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Exam Techniques: How to Check Your Answers, and Silly Mistakes to Avoid

There've been a couple of threads asking about how to check your answers properly recently. What I've done is collate a list of things you can do to check certain topics in each core module, and also some of the more common or easily made mistakes per section. I have no idea if this will actually be useful, but I was bored and needed something to do. If it does help someone then :smile:

, and if it doesn't at least I've passed some time :tongue:

ONLY CARRY OUT THESE CHECKS IF YOU HAVE FINISHED THE WHOLE PAPER

When you read through this, a lot of it will sound obvious, or stupid. There's a reason we call them silly mistakes.

Please feel free to suggest extra advice. I'll incorporate it and give you full credit. Not all of the topics are covered - I struggled to think of things to say in them, so didn't include them.

Also, if anyone wants to do a similar list for the applied modules, feel free, and I'll include it in here.

General Advice

It can be tempting to try and do large leaps of working in your head if you're confident. If you do, make sure you write the question the 'long way' when you're checking, minus signs have a horrible tendency to appear occasionally!

Something you may not be doing already: after doing a past paper, if you have access to it, read the examiner’s report. It usually has some interesting information on what students struggled with, common mistakes on parts of each question etc.

If you factorise something, multiply it back out to check it.

If you solve something, put your value back in to check it.

Redo any calculator work.

Always check your calculator is in the right mode (degrees or radians) before answering a question.

Check each line of your working as you do it.

Read the question twice before you start it.

The following was contributed by ttoby:

You can check definite integrals by typing them into a graphics calculator. You can do the same thing for trapesium/Simpson's rule questions but in this case you would expect your answer and the calculator's answer to be slightly different.

You can check if two expressions are equal by taking each variable and assigning it a value (e.g. 5) then evaluating the two expressions in your calculator. For example you could check the line (x+2)(x+3)=x^2+5x+6 by calculating (7+2)(7+3) and 7^2+5*7+6 in your calculator. In this case I was checking it for x=7.

If you've just calculated the equation of a line, and you're told that points A and B lie on the line, then substitute those points back into your equation to check.

If a formula you need to use is printed in the formula booklet, then even if you have it memorised, check that your formula matches what's in the formula booklet. If at any point you had to use a formula that's not in the formula booklet then check that your memorised formula is valid by deriving it from other formulae you know (a good example for this is the formula for cos2x and sin2x which you should check even if you have them memorised).

Keep any workings associated with checking completely separate from your actual answer. I usually scribble all over the question paper but if you do it in the answer booklet make sure it's all clearly crossed out to avoid confusing the examiner.

Core Mathematics 1

Coordinate Geometry
If you're finding the equation of a line, is your gradient positive or negative? Which did you expect it to be? If it's a rising line and you have a negative gradient, you'll want to check your answer for missed minus signs.

Remember the difference between a tangent and a normal: tangents have the same gradient, whereas the gradient of the normal multiplied by the gradient of the line/curve equals -1.

When you're using

y-y_1 = m(x-x_1)

and it asks for the equation where the curve is at its minimum/maximum point, remember that the gradient will be 0. It's as simple as that; the equation will be

y-y_1 = 0

(i.e

y = y_1

. Courtesy of Contrad!ction.

Differentiation
Don't panic if you're faced with something horrible looking, the rules are still the same!

Rewrite fractions where x is the denominator so that they're in power form, same principle for square roots, write them as fractional powers.

Inequalities
Sketch the graph. Seriously, sketch it. Mark the region you want. This should stop you writing your inequalities backwards.

Also remember that multiplying or dividing by a negative number will change the orientation of the inequality. ( > becomes <, and vice versa).

Do you know the difference between the following phrases: “At least”, “no more than”, “more than” “less than” and “at most”? If you have to write the inequalities yourself, make sure you choose correctly between using strict inequalities and not using them.

Simultaneous Equations
As with all questions which involve getting solutions of equations: put your values for x and y back into the equation, and see if they work.

Remember to calculate y using the original equations once you get x.

Solving Equations
As with simultaneous equations, substitute your answer into the original equation to check it.

