VCE Methods Question Thread!

[Einstein]

with this equation in attachement

whats the significance of the 2pi, is it multiplied by the square root....

say ive got square root 18, which is 9*2 which becomes 3square root 2. whats the 3? is it multiplied by the square root 2?

hows this whole thing exactly work - i know surds but ive never thought of it this way if i been given a question ill just do it.

thanks

[Einstein]

also when you got an equation like

The maximal (implied) domain of the function with rule f(x) = 7(x + 4)/2. is the domain and implied domain essentially the same thing?

[kinslayer]

also when you got an equation like

The maximal (implied) domain of the function with rule f(x) = 7(x + 4)/2. is the domain and implied domain essentially the same thing?

Note that we are not actually given the complete function, only its rule. So the actual domain could be anything; we don't know.

When we write down f(x) = 7(x + 4)/2 and leave out the domain/codomain, we're implying that the domain is R because that is the largest set of real numbers that makes sense, but there are lots of functions with the same rule but with different domains.

[kinslayer]

with this equation in attachement

whats the significance of the 2pi, is it multiplied by the square root....

say ive got square root 18, which is 9*2 which becomes 3square root 2. whats the 3? is it multiplied by the square root 2?

hows this whole thing exactly work - i know surds but ive never thought of it this way if i been given a question ill just do it.

thanks

There is nothing special going on here. If a and b are both positive numbers then . As you pointed out, .

In your attachment, the square root of the stuff under the radical is a number, and so is 2pi. Putting them next to each other like that just means to multiply one by the other.

[HEN_iP]

By the method of linear approximation we can find an approximate value for to be:

[b^3]

[HEN_iP]

Thanks! Forgot to take out as

[kinslayer]

If the determinant of the (square) coefficient matrix is non-zero, then there is a unique solution. If it is zero, then there could be zero or infinitely many solutions.

The usual method to solve this problem would be to find the reduced row echelon form of the augmented matrix and see how the solutions depend on k.

If you have all zeroes in a row on one side of the line and a non-zero number on the other, the system has no solutions. If you have all zeroes in a row, on either side of the line, the system has infinitely many solutions. If you have all* non-zero rows then the (square) coefficient matrix has an inverse and the system has a unique solution.

*fixed

[lzxnl]

Hey so for questions that ask you:

Solve the following system of equations and find the value of k, such that a unique solution, infinite number of solutions or no solutions are given.

Do we use the determinant of the inverse matrix? I was taught this briefly by my tutor but I forgot how to do it.. Can someone elaborate? Was it determinant equal zero means there is one unique solution or something like that?

Also, would it be better if we solved it without matrices? Ie. Actually solve the simultaneous equation by hand, and apply logic (i.e.. since there are an infinite number of solutions, gradient must be the same etc.) without using matrices?

Thanks!

If the determinant of the (square) coefficient matrix is non-zero, then there is a unique solution. If it is zero, then there could be zero or infinitely many solutions.

The usual method to solve this problem would be to find the reduced row echelon form of the augmented matrix and see how the solutions depend on k.

If you have all zeroes in a row on one side of the line and a non-zero number on the other, the system has no solutions. If you have all zeroes in a row, on either side of the line, the system has infinitely many solutions. If you have no non-zero rows then the system has a unique solution.

*ahem this is VCE. No row reducing allowed here
also 'no non-zero rows?' That one I believe IS a typo

For VCE purposes, as kinslayer has mentioned, check if the coefficient matrix determinant is zero. If it is zero, you should get a few values for your parameter. If you plug those into the original equations, you should either get identical lines (like x+y = 2 and 2x + 2y = 4) or parallel lines (x+y=2 and 2x+2y=2). Identical lines => infinitely many solutions
Parallel lines => no solution

As for the determinant of the inverse matrix, for a 2 x 2 matrix (which is all you'll need for Methods by hand) it's the product of the top left and bottom right minus the top right times bottom left.
If your matrix is
a     b
c     d

Your determinant is ad - bc

However if you're not comfortable with matrices, you CAN solve it by hand. Just make sure you're careful to not divide by any parameters. For instance, kx + ky = 0 doesn't necessarily mean x+y=0, it might just mean k=0, as an example.
You can go from x+y = 1 to kx + ky = k
but not the other way around. Why? If k happens to be zero, you will have divided by zero which you can't do.


As an example, consider kx + 3y = k-1
                                       2x + (k-1)y = 1
Double first equation, multiply second by k
2kx + 6y = 2k-2
2kx + k(k-1)y = k

What you can't do is double the first equation and divide it by k, for instance

No solutions when gradients are equal => same y coefficients
6 = k(k-1)
k=3 or k=-2
If k=3, our equations are 6x + 6k = 4
                                 and 6x + 6k = 3. Parallel lines

If k=-2, we have
-4x + 6y = -6
-4x + 6y = 2
Again parallel lines

We'd get the same line if the second equation were -4x + 6y = -6 as well

[kinslayer]

*ahem this is VCE. No row reducing allowed here

forgot what forum I was posting in

[Einstein]

if ive got the question

f: [-1,3] -> R, f(x) = 2x^2-4x-3     state the range of f

i sub both -1 and 3 in and both range values give me [3,3]

answer states [-5,3]. i know i havent done anything wrong with calculations but what happens when their both the same ie. ive got [3,3] what do i do now to get [-5,3].

thanks

[IndefatigableLover]

if ive got the question

f: [-1,3] -> R, f(x) = 2x^2-4x-3     state the range of f

i sub both -1 and 3 in and both range values give me [3,3]

answer states [-5,3]. i know i havent done anything wrong with calculations but what happens when their both the same ie. ive got [3,3] what do i do now to get [-5,3].

thanks

That's the range of the function from the restricted domain but sometimes the turning point lies between that (maybe try finding that)!

[Einstein]

what exactly do you mean? i dont quite understand.

Also ive come across another question:

what is the exact values of x for 3(2x-1)^2+2(2x-1)-8=0

ANSWER IS -1/2 OR 7/6

[TrueTears]

what exactly do you mean? i dont quite understand.

Also ive come across another question:

what is the exact values of x for 3(2x-1)^2+2(2x-1)-8=0

ANSWER IS -1/2 OR 7/6

Just expand out each term, you should get and solve as per usual.

[Zealous]

what exactly do you mean? i dont quite understand.

Also ive come across another question:

what is the exact values of x for 3(2x-1)^2+2(2x-1)-8=0

ANSWER IS -1/2 OR 7/6

Cool way! (unnecessary but looks nice)