VCE Methods Question Thread!

[Mr. Study]

I have seen the solutions to this question, and I have no idea where they pull off some of the numbers.

 Thanks for the help.

[Phy124]

Oh man, haven't done one of those since exam day

Firstly the area under the curve of a probability density function must be equal to one.

As 0.6 of this "one" is covered over the domain (0.3 x 2) and the function is 0 for

The remaining area of 0.4 must come from the domain .

This means that the function over this domain must be 0.1 (as 0.4/4 = 0.1) i.e. p = 0.1

Then to work out , you'll need to break down the function into it's separate parts (chosing the correct line segment for the domain you are integrating over) and integrate.



(* note:you don't actually have to include as anything where the function is 0 won't have any area under it)

Which you can just do using rectanges;

0 + 0.6 + 0.1 = 0.7

Alternatively you could do;



Which is 1 - 0.3 = 0.7

This latter method becomes more practical when you move into harder questions that instead of using straight line segments, use parabolas and other graphs in unison. In these you want to minimise the amount of separate functions you use when integrating, as it's quicker.

[Mr. Study]

Very very good explanation!

Thank you so much for that!

[soccerboi]

Could someone help me with this:
The simultaneous linear eqns
mx+12y=24
3x+my=m
have a unique solution for what values?

Answer is m=6 or m=-6
I dont understand how to get the above answers
Also, what does the term 'unique solution' mean? does it just mean 'have a solution'?
Thanks

[Phy124]

Could someone help me with this:
The simultaneous linear eqns
mx+12y=24
3x+my=m
have a unique solution for what values?

Answer is m=6 or m=-6
I dont understand how to get the above answers
Also, what does the term 'unique solution' mean? does it just mean 'have a solution'?
Thanks

So with simultaneous linear equations you can have three things;

Infinite solutions - this is when the two lines are the same (on top of each other [e.g. 2x + 2y = 2 is the same as x + y = 1]) and hence have the same y values for all x values and vice versa
No solution - this is when the two lines are parallel and do not have the same value at any point
Unique solution - this is when the two lines interact for one value (cross over).

So we are looking for a unique solution, so that they cross over each other.

A unique solution occurs when the determinant is not equal to zero.







You can show that the values of cannot give a unique solution, but instead no solution by subbing the values in to the simultaneous equations.

It will end up that the ratio of y's and x's is the same, but they have a different y-intercept.

[Lasercookie]

Here's an alternate way of working it out too.

mx+12y=24 ...(1)
3x+my=m ...(2)

- Rearrange for y




- Take the derivative of each




- so we're looking for when the derivative of (1) is not equal to (2), solve for m

[Hancock]

Could someone help me with this:
The simultaneous linear eqns
mx+12y=24
3x+my=m
have a unique solution for what values?

Answer is m=6 or m=-6
I dont understand how to get the above answers
Also, what does the term 'unique solution' mean? does it just mean 'have a solution'?
Thanks

Here's another way.

Find the determinent of this matrix.

mx+12y=24
3x+my=m

Which when you put in matrix form (for this question):

[m 12]
[3   m]

The determinent is m^2 - 36 and when this equals zero, those m-values will give a unique solution.

So m^2 - 36 = 0, m=+6 or -6.

[benapp]

Two dice are rolled and the outcomes noted. If X is the sum of the numbers showing on the two dice, find:
a. E(X), the expectation of X (Not sure what to put for X in distribution table)

[kamil9876]

Not sure what to put for X in distribution table.

... X is the sum of the numbers showing on the die ...

Anyway, you can do it like that i guess, find the distribution and then calculate the mean. But I guess a quicker way would be to just notice that X can be written as a sum of two random variables: where X_1, X_2 is the outcomes on the first and second die respectively. Hence

[Mr. Study]

Sorry if this is a noob question but I believe I am doing the question completely wrong.

All I am doing is:




Then just solve for , but I don't get the answer, which is and

Would someone be able to show me how they would do it?

[Hancock]

Sorry if this is a noob question but I believe I am doing the question completely wrong.

