VCE Methods Question Thread!

[DoubleZo7]

Hello!
Could someone please help me with this normal distribution problem that I have attached?
It's rather urgent
Thanks!

[hello876]

Try thinking this question out point by point. Say you translate, and then dilate, then the translation itself will be dilated from the axis, whereas if you dilate first, and then translate, the translation will stay constant. Let's say you have f(x)=x², and you translate 2 up and dilate by 2 from the x-axis. If you translate first, then dilate it comes to f(x)=2(x²+2)=2x²+4. However, if you dilate first it comes to f(x)=2x²+2, so I hope you can now see the importance of the order of your transformations. This is just a general rule, it may not always be important to do it right as it may not always affect the final function, but it is important to do it right as a principle.

Thank you!

[Jackson.Sprigg]

I'm stumped on how to tackle 11a (attached). I'm preeetty sure there's no common factors? Yeah idk, any help would be appreciated.

Thank in advance

[darkz]

I'm stumped on how to tackle 11a (attached). I'm preeetty sure there's no common factors? Yeah idk, any help would be appreciated.

Thank in advance

Try using long division
e.g. if you had 8/3 => 2 + 2/3
Hope this helps!

[Sine]

I'm stumped on how to tackle 11a (attached). I'm preeetty sure there's no common factors? Yeah idk, any help would be appreciated.

Thank in advance

[Jackson.Sprigg]

Yep, thanks to both of you. Legends. 

[MB_]

Hello!
Could someone please help me with this normal distribution problem that I have attached?
It's rather urgent
Thanks!

You know that the mean is 252 and the SD is 12 and you're wanting to find the P(X>x)=0.4. You can find x by using the inverse norm function on your CAS.

[Jimmmy]

Sine, mind if I ask what on earth you've done here? I mean, it's genius but I have no clue how you did it. 

[Sine]

Sine, mind if I ask what on earth you've done here? I mean, it's genius but I have no clue how you did it. 

Well it's something that would be quite hard to come up with in an exam so it's important to have done it before. In this case a handy trick is to know that the constant in the expression when simplified will be the coefficients of the x's in the original. E.g. (2x+1)/(3x+2) will have a constant of 2/3. This means that we need to isolate x first hence factoring out all the coefficients of x. From here to get it to simplify we need to find something to cancel so we do something akin to completing the square. We add enough to the numerator so that it will cancel with the denominator but since we can't start just adding random numbers we need to remove what we added. From there we split the fraction to two - one will be a constant and the other with an x in the denominator. From here it is just standard simplification.

[lzxnl]

Here's another way of approaching it.

You've got \(\frac{2x+1}{5x+3}\). OK. We want to divide them. They're both linear polynomials, so if we divide them, we should get a constant, right? If you divide a degree 3 polynomial by a degree 1 polynomial, you get a degree 2 polynomial + remainder.

So, if you were to divide them, you would look at the 2, look at the 5, and your result would be \(\frac{2}{5}\). Only problem is, there's a remainder. This remainder is \(\left(2x + 1\right) - \frac{2}{5}\left(5x+3\right) = 1 - \frac{6}{5} = -\frac{1}{5}\). Think about what a remainder is. 7/3 = 2 remainder 1, so we have 2*3 + 1 = 7. Let's use this pattern in reverse for the fraction. \(2x+1 = \frac{2}{5}(5x+3) - \frac{1}{5}\), so dividing gives \(\frac{2x+1}{5x+3} = \frac{2}{5} - \frac{1}{5}\times\frac{1}{5x+3} = \frac{2}{5} - \frac{1}{5(5x+3)}\).

That's all it is. Long division.

[Jimmmy]

Thanks so much to both of you! After a bit of time reading both I understand them now. 

Here's another way of approaching it.

You've got \(\frac{2x+1}{5x+3}\). OK. We want to divide them. They're both linear polynomials, so if we divide them, we should get a constant, right? If you divide a degree 3 polynomial by a degree 1 polynomial, you get a degree 2 polynomial + remainder.

So, if you were to divide them, you would look at the 2, look at the 5, and your result would be \(\frac{2}{5}\). Only problem is, there's a remainder. This remainder is \(\left(2x + 1\right) - \frac{2}{5}\left(5x+3\right) = 1 - \frac{6}{5} = -\frac{1}{5}\). Think about what a remainder is. 7/3 = 2 remainder 1, so we have 2*3 + 1 = 7. Let's use this pattern in reverse for the fraction. \(2x+1 = \frac{2}{5}(5x+3) - \frac{1}{5}\), so dividing gives \(\frac{2x+1}{5x+3} = \frac{2}{5} - \frac{1}{5}\times\frac{1}{5x+3} = \frac{2}{5} - \frac{1}{5(5x+3)}\).