If you're dealing with a disguised quadratic don't forget that after you solve the equation you still need to find x from whatever variable you used as a 'disguise'. (E.g.

x = t^{\frac{1}{2}}

- you still need to find x after getting t)
Remember that you *can* have a negative discriminant; it just means there are no real roots.

Core Mathematics 2

Arithmetic + Geometric Series
Make sure you’re using the right equations. My old classmate didn’t check his terms closely enough and tried to work out a geometric series question using arithmetic equations.

Integration
If you differentiate your answer, you should get the original equation.
+c. +c. +c. +c. You WILL lose a mark if you forget your constant in an indefinite integral.

Be very careful with minus signs when evaluating definite integrals. Don’t do them in your head.

Logarithms
Know your log laws: things like log(A+B) do NOT equal anything other than log(A+B). You have log(AB) = log(A) + log(B), log(A/B) = log(A) – log(B) and Alog(B) = log(B)^A, but that’s it.

Radians
Check your calculator mode. Convert degrees to radians.

Trapezium Rule
If you have n intervals, you’ll have (n + 1) lots of coordinates. (5 lines parallel to the y-axis, 4 areas to calculate)

Lay out your working well. If your working isn’t organised for a question with a large numbers of intervals you will potentially confuse both yourself and the examiner when you try and read through what you’ve done.

Trigonometry
You need to know your identities. It cannot be stressed how important they are.
Make sure you’re working in the right range, if you’re asked for an answer between 0 and pi, don’t say theta = –pi/4.

If the question uses degrees, use degrees. If it uses radians, use radians.

Core Mathematics 3

Differentiation
Know when to use the chain rule, when to use the product rule and when to use the quotient rule.

Write out things fully – you might be fully capable of differentiating

\frac{1}{x}(\sqrt{x} + 3x)^{-3}

in your head, but don’t risk making a mistake.

Also, you can check quotient rule answers by using the product rule on the original equation by putting the denominator to the power -1. It can get messy, but it does work.

Functions
Know the difference between domain and range. Domain is the set of numbers for which a function is defined, range is the numbers which a function can output.

Reverse Chain Rule (Integration)
You can differentiate your answer to check it.
Don’t forget to +C.

Solving Equations with Numerical Methods
If you’re using an iterative formula, use your calculator’s memory.

If you starting value isn’t given, you need to choose one that’s close to the root. Sketch the graph quickly if you aren’t sure.

Trigonometry: Addition formula
These are in the formula booklet. Double Angles aren’t (AFAIK), so make sure you know how to derive them.

If you can answer the last question on the January 2008 paper then you should be pretty good in this area. Just be careful with the minus signs.

Trigonometry: sec / cosec / cot
Remember which trig function is the reciprocal of which. If it helps, there’s the maths pickup line: “Hey, you’re rather 1/cos(c)”. I’ll let you work that one out.

Core Mathematics 4

Algebraic Long Division
Remember that x^3 + 3 = x^3 + 0x^2 + 0x + 3. Just because the question gives you something like ax^3 + b doesn’t mean you can ignore the x^2 and x terms.

Binomial Theorem (non-positive integer n)
To expand brackets with a non-positive integer power, they have to be in the form (1 + x)^n. If you originally have (a + x)^n, you’ll need to rewrite the function.

Pay careful attention to minus signs. Use brackets everywhere if necessary.

Differential Equations
Check the wording in the question carefully, it may hint towards a minus sign needing to be in your answer (e.g. rate of cooling questions).

Differentiating/Integrating Trigonometric Functions
When you differentiate: Sin(x) -> Cos(x) -> -Sin(x) -> -Cos(x) -> Sin(x)…

Follow that chain backwards when integrating.

You have to be in radians for this to work.

Implicit Differentiation
When you implicitly differentiate y with respect to x, you get a dy/dx. You don’t treat it as a constant.

Integration: By Parts, By Substitution
As always, remember the +C in an indefinite integral.

When integrating by parts, write out clearly which part of your integrand is u, and which is dv.