All I am doing is:




Then just solve for , but I don't get the answer, which is and

Would someone be able to show me how they would do it?

For equation 1:

x^2 - 5x - 6 > 0
Solve for x-intercepts:
x^2 - 5x - 6 = 0 (Note: I can't use greater or equal than)
(x-6)(x+1) = 0

x = 6, x = -1
Notice this is a positive parabola and it's asking you when this parabola is above the x-axis (once you move the 6 across to the left side of the equation).

This occurs when -1 >= x and x >= 6.


Equation 2:

-x^2 + 5x - 6 > 0
-x^2 + 5x - 6 = 0
x^2 - 5x + 6 = 0
(x-3)(x-2) = 0

x = 3, x = 2

We have now changed the question to ask when this negative parabola is above the x-axis. This occurs when 2 =< x =< 3

So the three regions where this equation is satisfied is:

2 =< x =< 3
-1 >= x
x >= 6.

[Mr. Study]

Oh ... that was a very noob question ...

I wasn't thinking about sketching the question ... and now I can blatantly see how easy that is ... 

Thanks for that Hancock!

[CommanderElahi]

Could I please have some help with these questions! (:



1.   The equation of the curve whose gradient is described by f’(x) = x^2+2x^2+x and passes through the point (1,2) is:

2.The growth rate of bacteria is given by equation dp/dt =2e^5t. Find the number of bacteria in 3hrs if at t=0, population =10

3.∫ x^3+3x+2/x  dx

4. Using calculus find exact area of pi/2---->2pi∫ sinx dx

5. The area enclosed by the curve f(x)=x^2+2x-3 the x axis and the lines x=0, x=3 equals=

Thanks guys!

@Mr Study, VegimitePi, Hancock

[pi]

Could I please have some help with these questions! (:



1.   The equation of the curve whose gradient is described by f’(x) = x^2+2x^2+x and passes through the point (1,2) is:

2.The growth rate of bacteria is given by equation dp/dt =2e^5t. Find the number of bacteria in 3hrs if at t=0, population =10

3.∫ x^3+3x+2/x  dx

4. Using calculus find exact area of pi/2---->2pi∫ sinx dx

5. The area enclosed by the curve f(x)=x^2+2x-3 the x axis and the lines x=0, x=3 equals=

Thanks guys!

@Mr Study, VegimitePi, Hancock

Just some tips, you can do the rest

1. Assuming you mean f'(x) = x^3+2x^2+x. Integrating gives f(x) = x^4/4 + 2x^3/3 + x^2/2 + c. Sub x=1 and y=2 and solve for c to find f(x).

2. integrating gives p(t) = 2e^5t/5 + c , sub in t=0 and p=10 to find c. Then, using your complete p(t), let t=3 to find p.

3. standard integration, you can do this! remember that the integral of 1/x = ln(x) or log_e (x)

4. Note that parts cancel out each other (draw the graph), so just solve 0 --> pi/2 ∫ sinx dx Remember that the integral of sin(x) is -cos(x)

5. There is an x-int at x=1, so divide the integral into two parts: 0---->1∫ x^2+2x-3 dx  and 3---->1∫ x^2+2x-3 dx. Add those together for the answer



Note: I'm very rusty on maths, so sorry in advance for the errors haha

@Mr Study, VegimitePi, Hancock

Thanks for the "tag", but there are MANY who are a lot better than me. Try b^3, The AN Dunce, DisaFear, kamil, TT, Humph, Mao, Thushan/t-bag/dilmah, paulsterio, istafa, samad, Aurelian, illuminati, laseredd, Somye, Ahmad, brightsky, enwiabe, taiga, pixon, romperait, funkyducky, luken93, Jenny_2108 and the list goes on! They're all more skilled than me

[sh00my]

Hey guys, I can't do this

Please help it add to my internal mark if I can get it.

Interal( (cos(x)^2)/(1-sin(x)) dx)

I've tried integration by parts, integration by substitution... EVERYTHING!!