A Level Maths : Common Mistakes/misconceptions Watch

Pangol
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#21
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You can't prove an identity by assuming it to start with, doing some valid working, ending up with something true (such as 0 = 0) and deducing that your starting place was therefore true. False statements can lead to true ones, so just because you've arrived at something true, it doesn't mean you started with something true.
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Sidd1
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(Original post by Sir Cumference)
I thought it would be useful for A Level maths students if we compile a list of common mistakes and misconceptions that occur in pure, stats and mechanics. Please post anything that you can think of.

Here are two to start things off and I'll add more at some point:

- The standard trig derivatives that you've learnt e.g. \frac{d}{dx}\left(\sin x\right) = \cos x are only true if the angle (x in this case) is measured in radians. So as a general rule it's best to always work in radians (especially if the question involves calculus) unless a question specifically mentions degrees. Of course if you haven't learnt radians yet then everything will be in degrees!

- For trig equations you should only divide the equation by a trig function if you have considered whether the function could be equal to 0. For example, if you have \sin x \cos x + \cos x = 0 and you divide by \cos x you will get \sin x + 1 = 0 but you will lose the solutions for when \cos x =0. Normally it's best to factorise instead of dividing, so here you would get \cos x\left(\sin x + 1\right) = 0. This doesn't just apply to trig equations e.g. x^2 + x = 0 \Rightarrow x + 1 = 0 but it's common for the mistake to occur with trig equations.
I am so happy this has been created so useful!!!
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Sir Cumference
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(Original post by Sidd1)
I am so happy this has been created so useful!!!
Great It will keep being added to during the year.
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Sidd1
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(Original post by Sir Cumference)
Great It will keep being added to during the year.
Does anyone have any useful tips for revising Mechanics and understanding it? Any predictions for this years papers?
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3pointonefour
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#25
The reaction force at a hinge is variable and typically has 2 components

a^x > b does NOT NECESSARILY imply x > ln(b)/ln(a), you have to check if 0 < a < 1. If so, then ln(a) is negative which would flip the inequality sign.


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Sorry but cba to latex
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ghostwalker
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#26
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Just been digging through the old threads on here and turned up.

1) Contrary to the belief that every number has only one representation, every non-zero terminating decimal has another representation with infinite trailing 9's.

E.g. 1, can be written as 0.999999..., or more succinctly as 0.\dot{9}


2) F=\mu R applies in all situations relating to friction.

NO. Only if you have limiting eqilibrium or motion between the two surfaces. Otherwise F&lt;\mu R

Plus one that just came to mind:

3) Assuming an angle in a diagram is a right-angle 'cause it looks about right.
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robertoooo
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(Original post by Sir Cumference)
I thought it would be useful for A Level maths students if we compile a list of common mistakes and misconceptions that occur in pure, stats and mechanics. Please post anything that you can think of.

Here are two to start things off and I'll add more at some point:

- The standard trig derivatives that you've learnt e.g. \frac{d}{dx}\left(\sin x\right) = \cos x are only true if the angle (x in this case) is measured in radians. So as a general rule it's best to always work in radians (especially if the question involves calculus) unless a question specifically mentions degrees. Of course if you haven't learnt radians yet then everything will be in degrees!

- For trig equations you should only divide the equation by a trig function if you have considered whether the function could be equal to 0. For example, if you have \sin x \cos x + \cos x = 0 and you divide by \cos x you will get \sin x + 1 = 0 but you will lose the solutions for when \cos x =0. Normally it's best to factorise instead of dividing, so here you would get \cos x\left(\sin x + 1\right) = 0. This doesn't just apply to trig equations e.g. x^2 + x = 0 \Rightarrow x + 1 = 0 but it's common for the mistake to occur with trig equations.
‘Maths is hard’
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Dancer2001
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(Original post by Sidd1)
Does anyone have any useful tips for revising Mechanics and understanding it? Any predictions for this years papers?
Loads of basic trig practice. Once you can resolve angles, a lot of it suddenly makes sense. If you understand how to do it by drawing triangles, you can derive the formulas if you forget them in the exam. Having a look at polar coordinates from further maths might help as well.
Remember that you have SUVATs in the formulae book, and do lots of practice questions.
Last edited by Dancer2001; 1 month ago
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3pointonefour
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Another one to do with logarithms:

