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March 29, 2024, 11:14:16 am

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RuiAce

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Re: Mathematics Question Thread
« Reply #3990 on: February 03, 2019, 11:45:27 am »
+1
Try using binomial probability i.e.
Let X be the number of birds with white feathers
Then, for i) Pr(X=1)
Then, for ii) Pr(X≥1)

(Success; p=1/12)
(Sample size; n = 3)
Please keep in mind that for the current syllabus they're not introduced to random variable notation. They simply refer to a "binomial probability" by the formula \( \binom{n}{k}p^k(1-p)^{n-k} \). That and it is only examined in Extension 1/2 in NSW.
Hey everyone, I need help with a probability question from HSC course.

Q) A certain type of bird has a probability of 1/12 of hatching a bird with white feathers. If a bird lays three eggs, find the probability of hatching:

i) exactly one bird with white feathers
iI) at least one bird with white feathers.


Any help or or solution will be appreciated guys. Thanks
\[ \text{Note that there are three different ways this can happen.}\\ \text{Either the first bird has white feathers, or the second does, or the third does.} \]
Denoting W - white feathers and N - no white feathers here:
\[ \text{Thus we have}\\ \begin{align*} P(\text{Exactly 1 with white}) &= P(WNN) + P(NWN) + P(NNW)\\ &= \left( \frac1{12} \right)\left( \frac{11}{12}\right)^2 + \left( \frac1{12} \right)\left( \frac{11}{12}\right)^2 + \left( \frac1{12} \right)\left( \frac{11}{12}\right)^2\\&= \frac{121}{576} \end{align*} \]
\[ \text{For the other one, we can just consider the complement.}\\ \begin{align*}P(\text{At least 1 white}) &= 1 - P(\text{No white})\\ &= 1 - P(NNN)\\ &= 1 - \left( \frac{11}{12}\right)^3\\ &= \frac{397}{1728} \end{align*}\]

alexnero7

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Re: Mathematics Question Thread
« Reply #3991 on: February 03, 2019, 02:47:00 pm »
0
Can somebody please explain to me how we go from:

x^2-\frac{x^3}{3}
to...
x^2\left(1-\frac{x}{3}\right)

If you don't understand the above... please let me know, and ill send a photo instead.. thanks

Opengangs

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Re: Mathematics Question Thread
« Reply #3992 on: February 03, 2019, 03:11:11 pm »
+3
Can somebody please explain to me how we go from:

x^2-\frac{x^3}{3}
to...
x^2\left(1-\frac{x}{3}\right)

If you don't understand the above... please let me know, and ill send a photo instead.. thanks
Hey there!

Notice that:

So, factorising \(x^2\) gives us:
as required.

alexnero7

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Re: Mathematics Question Thread
« Reply #3993 on: February 03, 2019, 03:19:21 pm »
0
Thank you Opengangs!! :)




Hey there!

Notice that:

So, factorising \(x^2\) gives us:
as required.

alexnero7

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Re: Mathematics Question Thread
« Reply #3994 on: February 03, 2019, 07:00:26 pm »
0
Hey can someone please solve these? Thanks


dream chaser

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Re: Mathematics Question Thread
« Reply #3995 on: February 03, 2019, 07:17:54 pm »
+2
Hey can someone please solve these? Thanks

Hi alexnero7,

I will try and answer your questions.

Q9. First of all, lets establish what maximum and minimum values are. They correspond to the y values. For instance, the graph y=x^2 +3 would have a minimum value of 3 and y=-x^ -3 would have a maximum value of -3.

Now to find these values. We have to differentiate the equation. Once you do that, you equal the differentiated equation to zero and solve for x. Once you solve for x, sub the x value into the original equation to obtain the maximum and minimum values. Note, make sure the x values fall under the implied domain.

Q10. For this question, they state their is no stationary points. Therefore, normally the maximum and minimum values will be at the endpoints. To make sure of this, sketch the graph.

Hope this helps.

meerae

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Re: Mathematics Question Thread
« Reply #3996 on: February 03, 2019, 07:45:09 pm »
+1
Hey can someone please solve these? Thanks

Hey!

I thought I'd give insight on how I'd solve them, but dream chaser's way should get you to the correct answer.

Q9: I'd first sketch the curve, so you'd differentiate (keeping in mind that y= (x+5)^1/2 and then using chain rule) make y'=0 and all the curve sketching
So, now when you attempt to find the maximum/minimum values you can cross reference with your graph to ensure you haven't made a silly algebraic mistake.
To find the max/min values, you would test your value when you made y'=0 as well as the end points (-4 and 4) by putting them into the y equation to find the max/min.

Q10: As the question doesn't ask to sketch, I wouldn't waste my time (especially in an exam). I would differentiate the equation, make it equal 0 and then conclude there is no stat. point.
Like in the previous question, you would have to test the stat. points and the end points (in this case, -3 and 3), but you do not have any stat points which means you would just sub in the end points and determine which is max and min.

