Hey peeps,

Thought that this thread should be here. Since it has been incredibly helpful to many others and me as well, on the old forums.

As usual, don't hesitate to ask because there are always people willing to help on AN.

So here's my first q:
Copies of an ad for a course in practical statistics are sent out to maths teachers in a large city. For each teacher that receives a copy, the probability of subsequently attending the course is 0.09. 20 teachers who receive a copy of the ad. What is the probability that the number who subsequently attend the course will be:
a. 2 or fewer.

b. exactly 4

Much appreciated,
beep boop

    beep_boop
    I think you'll be calculating a binomial distribution

    To find Pr(X=4), you evaluate (20! / [16! x 4!]) x (0.09)4 x (0.91)16

    To find Pr(X<=2), you'll have to add up individual probabilities of Pr(X=0), Pr(X=1) and Pr(X=2), which you calculate with the binomial distribution formula

      beep_boop

      If this is CAS enabled you can use binomial CDF to find out the answer for a without doing the individual working out. Otherwise, follow Billzene's instructions.

        17 days later
        a month later

        Hi everyone!!
        Can someone please help with this question in Methods Book:
        A four-digit number (with no repetitions) is to be formed from the set of digits {0, 1, 2, 3, 4, 5, 6, 7}. Find the probability that the number:
        a) is even

        I approach this question with four blanks ___ ___ ___ ___
        If a number is even it can end with 0,2,4 or 6
        Case 1: Ends with 0
        7 X 6 X 5 X 4
        Case 2: Ends with 2,4, or 6
        6 X 6 X 5 X 4
        = 1560 ways

        The total ways of creating a number would be:
        7 X 7 X 6 X 5
        =1470

        Probability = 1560/1470
        which is not possible

        The solutions say the answer is:

        25/49
        How?
        I am so confused.

        Please help.

        Thank You so much!!

          PizzaMaster
          I think you have to consider the cases slightly differently.

          Case 1: Ends with zero
          Since 0 must be at the end, there is only one possibility for the last digit. We can fill this in:
          _ _ _ 1
          After this, we are left with 7 numbers that could still come at the start, as 0 has been used.
          7 _ _ 1
          Continue filling this in to get 7 X 6 X 5 X 1 = 210 ways

          Case 2: Ends with 2, 4, or 6
          The last digit can be either 2, 4, or 6, so that’s three possibilities for the last position:
          _ _ _ 3
          After this, there are 6 options for the first digit (excluding 0 and whichever number we just used)
          6 _ _ 3
          Then 6 again for the next digit (excluding the two we already used, but including 0)
          6 6 _ 3
          Which is 6 X 6 X 5 X 3 = 540 ways

          As you already calculated, the total is 1470, so the probability that the number is even is (210+540)/1470 = 25/49

          Hope this helps!

            Oh so you have to consider zero as 1 in the blank and as a separate case.

            I get it now!!!
            Thank you so much.

            Hello everyone

            Just a quick question:
            Is radians only activated on CAS when dealing with circular functions/ trig
            OR
            does it effect your other day-to-day calculations and graphing as well?

            Thank you!

            If you look in the top right-hand corner, it should either say 'rad' or 'deg' next to the battery symbol.

            If you click on it with the mouse - it will switch to the other mode. (You will still need to re-do calculations though)
            Just be careful - if you press the up/down arrows while the mouse is still over it - it may switch back.

            Hope this helps!

            2 months later

            Hey guys,

            How do you find the domain of:
            √x-1/x+2 ?

            We know that x-1/ x+2 has to be >= 0

            I tried doing it in a way that I brought x+2 to the other side and then I had:
            x-1 >= 0

            But the answer says:
            x <-2 U x >=1

            How...
            Is there a method to do this?

            Please help.

            Thanks.

              PizzaMaster

              Function:
              √(x-1)/( x+2)

              √(x-1) implies that x≥1 as you cannot have a negative value inside the root (to get a real value). x+2 is the denominator and cannot equal zero. So the domain of x+2 would be any real number except -2. But since the domain of √(x-1) is x≥1, we can disregard the -2 and be left with the final answer of x≥1. I'm not sure if I interpreted your function correctly tho, my answer is correct tho according to my interpretation. Let me know if the function is listed above is the one you are talking about, I CAS verified it as well.

                Thank you for your help.
                However, I meant that the (x-1) and (x+2) are both under the square root sign.
                Sorry for the confusion.

                No worries, in that case we have a whole new situation. The same principle applies in that x+2 cannot equal zero and everything inside the bracket must be 0 or greater. To be honest, I would convert (x-1)/(x+2) into a partial fraction which would yield 1 - 3/(x+2) which can be obtained through long division. If you graph that you get a hyperbola (I think it's called that). You just gotta find when the graph is greater than or equal to zero. I would strongly suggest graphing it and you'll come to realise that from -2<x<1 the graph is negative. Thus the domain of that function would be the direct opposite, aka your aforementioned answer.

                You'll find graphing stuff is always gonna help you and you need to learn and get used to do doing it. Hope that Helps.

                Can a one-to-one function have a many-to one inverse function?
                For example:
                f(x) = 2x^ 3/4 + 1

                has an inverse:
                f-1(x) = [(x-1) / 2]^ 4/3

                How come?

                  PizzaMaster
                  If you have f(x) = x0.5

                  Then f-1(x) = x2, but restricted to the domain R+ u {0}, which means the inverse is also 1-1. The negative branch of x2 isn't a reflection of y = x0.5 in the line y = x, so it's not part of the inverse function

                    Ohh so you mean that the range of the original is the domain of the inverse?

