maths

• Partial Fractions


  • Be able to split a fraction whose denominator is a product of linear expressions, e.g. 2x+3/x(x+1)
  • Be able to split a fraction where one (or more) of the factors in the denominator are squared, e.g. 2x+3/x^2(x+1)
  • Deal with top-heavy fractions where the highest power in the denominator is greater or equal to the highest power in the denominator, e.g. x^2+2/x(x+1)
• Dealing with top heavy fractions


  1. Use algebraic long division method

  2. Quotient of 1 and remainder of -x + 2

  3. Split the -x+2/x(x+1) into partial fractions as normal
• When you have a squared factor in the denominator, you need two fractions in your partial fraction sum: 2/x^2(x+1) ≡ A/x + B/x^2 + C/(x+1)
• Dealing with three unknowns in partial fractions


  1. Use substitution to get two of them (e.g. the A and the B)

  2. Compare the coefficients of x^2 to get the C
• Partial Fractions


  • Forgetting the extra term when the denominator's factors are squared
  • Being sloppy at algebraic long division
  • Careful with substitution of negative values
  • May need to factorise the denominator first before expressing as partial fractions
  • Not realising the fraction is top heavy and therefore trying to incorrectly do: 2x^2/x(x+1) = A/x + B/(x+1)
• Parametric Equations


  • dy/dx = (dy/dt)/(dx/dt) (This makes sense as we have just divided numerator and denominator by dt)
  • Be able to integrate parametric equations
  • Be able to convert parametric equations into a single Cartesian one
• Converting parametric equations involving trig functions to Cartesian

  1. Make sin x and cos x the subject before using the identity sin^2 x + cos^2 x ≡ 1

  2. Often squaring one of the parametric equations helps so that we have sin^2 x and/or cos^2 x
• Parametric Equations


  • Hitting a dead end converting parametric equations to Cartesian
  • Forgetting to multiply by dx/dt when integrating parametric equations
• Binomial Expansion


  • Expanding out an expression of the form (1 + kx)^n, where n is negative or fractional
  • Expanding out an expression of the form (a + kx)^n, where a needs to be factorised out first
  • Finding the product of two Binomial expansions, e.g. √(1+x)/√(1-x) → (1+x)^(1/2)(1-x)^(-1/2)
• Binomial Expansion


  1. (1 + kx)^n = 1 + n(kx) + n(n-1)/2!(kx)^2 + n(n-1)(n-2)/3!(kx)^3 + ...

  2. Factorise the first term out if it is not 1, raised to the power outside the brackets

  3. When finding the product of two expansions, only consider terms up to x^2
• Binomial Expansion


  • Lack of brackets when squaring/cubing things
  • Forgetting to raise the factor you take out to the power
  • Forgetting the factorials in the binomial coefficients
  • Being careless with signs
  • Forgetting the minus in the power when expanding 1/(x+1)^2
• Differentiation


  • y = ax represents 'exponential growth' when a > 1, and 'exponential decay' when 0 < a < 1
  • d/dx(a^x) = a^x ln a
  • Be able to differentiate implicitly
  • Be able to set up differential equations
  • Connect different derivatives involving rates, e.g. dA/dx = dA/dt * dt/dx
• Implicit Differentiation


  Collect the dy/dx terms on one side and factorise it out
• Differential Equation


  An equation involving variables and derivatives of those variables
• Whenever you see the word 'rate', think /dt
• Differentiation


  • Accidentally treating x or y as constants rather than variables
  • Forgetting to put the dy/dx when differentiating implicitly
  • Exponential functions do not behave like polynomials when differentiated
  • Mistakes when connecting rates of change
• Vectors


  • Find the point of intersection of two lines or prove that two lines do not intersect
  • Find the angle between two vectors
  • Find the magnitude and direction of a vector
  • Find the equation of a line in vector form
  • Find the equation of a plane in vector form
  • Solve problems involving vectors
• A


  10 = 20π
• Differentiating A


  dA/dt = 20π × 2 = 40π
• A classic mistake is to accidentally treat x or y as constants rather than variables, when differentiating implicitly
• The correct differentiation of xy is x(dy/dx) + y, not just y
• When differentiating implicitly, you might forget to put the dy/dx
• Exponential functions do not behave like polynomials when differentiated
• Many students often get their equation wrong when connecting rates of change, often say dividing instead of multiplying, or vice versa
• Common vector questions (in rough descending order of frequency)


  • Find the point of intersection of two lines or prove that two lines do not intersect
  • Find the angle between two lines
  • Finding a missing x/y/z value of a point on a line
  • Find the length of a vector or the distance between two points
  • Find the nearest point on a line to a point not on the line (often the origin) – note: not in your textbook!
  • Show lines are perpendicular
  • Show a point lies on a line
  • Show 3 points are collinear (i.e. lie on the same straight line)
  • Find the area of a rectangle, parallelogram or triangle formed by vectors
  • Find the equation of a line
  • Find the reflection of a point in a line
  • Find the point after going a specific distance in the direction of a given vector
• When you see the i, j, k unit vectors used in an exam question, never actually use this notation yourself: always just write all vectors in conventional column vector form
• Almost always draw a suitable diagram
• When finding the area of a shape, you can almost always use your answers from previous parts of the questions, including lengths of vectors and angles between two vectors
• Area of non-right angled triangle

  1/2 ab sin C
• A parallelogram can be cut in half to form two congruent non-right angled triangles (i.e. multiply by 2)
• To show 3 points A, B, C are collinear, just show that AB is a multiple of BC (i.e. vectors are parallel)
• When finding the angle between two lines, accidentally using the full vector representation of the line (in your dot product), and not just the direction component
• Making sign errors when subtracting vectors, particularly when subtracting an expression involving a negative
• Once finding out s and t (or μ and λ) when solving simultaneous equation to find the intersection of two lines, forgetting to show that these satisfy the remaining equation
• Forgetting the square root when finding the magnitude of a vector
• Integration topics


  • Integrating trig functions, including reciprocal functions and squared functions
  • Integrating by 'reverse chain rule' (also known as 'integration by inspection')
  • Integrating by a given substitution
  • Integration by parts
  • Integrating by use of partial fractions
  • Integrating top heavy fractions by algebraic division
• One often forgotten integration is exponential functions such as 2^x
• Know the two double angle formulae for cos like the back of your hand, for use when integrating sin^2 x or cos^2 x
• For integration by 'reverse chain rule', always 'consider' some sensible expression to differentiate, then adjust for the factor difference
• For integration by substitution, the official specification says "Except in the simplest of cases, the substitution will be given"