When integrating by substitution you need to change the limits of integration if you leave your answer in terms of your substitution. If you don’t, then you’ll need to rewrite your answer in terms of the original variable.

Courtesy of nuodai:

I wrote a couple of guides about integration by substitution and integration by parts over the summer.

Parametric Equations
When you’re rewriting parametrics in Cartesian form, you’re trying to remove the parameter.

Partial Fractions
You can sub in a value of x into both the original fraction and your answer to see if they match.

Vectors
If two lines are perpendicular, there’s a right angle between them. Hence, the scalar product (or dot product) is equal to 0.

EDIT1: Don't you just love it when you submit a post and then go "Oops forgot to do x"
EDIT2: FML - I just saw the time!

(edited 13 years ago)

Reply 1

infernalcradle

any chance you could do one thats for exams in general and not specific for maths

(bio and chem would be most appreiciated :biggrin:

)

Reply 2

EEngWillow

Original post by infernalcradle

any chance you could do one thats for exams in general and not specific for maths

(bio and chem would be most appreiciated :biggrin:

)

I doubt that I'd be able to, sorry. It's a lot easier to give a detailed list like this for a subject you actually know. I could do it for Physics or Electronics...but that's about it really. You might find it a useful revision exercise to try and make your own list. Stopping to think about where you can make mistakes in each topic, or what areas are confusing for most students will help you avoid those mistakes in the future...hopefully.

That said, the general advice is pretty universal: any calculations you've done, redo them. Read the question twice before you start it, and again when you're done - ask yourself "have I actually answered the question?".

Reply 3

nuodai

This would be a good thing to make a wiki page about.

FWIW, I wrote a couple of guides about integration by substitution and integration by parts over the summer, and intend to write more... so they're there if anyone finds them useful. They're a bit waffly though... I could do with improving my writing style a bit :p:

Reply 4

EEngWillow

Original post by nuodai

Didn't think about that, wiki page could be a good idea. Never made one though. And you, waffly? Your English seems so good! :P

Reply 5

Contrad!ction.

I'm good at mistakes! Guilty of so many of those. Here's another thing:

This came up in a past paper I did

Core 1: when you're using

y-y_1 = m(x-x_1)

and it asks for the equation where the curve is at its minimum/maximum point, remember that the gradient will be 0. It's as simple as that; the equation will be

y-y_1 = 0

y = y_1

I didn't make the connection and ended up with the gradient as some disgusting fraction.

I think personally, with no influence from anyone else, especially not the thread starter :wink:

, that this would benefit from a sticky. :smile:

Reply 6

EEngWillow

Thanks to the moderator who moved this to the proper forum :smile:

Getting messages from people saying this has been really useful, so I'll continue working along these lines and maybe try and make other 'checking / silly mistake' related resources iff they're wanted.

Original post by nuodai

FWIW, I wrote a couple of guides about integration by substitution and integration by parts over the summer, and intend to write more... so they're there if anyone finds them useful. They're a bit waffly though... I could do with improving my writing style a bit :p:

Had a quick look at your two articles and they're really good - have included them under the C4 section. Hope you don't mind. If you ever do more please let me know, they were a good read! :smile:

Original post by Contrad!ction.

I'm good at mistakes! Guilty of so many of those. Here's another thing:

Thanks for the contribution, added it.

Reply 7

ttoby

I have some to add:

If a question ever asks you to calculate an angle shown on a diagram, check your answer with a protractor. Most A-level diagrams that I've seen are drawn to scale.

You can check definite integrals by typing them into a graphics calculator. You can do the same thing for trapesium/Simpson's rule questions but in this case you would expect your answer and the calculator's answer to be slightly different.

Check maximum/minimum questions by sketching a graph. Use a graphics calculator to do this if permitted; otherwise sketch the graph by hand. Sketch graphs themselves can be checked by making a table of x,y values.

You can check if two expressions are equal by taking each variable and assigning it a value (e.g. 5) then evaluating the two expressions in your calculator. For example you could check the line (x+2)(x+3)=x^2+5x+6 by calculating (7+2)(7+3) and 7^2+5*7+6 in your calculator. In this case I was checking it for x=7.

