When you read through this, a lot of it will sound obvious or stupid. There's a reason we call them silly mistakes.
General advice for A-level Maths
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.
Re-do 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.
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.
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.
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.
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.
Do not forget to calculate y using the original equations once you get x.
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^(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 and 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.
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.
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.
Check your calculator mode. Convert degrees to radians.
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.
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
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 a function like 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.
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.
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.
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.
When you’re rewriting parametrics in Cartesian form, you’re trying to remove the parameter.
You can sub in a value of x into both the original fraction and your answer to see if they match.
If two lines are perpendicular, there’s a right angle between them. Hence, the scalar product (or dot product) is equal to 0.