

[Riviet]

Yeah, i forgot that cosx is negative in the 2nd quadrant and sin30 is 1/2, i was kinda suspicious when i got the irrational y-coordinate

 

[jake2.0]

Question 9

Question 10
Prove x²+y²>=2xy, where x and y are integers

(x-y)² >=0
x² -2xy + y² >=0
x²+y²>=2xy

 

[Riviet]

Jake you didn't do 2 important things:
1) You didn't hide the solution in a

Spoiler

tag
2) You didn't provide the next question

 

[word.]

kickstart:

Question 11
Consider the geometric series: 1 + lnx + (lnx)² + ...
Find the values of x for which the limiting sum of this series exists.

 

[klaw]

Question 11

Spoiler

S_n=a(1-rⁿ)/(1-r)
S_infinite=a/(1-r), |r|<1
.: |ln x|<1
.:1/e is less than x which is less than e

Question 12
Show that d/dx (3^3x+1)=3^3x+2ln3