Mathematics Quick Revision Notes │ Helpful Tips for Success │ Study Guide for Exams

1. C4 REVISION

2. Partial Fractions • Set up as 3 separate fractions – usually one its own, one squared and one of each next to other • Sub in values of x to determine A, B and C (first 2 picks should make a bracket worth 0) • If you have to integrate at the end don’t forget to look for logs and to put +C on the end!!

3. Binomial • Formula in booklet – no need to learn • Don’t forget (-3x)² = + 9x² • To get range of valid values, reciprocate coefficient of x. e.g. (1 + 4x)½ is |x| < ¼

4. R - α • If they don’t give you an identity to use then pick Rcos(θ – α) • Quick trick – square, add and root coefficients to get R and use tan -1 sin coefficient to get α cos coefficient

5. Double angle formulae • You MUST learn the following: • sin 2θ = 2sinθcosθ Generally, pick the formula that will eventually • cos 2θ = cos2θ – sin2θ set up a quadratic (or something that factorises). • cos2θ = 1 – 2sin2θ Sometimes, these will also have to be rearranged - • cos2θ = 2cos2θ – 1 we cant ∫ sin2 or cos2 so rearrange to get • tan2θ = 2tanθ ½ + ½ cos2θetc 1- tan2θ

6. Parametric Equations • Find dx/dt and dy/dt then calculate dy/dt x dt/dx to get dy/dx • This gives us the gradient of the Tangent. For a ‘normal’ reciprocate and negate. • You will probably then have to sub in some value of p to create a given expression. If this is a cubic that needs to be solved, do this by trial and improvement (its usually either 1, -1, 2 or -2)

7. Volume of Solids of Revolution For y = ƒ(x) then vol = π ∫ y²dx. The limits are the coords that cut the x axis. Look out for Trig!! Remember we cant ∫ sin² or cos² so swap it using the ‘double angle formulae’ If the function is a y = 3+x² type, when you square it don’t forget to write as (3+x²)(3+x²) to get 9+6x²+x4 NOT just 9+x4 Leave in terms of π unless otherwise stated. Look out for top being differential of bottom – Logs!!!

8. Integration by parts • Formula in booklet • ALWAYS use ln x = u and ex = dv/dx

9. Integration by Substitution • LEARN THIS ROUTINE : • Step 1 – find du/dx and rearrange so we can get rid of the dx • Step 2 – find the new limits by substituting the original limits into the u = equation • Step 3 – re-write the whole integral with new limits, new u and new du • Step 4 – integrate this and finish

10. Differential Equations • If a simple direct proportion, write as dp/dt = kp then separate and ∫ to get ∫ 1/p dp = ∫ k dt which gives ln |p| = ½kt • Now take “e’s” to get p = Aekt (not always a simple direct proportion though!!) • They’ll give you an initial value (i.e. t = 0) which you sub in to get A then another value to work out k. • Check your final answer make sense in context!!

11. Vectors • Square, add, root to find magnitude of vector |a| • AB = b – a • Parallel vectors are multiples of each other e.g. 3i + 2j + 5k and 9i + 6j + 15k are parallel • Dot product a.b = aibi + ajbj + akbk and = 0 if PERPENDICULAR • Angle between vectors cos θ = a.b |a||b| • Intersecting lines: set up simultaneous equations to work out λ and μ : If they intersect λ and μsatisfy all 3 equations If skew they only satisfy 2 equations If parallel there are no solutions for λ and μ

12. Proof by contradiction • Learn √2 proof from notes. √3 and √5 proofs are the same except swap all 2’s for 3’s/5’s. • Other proofs usually involve getting a quadratic and showing that they have no roots.