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FlammaZ

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Re: Mathematics Question Thread
« Reply #4500 on: September 24, 2020, 07:16:06 pm »
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apologies, not so easy to write on a computer with your finger  ;D . could you explain what you mean by the c=0 part. fyi, V(0) = -14.72, however this is not logical. (But it does not = 0)   
« Last Edit: September 24, 2020, 07:17:45 pm by FlammaZ »

keltingmeith

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Re: Mathematics Question Thread
« Reply #4501 on: September 24, 2020, 07:38:42 pm »
+3
apologies, not so easy to write on a computer with your finger  ;D . could you explain what you mean by the c=0 part. fyi, V(0) = -14.72, however this is not logical. (But it does not = 0)   

Wait, HSC uses CAS calculators now? I thought they weren't allowed

Okay, so, if we take:



And integrate with respect to t, we get:



Now, this ONLY matches what you wrote if C=0. Watch what happens if I say V(0)=0, though:



Which is very different to what you wrote.

---

After writing this, I think I've realised the confusion - you do know that after every indefinite integral, you MUST include the constant of integrate (often referred to as, "always remember your +C"), right?

FlammaZ

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Re: Mathematics Question Thread
« Reply #4502 on: September 24, 2020, 07:49:23 pm »
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i thought i was missing something like that, i completely forgot the C,  :P forgive me for my stupid mistake. Keltingmeith and Fun_jirachi. I think now i will always remember it.
Also im not doing HSC im in a different state, i am really new to the site and don't know how to really use it thank you for helping me out though.
« Last Edit: September 24, 2020, 07:51:20 pm by FlammaZ »

fun_jirachi

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Re: Mathematics Question Thread
« Reply #4503 on: September 24, 2020, 07:50:01 pm »
+3
Wait, HSC uses CAS calculators now? I thought they weren't allowed

We don't - evaluating this by hand requires IBP, which is outside the scope of Maths Advanced. Must be either the wrong board or some textbook exercise :)

apologies, not so easy to write on a computer with your finger  ;D . could you explain what you mean by the c=0 part. fyi, V(0) = -14.72, however this is not logical. (But it does not = 0)   

Don't like answering questions that have already been answered, but I feel like this is important enough to warrant an additional response :)

This sort of working is sort of what we require to help you adequately - in hindsight the confusion wasn't really about methodology as previously thought (I am no longer confused :D). Assuming the initial condition isn't a requirement since it's given in the question (and I quote from the question 'Initially the tank is empty.') - you've got to do what keltingmeith has already said you should do, as this implies that V(0) = 0, not -14.72 as you've said. Watch out for these sorts of statements as these give you the initial condition that is required to get the proper solution.

You either have two choices when finding V(t) - a) integrate as an indefinite integral, then substitute the initial condition in as shown by keltingmeith (don't forget +C), or b) integrate as a definite integral with v and t as upper bounds and the initial condition as a lower bound (or vice versa). If you find yourself forgetting the constant of integration, definitely go with the second method - this is the reason you have negative volume as an initial condition (ideally something this weird should tip you off that's something's gone wrong in your calculation rather than your methodology - since methods are something people tend to get correct in my experience).

Hope this helps :)
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Ookei

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Re: Mathematics Question Thread
« Reply #4504 on: September 30, 2020, 06:32:03 pm »
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When determining nature of points and testing concavity, can we just write +ve and -ve instead of actually writing in the values?

fun_jirachi

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Re: Mathematics Question Thread
« Reply #4505 on: October 01, 2020, 03:50:39 pm »
+3
Hey :)

You could probably get away with it if using a sign table. If you're using the second derivative, definitely write the evaluated value of the second derivative at the used point. That being said, it's always safer to write the value down.

For school assessments, regardless of what's said here or anywhere else, go by your teacher's directives. Otherwise the above is a good rule of thumb.
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Coolmate

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Re: Mathematics Question Thread
« Reply #4506 on: October 11, 2020, 11:55:58 pm »
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Hey Everyone! ;D

I am currently working through the NESA maths past papers and have come across some questions and was wondering if someone could please let me know if these questions are applicable to the new syllabus?

- Q15c (iii) and (iv) (2018)
- Q16a (i), (ii) and (iii) (2018)
- Q16c (2017)
- Q12 (2016)
- Q15b (2015)

Thanks in advance!
Coolmate 8)
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Coolmate

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Re: Mathematics Question Thread
« Reply #4507 on: October 22, 2020, 02:24:29 pm »
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Hi everyone! :)

Could someone please step through on how to do the following questions as I am unsure how to do them:
- 12a (2019)
- 12d (2019)
- 15bi and ii (2017)

Thanks in advance!
Coolmate 8)
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fun_jirachi

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Re: Mathematics Question Thread
« Reply #4508 on: October 22, 2020, 05:20:39 pm »
+5
Hey :)

12a)i)
- Find the gradient for the line AC. Recall that for lines \(l_1\) and \(l_2\), the respective gradients \(m_{l_1}\) and \(m_{l_2}\) must have product -1.
- Use the point C (which has coordinates given), and substitute the previous result into the point-gradient form of the line.

12a)ii).
- You can find the x-intercept of the line calculated above, then find the length of BC (reasonably easy since they both have y coordinate 0), then use formula \(A = \frac{bh}{2}\) (think about where you can find the height!)

12d)
Basically asking you to evaluate \(\int_0^3 \frac{3x}{x^2+1} \ dx\). Recall that \(\int \frac{f'(x)}{f(x)} \ dx = f(x) + C\) - try manipulating this integral into a better form so you can use this formula.