Check any geometrical questions by doing a scale drawing.

If you've just calculated the equation of a line, and you're told that points A and B lie on the line, then substitute those points back into your equation to check.

Do a whole question again, but working with rounded values rather than exact values so that you work it out faster.

If a formula you need to use is printed in the formula booklet, then even if you have it memorised, check that your formula matches what's in the formula booklet. If at any point you had to use a formula that's not in the formula booklet then check that your memorised formula is valid by deriving it from other formulae you know (a good example for this is the formula for cos2x and sin2x which you should check even if you have them memorised).

Keep any workings associated with checking completely separate from your actual answer. I usually scribble all over the question paper but if you do it in the answer booklet make sure it's all clearly crossed out to avoid confusing the examiner.

If you do the same question a second time and get a different answer, don't immediately assume that the second attempt is correct and cross out your first attempt. You should carefully compare your different attempts line by line in order to find the mistake. Often I have found that the original attempt was correct but a second attempt was wrong simply because I did the second attempt much faster and less neatly.

Reply 8

EEngWillow

With the exams starting soon I'd like to try and keep this near the top of the forum, to help as many people as possible.

...So shameless bump.

Hope you mods don't mind,

DFranklin

(edited 13 years ago)

Reply 9

DFranklin

Original post by ttoby

I have some to add:

Some good advice here, but also some I'd argue with I'm afraid...

If a question ever asks you to calculate an angle shown on a diagram, check your answer with a protractor. Most A-level diagrams that I've seen are drawn to scale.

This sounds dangerous to me; I've seen quite a few questions where the diagram was very obviously not to scale.

Check maximum/minimum questions by sketching a graph. Use a graphics calculator to do this if permitted; otherwise sketch the graph by hand. Sketch graphs themselves can be checked by making a table of x,y values.

If you're going to make a table of x, y values you might as well do that first and use it to guide your sketch.

Check any geometrical questions by doing a scale drawing.

Unless you're sufficiently stuck that you can't see how to do the problem, then I don't see this as being cost effective. Rough sketch to make sure you haven't done any blatant 'calculate 180-x instead of x' type errors is enough.

Do a whole question again, but working with rounded values rather than exact values so that you work it out faster.

I can't see rounded values save any significant time, and unless you have a good understanding of rounding, it's hard to be sure whether a final difference is because a calculation is wrong or just due to rounding.

If a formula you need to use is printed in the formula booklet, then even if you have it memorised, check that your formula matches what's in the formula booklet. If at any point you had to use a formula that's not in the formula booklet then check that your memorised formula is valid by deriving it from other formulae you know (a good example for this is the formula for cos2x and sin2x which you should check even if you have them memorised).

In an exam situation, you should know these formulae well enough that you don't need to check. (I'd allow an exception for the sum/product trig formulae, although I certainly knew them off by heart when doing A-levels).

If you do the same question a second time and get a different answer, don't immediately assume that the second attempt is correct and cross out your first attempt. You should carefully compare your different attempts line by line in order to find the mistake. Often I have found that the original attempt was correct but a second attempt was wrong simply because I did the second attempt much faster and less neatly.

Again, in an exam, you have to question whether there's any point doing a sloppy second attempt.

Personally, I always did "long arithmetic" sections twice (things like adding 20 numbers, or a 5-vector dot product), but on the other hand I could do the calculations very quickly both times (because I knew each calculation would be checked). It's better to do a calculation twice with a 95% accuracy rate than once with a 99% accuracy rate.

But for more 'sophisticated' work, I found it a better time trade off to check each line before going on to the next. You won't catch as many errors, but it's a fair bit quicker, and you won't have many 'false-positives' (where you were right to start with, but doing the calculation again you get a different answer that you have to troubleshoot).

Reply 10

EEngWillow

Original post by DFranklin

Some good advice here, but also some I'd argue with I'm afraid...

This sounds dangerous to me; I've seen quite a few questions where the diagram was very obviously not to scale.

If you're going to make a table of x, y values you might as well do that first and use it to guide your sketch.

