# FP1

### Description

Mind map of the entire FP1 module including equations and diagrams
Mind Map by rb.russell1, updated more than 1 year ago
 Created by rb.russell1 about 9 years ago
34
1

## Resource summary

FP1
1. Complex Numbers
1. i= √(-1)
1. i^2= -1
1. i^odd= ± i
1. i^even= ± 1
2. Dividing
1. "realise" the denominator
1. if you have x/(a+bi)
1. times by (a-bi)/(a-bi)
1. sames as x1
1. complex conjugate
1. if z = a+ib
1. z*=a-ib
1. zz*= real no.
1. difference of 2 squares
2. make bottom a real no.
2. modulus and argument
1. argand diagram
1. vectors
1. magnitude
1. modulus, r or |z|
1. length
1. pythag
2. direction
1. argument, Arg(z)
1. angle between vector and x-axis
1. tanθ= y/x
2. modulus-argument form
1. if z=x+iy
1. by trig
1. x=rcosθ
1. y=rsinθ
2. z=r(cosθ+i sinθ)
2. 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
1. 15+8i=(a+ib)^2
1. etc.
1. if z is a root
1. so is z*
2. Co-ordinate Systems
1. Parabola
1. equation
1. cartesian
1. y^2=4ax
2. parametric
1. x=at^2
1. y=2at
2. Rectangular Hyperbola
1. equations
1. cartesian
1. xy=c^2
2. Parametric
1. x=ct
1. y=c/t
2. 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)
2. Proof by Induction
1. method
1. prove true for n=1
1. assume true for n=k
1. show true for n=k+1
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
1. therefore f(k+1) is also divisible by a
1. remember rules of indicies
1. a^n x a^m= a^n+m
1. (a^n)^m= a^nm
2. Matrices
1. Sub step 2 into step 3
1. M^k+1
1. same as M(M^k)
2. recurrence relationships
1. show true for
1. n=1 AND n=2
1. for step 3
1. use recurrence formula for U k+1
1. sub U k in
4. Series
1. General formulae
1. r
1. r^2
1. r^3
1. 1
2. sum between 2 limits
1. sum up to top limit
1. minus sum up to bottom limit -1
2. to show a summation formula = ....
1. take out common factors
3. 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
1. between a and b
1. where f(a) -ve and f(b) +ve
2. linear interpolation
1. if root lies in [a,b]
1. use similar triangles
1. or formula
1. not in data book
2. 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
2. Matrices
1. multiplying
1. only multiply if
1. no. of columns of 1st matrix
1. same as no. of rows of 2nd
1. product dimensions
1. same no. rows as 1st matrix
1. Same no. of columns as 2nd
2. not comutative
1. AB≠BA
3. dimensions
1. (nxm)
1. n= rows
1. m=columns
1. simultaneous equations
2. Transformations
1. vectors
1. position vector
1. from origin
1. can be written (x,y)
1. translation vector
1. from given point
2. linear
1. linear expressions
1. point (0,0) unchanged
1. reflections
1. y-axis
1. x-axis
2. y=x
1. y=-x
3. enlargement
1. scale factor a
2. rotation
1. 180°
1. 90°
1. clockwise
1. anticlockwise
2. 45°
1. clockwise
1. anticlockwise
2. inverse matrix
1. transforms back to original
3. Inverse
1. A^-1
1. AA^-1=I
1. I= identity matrix
1. determinant
1. singular if =0
1. transform shape
1. straight line
1. 0 area

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