Hope this helped!
meerae :)
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alexnero7

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Re: Mathematics Question Thread
« Reply #3997 on: February 03, 2019, 09:48:21 pm »
0
Hi everyone.. can someone please help me understand what is the significance of the 45 degree angle in this question? thank you. :)

RuiAce

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Re: Mathematics Question Thread
« Reply #3998 on: February 03, 2019, 09:50:32 pm »
+4
Hi everyone.. can someone please help me understand what is the significance of the 45 degree angle in this question? thank you. :)
\[ \text{We have }\frac{dy}{dx} = 4x^2+C\\ \text{but that extra piece of information says that}\\ \text{at the point }(-2,5)\text{, we have }\frac{dy}{dx}=\tan 45^\circ = 1. \]
This is because \( \frac{dy}{dx} \) is just the gradient of the tangent. And we also saw the formula \(m = \tan \theta\) in coordinate geometry, where \(\theta\) is the angle made between a line and the \(x\)-axis. The significance is that here, that \(45^\circ\) is the \(\theta\).
« Last Edit: February 03, 2019, 09:52:28 pm by RuiAce »

UStoleMyBike

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Re: Mathematics Question Thread
« Reply #3999 on: February 04, 2019, 11:52:57 pm »
0
Hi, these stupid probability questions are triggering me, help me find out what I'm doing wrong in this question:

In lotto, a machine holds 45 balls, each with a number between 1 and 45 on it. the machine draws out one ball at a time at random. Find the probability that the first ball drawn out will be
(a) less than 10 or an even
(b)between 1 and 15 inclusive
 or divisible by 6
(c)greater than 30 or an odd
 number.

jamonwindeyer

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Re: Mathematics Question Thread
« Reply #4000 on: February 05, 2019, 12:02:21 am »
+2
Hi, these stupid probability questions are triggering me, help me find out what I'm doing wrong in this question:

Hey! All of these look at the probabilities of mutually non-exclusive events. The formula you need:



That is, the probability of one thing OR another thing happening is the sum of each individual probability, minus the probability that they BOTH happen at once.  Let's take a)

Numbers Less than 10: 1,2,3,4,5,6,7,8,9
Evens: 2,4,6,8...,38,40,42,44
Numbers Less than 10 THAT ARE ALSO Evens: 2,4,6,8

So to calculate this probability:



Try and apply this same idea to the other two! :)

jamonwindeyer

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Re: Mathematics Question Thread
« Reply #4001 on: February 05, 2019, 12:09:02 am »
+2
Hey everyone!

Can anyone help with this question including worked solutions?
Find the area bounded by the curve y=(x+3)[squared], the y axis and the lines y=9 and y=16 in the first quadrant.
Find the volume of the solid formed if this area is rotated about the y axis.

Many thanks!

Hello! So drawing a bit of a sketch might help you visualise this, but essentially we are looking at the area between the right arm of that parabola and the y-axis, between the two y-values given. We need to rearrange to make \(x\) the subject:



So the area would be:



To swap to the volume, you use \(x^2\) instead of just \(x\), and add a \(\pi\) out front:



Both of these are just definite integrals using the regular rules, let us know how you go with them! :)

alexnero7

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Re: Mathematics Question Thread
« Reply #4002 on: February 12, 2019, 06:02:01 pm »
0
Hey everyone, hope you all well. Can somebody please help me find the primitive function of this attached question? Thanks. :)

myopic_owl22

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Re: Mathematics Question Thread
« Reply #4003 on: February 12, 2019, 06:40:33 pm »
+1
Hey everyone, hope you all well. Can somebody please help me find the primitive function of this attached question? Thanks. :)

Hi there,
To integrate functions like these, I'd recommend moving the (1/2) bit to the left side of the integral, as it's a constant multiplier. Dealing with the rest...
1. Convert into index notation
 
2. Integrate as usual. Treat the bracket as a term - power up by one, divide by new power.

We divide through by 4 as well to essentially 'reverse' the effects of the function of a function rule if the answer was to be diffed. General formula:

Note that this will only work for functions whose leading power of x is 1. Quadratics, etc. do not follow this rule.

Thankfully, this is about as tricky as 2U integration will get. Hopefully this helps!

Oh and don't forget +c :)

myopic_owl22

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Re: Mathematics Question Thread
« Reply #4004 on: February 12, 2019, 07:09:16 pm »
0
Can anyone explain the working out for this? Is there a method that you are supposed to use?
Question: Write this series in sigma notation.
1+1/2+1/4+...+1/512
Also in some examples I have seen like 3+6+12+...+3×2(power n) you are supposed to change the 'n' to a 'k'. Why is that?

Hi there,
Noticing that the terms in your series are all 1 divided by increasing powers of 2 (i.e. 2^0, 2^1, 2^2, up to 2^9), we'd write something like this:


The bottom is where n begins (0 in this case for our first term of 1) and the top number is when we finish. You can use trial and error without too much fuss to see what power of 2 that 512 is raised to. Alternatively, you could use logarithms, which is a later topic in the maths course. Here, log2512 = 9. We use n as our only variable as this is a finite series, which will stop when we finish adding 1/512 to our sum.

On the right, is what we're summing over and over again, with different values of n for each time we do it. Hopefully I'm making some sense.

We generally use k if the sum's final term is in general form (i.e. all the terms in this series follow this rule, where k is replaced by the term number). It will either denote some value that we either need to find, or don't need to worry about. It's essentially the math's conventional placeholder, which isn't a major focus of the course anyway, but to see an example of it in action:

So yeah, it's more focusing on the process of summation rather than the answer. Let me know if I'm not making sense :)
« Last Edit: February 12, 2019, 08:04:07 pm by myopic_owl22 »