                    And so that's why
                    f(x) = 2x^ 3/4 + 1

                    does have an inverse:
                    f-1(x) = [(x-1) / 2]^ 4/3
                    BUT with domain [1, infinity)

                      How do we know whether the question wants us to give the dilation from x axis or y axis when given both the original and transformed function.

                      For example:
                      f(x)= 1/x2
                      f1(x)= 5/x2

                      I got dilation by a factor of 1/5 from the y axis but the answers say factor of 5 from the x axis. Is my answer still valid?

                        PizzaMaster

                        Correct that's why you need to restrict the domain of the original function

                          snowflake Your answer wouldn't be considered correct, if the coefficient of x2 were 5 ie y = 1 / (5x2) then it'd be correct since that would be a dilation from the y-axis

                              Billzene

                              Ohh ok but how do we whether to work out it’s dilation from the X or Y axis if it doesn’t say in the question?

                                  snowflake by recognition of the form y = a/(1/b(x-h))n + k

                                  a = dilation factor from x axis
                                  b = dilation factor from y axis
                                  h = translation in +ve direction of x axis
                                  n = power
                                  k = translation in the positive direction of y axis

                                  This is similar to turning point form of parabolas

                                      5 days later

                                      find values of a and b such that a(x+2)+b(x+3)=18x+8 for all values of x
                                      if someone could help that would be great

                                        chemistry1111 Expand then factorising LHS gives (a+b)x + (2a + 3b) = 18x + 8 --> by comparing coefficients, you obtain a+b = 18 and 2a+3b =8, solve simultaneously for a and b, giving a=46 and b=-28

                                          thank you so much that makes sense

                                          also this question
                                          For the polynomial P(x)=(a+1)x3 + (b-7)x2 +c+5, find values for a,b and c if P(x) has a degree 2, a leading coefficient of 3 and the constant term is -1

                                          when factorizing polynomials in methods unit 3/4, what method do most people use vcaa prefer, long division or equating coefficients

                                          9 days later

                                          Do we need to buy the new Methods 3/4 book for the new study design (2023)?
                                          I feel like the textbook questions are the same...

                                          4 days later

                                          PizzaMaster
                                          Howdy, I did QCE Methods, so if I mess this up, I’m really sorry.

                                          https://ibb.co/m0Q1xdV

                                          For b) may I know what the answers are? I may be able to work backwards 🙂

                                          I didn’t do d) because I wasn’t that confident in doing the question.

                                          Hope this helps,
                                          -jinx_58

                                            20 days later

                                            Hi jinx_58,

                                            Thank you so much for solving!
                                            The answer to part b is Domain = [0, 6]; Range = [0, 9/2]
                                            The answer to part c is 9/2

                                            The graph of y = 5/x -6 is reflected in the x-axis and
                                            then in the y-axis. The equation of the final image is:

                                            could someone help with this question

                                              chemistry1111

                                              1. reflected in the x-axis
                                              2. reflected in the y-axis

                                              **I'm not quite sure if y = (5/x) -6 or if it's y = 5/(x-6) so I'll show the working for both

                                              let f[x] = (5/x) -6

                                              1. x reflection: -f[x] = f1[x] = -[ (5/x) - 6 ] = -( 5/x ) + 6

                                              2. y reflection: f1[-x] = f2[x] = -(5/ [-x] ) +6 = -(5/ x ) +6

                                              let g[x] = 5/(x-6)

                                              1. x reflection: -g[x] = f1[x] = -[ 5/(x-6) ] = -5/(x-6)

                                              1. y reflection: g1[-x] = f2[x] = -5/( [-x] - 6) = -5/( -x - 6)

                                              hope that helps!

                                                Hi everyone,

                                                Is (2x-1) the same as (x-1/2)
                                                I mean they both give x= 1/2
                                                But let's say if we had to factorise 12x3 + 20x2 − x − 6
                                                and we get (2x-1)(2x+3)(3x+2)

                                                Can I also write this as:
                                                (x-1/2)(2x+3)(3x+2)?

                                                ALSO can some please help and find the roots of this: 12x3 + 20x2 − x − 6
                                                using rational root theorem

                                                much appreciated!!!

                                                  PizzaMaster
                                                  good question! to answer your first one, (2x-1) = 2 (x-1/2). they're not the same but the same x value will make both of them equal (0). I'm not home rn so I'll reply to the second one when I have access to my iPad.

                                                  EDIT
                                                  I think it's easier if you watch a video, here's a good one I found (watching 0:00-2:40 should be enough): https://www.youtube.com/watch?v=4XytYH35AP0

                                                    So I can write (2x-1)(2x+3)(3x+2) as 2(x-1/2)(2x+3)(3x+2) NOT (x-1/2)(2x+3)(3x+2)
                                                    Great, thanks crumblycupcakes!

                                                    With rational root theorem, I find that it is super time consuming... after we find the possible roots, is there a shortcut to know which ones to try first?
                                                    For example with 12x3 + 20x2 − x − 6:
                                                    the possible roots are +- [1/2, 1/3, 1/4, 1/6, 1/12, 2/3, 3/4, 3/2]
                                                    That is a lot! Is there a way to know which one we can try first?

                                                    Also does simplifying work?
                                                    for example with 12x3 + 20x2 − x − 6, we do like factors of 6/ factors of 12. Can we do factors of 1/ factors of 2 (since 6/12 = 1/2?)