A Level Maths : Common Mistakes/misconceptions Watch

ghostwalker
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#41
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Complex numbers:

There are two general approaches; geometric and algebraic. Some problems will yield to one approach orders of magnitude more easily than to the other. Sometimes a combination.
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RDKGames
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#42
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 \ln(x^2) = 2\ln(x) only if x&gt;0. Note that this equality makes no sense when x<0.

Otherwise, if x&lt;0 then  \ln(x^2) = 2\ln(-x).

More generally, the equality which holds for all real x (except zero) is actually
 \ln(x^2) = 2\ln|x|.

Similarly for other even powers of x.
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Sir Cumference
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#43
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A common mistake is to not use your formula book properly. As an example, recently I watched as a student was faced with this integral:

\displaystyle \int \cot^2 x \ dx

The student had a quick read through their formula book and couldn't find the integral (this was a good first step) so they put it to one side. Then they tried different methods to solve it including substitution and IBP (*) but none were successful. As a hint I told them to look at their formula book again and they finally realised that this is included:

\displaystyle \frac{d}{dx} \cot kx = -k \ \mathrm{cosec}^2 kx

And so they could change \cot^2 x into \mathrm{cosec}^2 x - 1 and use the above to integrate it. Easy!

In summary, sometimes a good approach with trig integrals is to attempt to use identities to change them into integrals that are in the formula book. This is all assuming that you don't know all the integrals in your formula book off-by-heart - most don't.


(*) With practice it should be clear that substitution/IBP are clearly not good approaches for this integral. The more integrals you do, the better you'll get at spotting things like this. A lot of A Level students don't practice integration nearly enough. It requires a lot more practice than most other topics.
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IsMo987
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Can you please tag me in the group
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Sir Cumference
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(Original post by IsMo987)
Can you please tag me in the group
Now you've posted in it, it will appear in your watched threads so you'll see whenever there are new posts (possibly with a notification).
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Sir Cumference
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#46
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A lot of students standardise in normal distribution questions when there's often no need to. E.g. if you have

X\sim N(50, 4^2)

and are asked to calculate P(X<53) then you can just type the values into your calculator function and you'll get full marks in the exam. There is no need to standardise first and use the percentage points table. Doing this is slow and unnecessary. The only time when it is necessary to standardise is if the question specifically tells you to or if the mean and/or variance are unknown.

Also, there are no occasions where you have to use your formula book tables, for the binomial or the normal distribution - you can always use your calculator instead. So personally I recommend ignoring tables completely and just use your calculator for everything. There are a lot of teachers who still use tables and they would disagree with me!
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Pangol
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#47
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(Original post by Sir Cumference)
Also, there are no occasions where you have to use your formula book tables, for the binomial or the normal distribution - you can always use your calculator instead. So personally I recommend ignoring tables completely and just use your calculator for everything. There are a lot of teachers who still use tables and they would disagree with me!
Great advice for the vast majority of cases. The only time where I think that the tables still have the edge is when doing binomial significance tests. It is very quick and easy to see where the cutoff point is between a probability of <0.05 and >0.05 is (or whichever significance level you are going for), so finding critical regions is faster this way.
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Gent2324
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(Original post by Sir Cumference)
A lot of students standardise in normal distribution questions when there's often no need to. E.g. if you have

X\sim N(50, 4^2)

and are asked to calculate P(X<53) then you can just type the values into your calculator function and you'll get full marks in the exam. There is no need to standardise first and use the percentage points table. Doing this is slow and unnecessary. The only time when it is necessary to standardise is if the question specifically tells you to or if the mean and/or variance are unknown.

Also, there are no occasions where you have to use your formula book tables, for the binomial or the normal distribution - you can always use your calculator instead. So personally I recommend ignoring tables completely and just use your calculator for everything. There are a lot of teachers who still use tables and they would disagree with me!
what about with pmcc? how can you find those values on the calculator?
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Sir Cumference
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(Original post by Gent2324)
what about with pmcc? how can you find those values on the calculator?
https://www.youtube.com/watch?v=D0EBJAQ7mUI
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Muttley79
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https://www.drfrostmaths.com/resource.php?rid=56

Some misconceptions and common errors.
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Sir Cumference
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(Original post by Pangol)
Great advice for the vast majority of cases. The only time where I think that the tables still have the edge is when doing binomial significance tests. It is very quick and easy to see where the cutoff point is between a probability of <0.05 and >0.05 is (or whichever significance level you are going for), so finding critical regions is faster this way.
With lots of calculator practice you can get the critical region very fast but for the average student I can see that using tables may be a bit faster in this case. I'm just not a fan of switching between between calculator/tables and would have liked the exam boards to abandon tables altogether - I find it old fashioned to list values in a table when they can be calculated using a calculator/computer.
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Pangol
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(Original post by Sir Cumference)
With lots of calculator practice you can get the critical region very fast but for the average student I can see that using tables may be a bit faster in this case. I'm just not a fan of switching between between calculator/tables and would have liked the exam boards to abandon tables altogether - I find it old fashioned to list values in a table when they can be calculated using a calculator/computer.
I have to admit that I was a bit surprised to see binomial tables in the formula book considering the new stipulation that calculators had to be able to calculate binomial probabilities.
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Gent2324
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i mean with like the hypothesis testing where you have the pmc and the sample size, page 9 in the stats book year 2 .
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Sir Cumference
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#54
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(Original post by Gent2324)
i mean with like the hypothesis testing where you have the pmc and the sample size, page 9 in the stats book year 2 .
For that use the tables if you're Edexcel. For other exam boards you'll be given the data you need if there are no stats tables in the formula book.
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NotNotBatman
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One mistake is to look at the integral \displaystyle \int \frac{\mathrm{d}x}{a^2-x^2} and think you need to use some form of a hyperbolic substitution, but no, it is a much simpler partial fraction type integral.

(Mobile so feel free to edit this post)
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NotNotBatman
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One mistake i see is not understanding how intersection and union of events work and what the rules are for independent and dependent events.

One trick for independent events is to turn english into maths . and becomes multiplication or becomes addition. I cant remember if the law of total probability ia taught at A level but essentially you need to find the probability of some thing and there are different ways to get the outcome of that event, you need to sum the possibilities. The trick above helps.

For example rolling two dice; whats the probability that the sum is even.

Well thats getting 1 and 1 or 1 and 3 or ...

Becase the events are independent you can say that p(1)p(1) + p(1)p(3) +...

Can you see how the trick has been used above? Only works for independent events.
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NotNotBatman
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One common mistake Ive seen is in using the cover up method for partial fractions. This only works if there are linear terms in the denominator, you cant do it with a quadratic denominator for example. This is not necessary to learn so dont panic if you dont know it. it's just a trick some students use.
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NotNotBatman
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Confusion of notation.
Thinking that  \dfrac{\mathrm{d}y}{\mathrm{d}x} is different from \frac{\mathrm{d}}{\mathrm{d}x} (y)

Saying things such as" find the dy/dx of y=... "
"Find the differential of ..."
These seem nitpicky, but it may cause some to write things down that don't make any sense.
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laurawatt
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Don’t forget +C :shakecane:

Started integration today and must have left it out in at least half of my answers :lol:
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NotNotBatman
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So vectors is only further maths now i believe.

The vector  \overrightarrow{AB} \neq b-a in general.

Im not sure why many teachers say this without even defining what a and b are.

It may be true, but you'd need to define the vector a to mean OA and b to be OB. Instead remember its "O to the last letter subtract O to the first letter", then you can define those vectors or usr it if it's labelled in the question.

To see why this is wrong. What if i made a question and called the vector joining the origin to A , say, b?
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