Unless you're sufficiently stuck that you can't see how to do the problem, then I don't see this as being cost effective. Rough sketch to make sure you haven't done any blatant 'calculate 180-x instead of x' type errors is enough.

I can't see rounded values save any significant time, and unless you have a good understanding of rounding, it's hard to be sure whether a final difference is because a calculation is wrong or just due to rounding.

In an exam situation, you should know these formulae well enough that you don't need to check. (I'd allow an exception for the sum/product trig formulae, although I certainly knew them off by heart when doing A-levels).

Again, in an exam, you have to question whether there's any point doing a sloppy second attempt.

Personally, I always did "long arithmetic" sections twice (things like adding 20 numbers, or a 5-vector dot product), but on the other hand I could do the calculations very quickly both times (because I knew each calculation would be checked). It's better to do a calculation twice with a 95% accuracy rate than once with a 99% accuracy rate.

But for more 'sophisticated' work, I found it a better time trade off to check each line before going on to the next. You won't catch as many errors, but it's a fair bit quicker, and you won't have many 'false-positives' (where you were right to start with, but doing the calculation again you get a different answer that you have to troubleshoot).

Hey DFranklin, thanks for dropping by to give your input.

I agree with these points, and am going to refine the first post over the next hour or so anyway since there are a couple of things I'd like to add or reword.

While you're here, do you think there's any chance of this thread being stickied? It will inevitably be pushed down a page or so each exam season, where it will do little good.

Reply 11

XeNOmEX

Very useful.

The amount of times i missed out the (-) sign on negative gradients when doing C1 past papers is CRAZY!!!!

Reply 12

ttoby

Original post by DFranklin

Some good advice here, but also some I'd argue with I'm afraid...

This sounds dangerous to me; I've seen quite a few questions where the diagram was very obviously not to scale.

Obviously I'd only do this when the diagram looks like it's to scale, and if in fact it turns out that the diagram wasn't to scale then the worst that can happen is that I waste 5 minutes checking something that was right in the first place.

If you're going to make a table of x, y values you might as well do that first and use it to guide your sketch.

It depends how complicated the sketch is. If it's a really horrible one involving cubics or polar coordinates or something then yes, it's definitely a good idea to make a table of values or run it through a graphical calculator first. But if it's something like y=(x-3)^2+2 then a graphical calculator would be a waste of time, unless you have loads of time at the end when it would be helpful to make sure you didn't translate the parabola the wrong way or something.

That was the sort of thing I was thinking of. Obviously I wouldn't waste time making the sketch all neat but a scale drawing would be helpful to check for this kind of mistake.

Yeah you're right. It's not a method I use often anyway.

I don't like to rely just on my memory for these sorts of questions and if I can quickly derive a formula that I'm supposed to remember then I will, in order to prevent any mistakes in how the formulas look.

Again, in an exam, you have to question whether there's any point doing a sloppy second attempt.

If it's a question that I can't check by quicker methods and I have time at the end then I'll do anything to make sure I get every mark. It's certainly not as efficient but it's better than sitting there doing nothing.

Personally, I always did "long arithmetic" sections twice (things like adding 20 numbers, or a 5-vector dot product), but on the other hand I could do the calculations very quickly both times (because I knew each calculation would be checked). It's better to do a calculation twice with a 95% accuracy rate than once with a 99% accuracy rate.

I like that method.

But for more 'sophisticated' work, I found it a better time trade off to check each line before going on to the next. You won't catch as many errors, but it's a fair bit quicker, and you won't have many 'false-positives' (where you were right to start with, but doing the calculation again you get a different answer that you have to troubleshoot).

I often do a brief check like that by eye as I go and yeah it does catch most errors.

Reply 13

Jono404

No one in my maths class seemed to be aware of this so I'll post it here incase it's of use to anyone, you can do the modulus function on a graphical calc (fx9750 GII in this case) by pressing [optn] [f5 (num)] [f1 (ABS)] then putting the graph function in in brackets after the ABS, then putting say y=whatever on the next line down and using the intersection solve to get the required values.