4U Maths Question Thread

[RuiAce]

I think it should be sufficient to differentiate and sub as long as you justify it (i.e tangent only intersects the ellipse once)

There's no ellipse here. That's a 5th power on y there.

I'm not sure if there was a typo in the original question or not. Because if the 5 is replaced by two we have \(x^2+2xy+y^2=4\) which is apparently a pair of parallel lines



Edit: Actually it does work, but we can't use any conic theorems because of the \(y^5\)

[RuiAce]

___________________________

[hanaacdr]

Hi
Could i get some help on question 3b

thank you

[RuiAce]

Hi
Could i get some help on question 3b

thank you

[hanaacdr]

Hi,
could i get some help on this question please

[jakesilove]

Hi,
could i get some help on this question please

Hey! Firstly, since all the coefficients are real, we know that either ONE of the roots is real, or THREE of the roots are real (as imaginary roots will come in complex conjugates).

The easiest way to answer the first half is by differentiating the function, it proving that it is non-decreasing. So,



For a>0, this derivative is ALWAYS positive. Thus, there will only be one real root, as the graph will never 'turn around'. It's helpful to sketch an example of a non-decreasing function (ie. gradient is always positive), to prove to the marker that you know why this fact results in one real root.

Now, if two roots are equal, we can right them as



Using sum and product rules.



and



We know from above that



so




Now, subbing alpha into the original equation, we get





Not sure where we are going with this; let's keep going?
 
We are trying to prove that



Subbing it what we've found above;



Which looks like it equals zero! Bit of a round about way of getting there, but hey, it works!

[ellipse]

Hi,
could i get some help on this question please

Here's an alternate method for part ii

[jakesilove]

Here's an alternate method for part ii

That's a heaps better method. Cheers for posting it up!

[ellipse]

That's a heaps better method. Cheers for posting it up!

No problem

[hanaacdr]

Hey! Firstly, since all the coefficients are real, we know that either ONE of the roots is real, or THREE of the roots are real (as imaginary roots will come in complex conjugates).

The easiest way to answer the first half is by differentiating the function, it proving that it is non-decreasing. So,



For a>0, this derivative is ALWAYS positive. Thus, there will only be one real root, as the graph will never 'turn around'. It's helpful to sketch an example of a non-decreasing function (ie. gradient is always positive), to prove to the marker that you know why this fact results in one real root.

Now, if two roots are equal, we can right them as



Using sum and product rules.



and



We know from above that



so




Now, subbing alpha into the original equation, we get





Not sure where we are going with this; let's keep going?
 
We are trying to prove that



Subbing it what we've found above;



Which looks like it equals zero! Bit of a round about way of getting there, but hey, it works!

THANK YOUU!

[beau77bro]

Hey guys could someone help me with 3b and c, I can't quite work it out algebraically

[RuiAce]

(Image removed from quote.)

Hey guys could someone help me with 3b and c, I can't quite work it out algebraically

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Side note: It may be worth mentioning that the given questions are the inverse hyperbolic sine and tangent functions, i.e. \( \sinh^{-1}x\), \( \tanh^{-1}x\). What we are trying to find are the exponential forms of the original hyperbolic functions \(\sinh x\) and \(\tanh x\)

[beau77bro]

thanks rui, i didnt even see that you could just cancel out the y squared, rookie mistake sorry ahhaha

[beau77bro]

I hope this question is better, not sure how to do 8a, the rest is fine but just abit confused on whether its algebraic or word and where to go? Sub in value less than 0 and show it doesn't work because it x^2 +3x/2 has to be greater than 0?

[RuiAce]

(Image removed from quote.)

I hope this question is better, not sure how to do 8a, the rest is fine but just abit confused on whether its algebraic or word and where to go? Sub in value less than 0 and show it doesn't work because it x^2 +3x/2 has to be greater than 0?

The natural domain of the first is x>0 but for the second it's x>0 AND x<-3/2