15b)i)
Note that \(M_0 = 0\), and that \(M_1 = X(1+\frac{\frac{4.2}{12}}{100})-2500 = (1.0035)X - 2500\). Always start with \(M_0\), and find \(M_1\) after, executing each successive action in order. Things to note here are that \(M_0 = 0\) because the withdrawal happens the day before the deposit (withdrawal is on the last day, deposit on the first day of the next month), and \(M_1\) is that expression because the month's deposit has been placed, interest has been tacked on, then the withdrawal occurs.

Executing instructions in this order on \(M_1\) means that \(M_2 = 1.0035(M_1) - 2500\). You can work it out from there - take away is the setup is so important and order matters.

ii)
After this, you can generalise the formula for \(M_n\) based on the pattern. Try working out \(M_3\) and even \(M_4\) if you have to, so you can see what it is. This can be applied for any question.

here, they want you to use n = 4, and \(M_4 = 80000\). After this, you have an equation in X that you can rearrange and solve.

Hope this helps :)
« Last Edit: October 22, 2020, 05:29:45 pm by fun_jirachi »
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HS26

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Re: Mathematics Question Thread
« Reply #4509 on: October 23, 2020, 06:44:37 pm »
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Hi, can someone please help me with this question? I don't understand the highlighted part... Thank you so much  :D
:)

keltingmeith

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Re: Mathematics Question Thread
« Reply #4510 on: October 23, 2020, 07:11:58 pm »
+5
Hi, can someone please help me with this question? I don't understand the highlighted part... Thank you so much  :D

Note that I'm going to be answering this for discrete random variables, because that's what's in this question - however, all of this also applies for continuous random variables, just replace the summation with an integration. I also don't know which part in there you're confused by, so forgive me if I cover something you already know. In general:



You are most familiar with this in terms of the mean, or the expected value of X. That is:



But I could put any function in there. Here's some examples:



So that's what the E[X^2] is. As for the formula:



This one is on your formula sheet, and is just a different way of calculating the variance. Let me know if I didn't answer what you were confused by

Coolmate

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Re: Mathematics Question Thread
« Reply #4511 on: October 23, 2020, 07:54:56 pm »
+1
Hey :)

12a)i)
- Find the gradient for the line AC. Recall that for lines \(l_1\) and \(l_2\), the respective gradients \(m_{l_1}\) and \(m_{l_2}\) must have product -1.
- Use the point C (which has coordinates given), and substitute the previous result into the point-gradient form of the line.

12a)ii).
- You can find the x-intercept of the line calculated above, then find the length of BC (reasonably easy since they both have y coordinate 0), then use formula \(A = \frac{bh}{2}\) (think about where you can find the height!)

12d)
Basically asking you to evaluate \(\int_0^3 \frac{3x}{x^2+1} \ dx\). Recall that \(\int \frac{f'(x)}{f(x)} \ dx = f(x) + C\) - try manipulating this integral into a better form so you can use this formula.

15b)i)
Note that \(M_0 = 0\), and that \(M_1 = X(1+\frac{\frac{4.2}{12}}{100})-2500 = (1.0035)X - 2500\). Always start with \(M_0\), and find \(M_1\) after, executing each successive action in order. Things to note here are that \(M_0 = 0\) because the withdrawal happens the day before the deposit (withdrawal is on the last day, deposit on the first day of the next month), and \(M_1\) is that expression because the month's deposit has been placed, interest has been tacked on, then the withdrawal occurs.

Executing instructions in this order on \(M_1\) means that \(M_2 = 1.0035(M_1) - 2500\). You can work it out from there - take away is the setup is so important and order matters.

ii)
After this, you can generalise the formula for \(M_n\) based on the pattern. Try working out \(M_3\) and even \(M_4\) if you have to, so you can see what it is. This can be applied for any question.

here, they want you to use n = 4, and \(M_4 = 80000\). After this, you have an equation in X that you can rearrange and solve.

Hope this helps :)

Hey fun_jirachi! ;D

Thankyou so much for the help, it now makes sense!

Coolmate 8)
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rirerire

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Re: Mathematics Question Thread
« Reply #4512 on: October 23, 2020, 11:28:02 pm »
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could someone pls help with HSC 2014 Question 14 b ii?
Equation: 2x^2 +8x +k=0
the roots are: a and B (alpha beta signs)
 
if a + B= -4
And a^2B + aB^2=6
Find k

Answer is: k/2(-4)=6
-2k=6
k= 6

I don’t get the working out at all... why it’s k/2, shouldn’t it be 2k?
thanks!

keltingmeith

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Re: Mathematics Question Thread
« Reply #4513 on: October 23, 2020, 11:58:15 pm »
+5
could someone pls help with HSC 2014 Question 14 b ii?
Equation: 2x^2 +8x +k=0
the roots are: a and B (alpha beta signs)
 
if a + B= -4
And a^2B + aB^2=6
Find k

Answer is: k/2(-4)=6
-2k=6
k= 6

I don’t get the working out at all... why it’s k/2, shouldn’t it be 2k?
thanks!

Nope. We want to solve:



We also know that alpha and beta are roots. So you'd think that should be equivalent to saying:



But if you expand the RHS, you'll notice that the x^2 doesn't have a two out the front! So instead, you need to get the coefficient of the x^2 to be 1. To do that, divide everything by 2:



Next, we can do the equality as you were thinking of:



And substituting into the third equation you gave:



Which is exactly what is given by the supplied answer.

Tl;dr, you might think it's 2k because the coefficient in front of the x^2 is 2, but remember that you want to undo that coefficient, as you can see by my working from first principles - hence why it's k/2, not 2k.

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Re: Mathematics Question Thread
« Reply #4514 on: October 24, 2020, 08:32:34 pm »
0
Hi Everyone!

Could I please have help with working out the correlation coefficient (r) and part ii, in the attached question?

Thanks in advance,
Coolmate 8)
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