That's all it is. Long division.

@Izxnl, with you solution, are you using the same knowledge as Sine that the coefficients of x will be the constant? I was able to work it out using handwritten long division, but totally blanked out when I read your solution and saw "Dividend - Quotient * Divisor = Remainder". Another handy thing to note that I didn't realise could be done.  Once you find that remainder, you can just divide it by the denominator in the original expression and simplify alongside your quotient, but thanks for explaining it in more detailed terms because I doubt I would've understood it as well if you hadn't. 

I love your use of Dividend form too, last year our teacher actually instructed us to use it when answering polynomial division questions, but I prefer how you've used it as a connection to your final answer, albeit in this one you need to finally divide by 5x+3 to get an equivalent.

[aspiringantelope]

Mary throws a fair, 6-sided dice. If it comes up greater than 3, she wins. If not, she throws again and if it comes up greater than 4, she wins. Calculate the probability that she wins.
Suppose that N is a positive, sixteen digit integer. Show that we can find some consecutive digits of N such that the product of these digits is a perfect square.

I hope someone is able to help me out with these >.< I'm probably sure some of you will know where I got these questions from but I just want to know what are the steps to get the answers.
Thanks!

[Jimmmy]

Mary throws a fair, 6-sided dice. If it comes up greater than 3, she wins. If not, she throws again and if it comes up greater than 4, she wins. Calculate the probability that she wins.
Suppose that N is a positive, sixteen digit integer. Show that we can find some consecutive digits of N such that the product of these digits is a perfect square.

I hope someone is able to help me out with these >.< I'm probably sure some of you will know where I got these questions from but I just want to know what are the steps to get the answers.
Thanks!

Hi AA,

In terms of the Mary question, think of it in terms of a tree diagram. So, the first throw, if it comes up 4,5 or 6, Mary wins! So that's a 1/2 probability.

If she loses, that's ok! We then need to find the probability of her 'winning'  the second roll, after losing the first. So she loses the first, there's also a 1/2 chance of that occurring. If she throws 5 or 6 on the second roll, Mary wins! So a 1/3 probability. To work out the probability of success on the second roll, you multiply along the tree diagram. So 1/2 * 1/3 = 1/6

From there, you simply need to find a common denominator for the two fractions and add them. So 1/2 = 3/6 and 1/6 = 1/6. Therefore 3/6 + 1/6 = 4/6, so there's a 2/3 probability of Mary winning. I hope that makes sense! I always like to draw diagrams/tables when working these analysis questions out.

I have no clue about the consecutive digits question though 

[aspiringantelope]

Hi AA,

In terms of the Mary question, think of it in terms of a tree diagram. So, the first throw, if it comes up 4,5 or 6, Mary wins! So that's a 1/2 probability.

If she loses, that's ok! We then need to find the probability of her 'winning'  the second roll, after losing the first. So she loses the first, there's also a 1/2 chance of that occurring. If she throws 5 or 6 on the second roll, Mary wins! So a 1/3 probability. To work out the probability of success on the second roll, you multiply along the tree diagram. So 1/2 * 1/3 = 1/6

From there, you simply need to find a common denominator for the two fractions and add them. So 1/2 = 3/6 and 1/6 = 1/6. Therefore 3/6 + 1/6 = 4/6, so there's a 2/3 probability of Mary winning. I hope that makes sense! I always like to draw diagrams/tables when working these analysis questions out.

I have no clue about the consecutive digits question though 

/( I see! Thanks for your detailed explanation!!!!! Probability isn't my strongest part xD /)

[lzxnl]

Thanks so much to both of you! After a bit of time reading both I understand them now. 
@Izxnl, with you solution, are you using the same knowledge as Sine that the coefficients of x will be the constant? I was able to work it out using handwritten long division, but totally blanked out when I read your solution and saw "Dividend - Quotient * Divisor = Remainder". Another handy thing to note that I didn't realise could be done.  Once you find that remainder, you can just divide it by the denominator in the original expression and simplify alongside your quotient, but thanks for explaining it in more detailed terms because I doubt I would've understood it as well if you hadn't. 

I love your use of Dividend form too, last year our teacher actually instructed us to use it when answering polynomial division questions, but I prefer how you've used it as a connection to your final answer, albeit in this one you need to finally divide by 5x+3 to get an equivalent.

Suppose you're dividing 1359 by 11. We need to pick a first digit, and it is going to be 1 because 1x11 = 11, closest to 13. Same idea above. When dividing 2x+1 by 5x+3, 2/5 gets you the 'closest' match as you then match the x's.