Remember that a logarithm can only take in positive values, so when solving an equation and you find a solution that leads to negative values going into a logarithm, you have to discard that value. For example:
(Here log(x) is to the base 2)
log(x-2) + log(x-1) = 1
After doing the log rules and solving the quadratic, you'll get x = 0 and x = 3 as solutions. However notice that subbing in x = 0 will give a log(-2) which is undefined for real numbers, hence DISCARD x = 0 and thus your only solution is x = 3


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Wish I had the energy to latex it probably would've looked much more readable
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Gent2324
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(Original post by Sir Cumference)
The dreaded "reverse chain rule": some students take it literally and say things like this

\displaystyle \int (x^2+3)^3 \ dx = \frac{1}{8x}(x^2+3)^4 + c

\displaystyle \int e^{x^2} \ dx = \frac{1}{2x}e^{x^2}+ c

These are both wrong. The only time you can do this kind of thing is if the inner function is linear e.g.

\displaystyle \int (2x+3)^3 \ dx = \frac{1}{8}(2x+3)^4 + c

If in doubt, use a substitution.
just to confirm we arent suppose to know how to integrate e^x^2 right?
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Sir Cumference
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#31
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#31
(Original post by Gent2324)
just to confirm we arent suppose to know how to integrate e^x^2 right?
No you can’t integrate that “in terms of elementary functions” which basically means you can’t integrate it at A Level.
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the bear
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SOH CAH TOA does not work unless there is a right angle in the triangle

:facepalm2:
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Sir Cumference
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#33
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For stats I find that it's common for students to forget to apply continuity correction when approximating a binomial distribution using a normal distribution.
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Sidd1
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(Original post by Dancer2001)
Loads of basic trig practice. Once you can resolve angles, a lot of it suddenly makes sense. If you understand how to do it by drawing triangles, you can derive the formulas if you forget them in the exam. Having a look at polar coordinates from further maths might help as well.
Remember that you have SUVATs in the formulae book, and do lots of practice questions.
Resolving to work out the resultant force? Oh goddd I hate that method I just resovle the x's and y's and find the resultant. Oh okayy, I'll try doing more questions thanks I don't do Further Maths lmaooo I'm already mentally drained from A level Maths.
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Sidd1
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(Original post by Sir Cumference)
For stats I find that it's common for students to forget to apply continuity correction when approximating a binomial distribution using a normal distribution.
Yess that's true I'm one of those students
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Pangol
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Don't try to solve all of the parts of circle questions just by using algebra. They are usually set up so that if you draw a sketch, the answers are obvious. Playing with the equations can often get you there as well, but it can involve a huge amount of unnecessary work, way out of proportion to the marks on offer.
Last edited by Pangol; 1 month ago
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Glaz
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Watching this, I love you all so much
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Sir Cumference
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If considering changes of sign to locate roots, a change of sign does not necessarily mean a single root. There could also be 3 roots, 5 roots etc., i.e. an odd number of roots. If the function isn't continuous then there could be no roots.

Also if there is no change of sign then don't assume that there are no roots. There could be 2 roots, 4 roots etc., i.e. an even number of roots.

A sketch of the different scenarios should convince you of all this and it will stick in your mind.
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Pangol
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Your calculator can do a lot of things, in particular work out the values of definite integals. You won't get any marks if you just use your calculator to get an answer, but if you have spent ages doing a complicated definite integral, check it on your calculator. If you are right, that's a big question you don't have to double check.
Last edited by Pangol; 1 month ago
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ghostwalker
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(Original post by Sir Cumference)
Also if there is no change of sign then don't assume that there are no roots. There could be 2 roots, 4 roots etc., i.e. an even number of roots.
To expand on that slightly:

The roots can be repeated. E.g. a double root. A simple example being y=(x-1)^2 which kisses the x-axis, but doesn't cross it. No sign change (comparing either side of x=1) and only 1 root, x=1, though we can think of it as a double root due to the square; so in that regard, yes, an even number of roots.
Life is rarely simple.

Edit: Similar generalisations apply to the odd number of roots, and to triple roots, etc.
Last edited by ghostwalker; 1 month ago
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