FP1

FP1

Complex Numbers
1. i= √(-1)
  1. i^2= -1
  2. i^odd= ± i
  3. i^even= ± 1
2. Dividing
  1. "realise" the denominator
    1. if you have x/(a+bi)
      1. times by (a-bi)/(a-bi)
        sames as x1
        complex conjugate
        if z = a+ib
        z*=a-ib
        zz*= real no.
        difference of 2 squares
    2. make bottom a real no.
3. modulus and argument
  1. argand diagram
    1. vectors
      1. magnitude
        modulus, r or |z|
        length
        pythag
      2. direction
        argument, Arg(z)
        angle between vector and x-axis
        tanθ= y/x
  2. modulus-argument form
    1. if z=x+iy
      1. by trig
        x=rcosθ
        y=rsinθ
      2. z=r(cosθ+i sinθ)
4. in equations
  1. equating real and imaginary
    1. if a+ib=c+id
      1. a=c and b=c
    2. find √(15+8i)
      1. let √(15+8i)= a+ib
        15+8i=(a+ib)^2
        etc.
  2. quadratic roots
    1. if z is a root
      1. so is z*
Co-ordinate Systems
1. Parabola
  1. equation
    1. cartesian
      1. y^2=4ax
    2. parametric
      1. x=at^2
        y=2at
2. Rectangular Hyperbola
  1. equations
    1. cartesian
      1. xy=c^2
    2. Parametric
      1. x=ct
        y=c/t
3. tangents and normals
  1. m(t)=dy/dx
    1. m(n)=-1/m(t)
      1. m(n)x m(t)=-1
  2. y-y1=m(x-x1)
Proof by Induction
1. method
  1. prove true for n=1
    1. assume true for n=k
      1. show true for n=k+1
        state true for all n≥1 where nϵ Ζ+
2. types
  1. series
    1. swap k for n
      1. NOT r
  2. divisibility
    1. consider f(k+1)-mf(k)
      1. show difference is divisible by a
        therefore f(k+1) is also divisible by a
        remember rules of indicies
        a^n x a^m= a^n+m
        (a^n)^m= a^nm
  3. Matrices
    1. Sub step 2 into step 3
      1. M^k+1
        same as M(M^k)
  4. recurrence relationships
    1. show true for
      1. n=1 AND n=2
        for step 3
        use recurrence formula for U k+1
        sub U k in
3. write your target!
Series
1. General formulae
  1. r
  2. r^2
  3. r^3
  4. 1
2. sum between 2 limits
  1. sum up to top limit
    1. minus sum up to bottom limit -1
3. to show a summation formula = ....
  1. take out common factors
Numerical Methods
1. show root in interval [a,b]
  1. find f(a) and f(b)
    1. change in sign
      1. root between a and b
2. Interval bisection
  1. next estimate
    1. midpoint
      1. (a+b)/2
      2. between a and b
        where f(a) -ve and f(b) +ve
3. linear interpolation
  1. if root lies in [a,b]
    1. use similar triangles
    2. or formula
      1. not in data book
4. Newton Raphson
  1. find f'(x)
    1. write out f(a) and f'(a)
  2. in formula book
    1. doesn't work @ turning point
      1. f'(x)=0
Matrices
1. multiplying
  1. only multiply if
    1. no. of columns of 1st matrix
      1. same as no. of rows of 2nd
        product dimensions
        same no. rows as 1st matrix
        Same no. of columns as 2nd
  2. not comutative
    1. AB≠BA
2. dimensions
  1. (nxm)
    1. n= rows
      1. m=columns
        simultaneous equations
3. Transformations
  1. vectors
    1. position vector
      1. from origin
        can be written (x,y)
        translation vector
        from given point
  2. linear
    1. linear expressions
      1. point (0,0) unchanged
        reflections
        y-axis
        x-axis
        y=x
        y=-x
        enlargement
        scale factor a
        rotation
        180°
        90°
        clockwise
        anticlockwise
        45°
        clockwise
        anticlockwise
  3. inverse matrix
    1. transforms back to original
4. Inverse
  1. A^-1
    1. AA^-1=I
      1. I= identity matrix
      2. determinant
        ad-bc
        singular if =0
        transform shape
        straight line
        0 area

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