# Further Pure 1 FP1 Complete Revision Notes AQA

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FP1 Key Notes
Chapter 1 ­ Complex Numbers
i2 = -1
and imaginary parts
i.e. (4+3i) + (7+8i) = 11+11i
(17+2i) ­ (9+8i) = 8-6i
The complex conjugate of a+bi is a-bi. You multiply an imaginary
number by its complex conjugate pair to rationalise the denominator.
The complex conjugate of z is z*
If the roots of a quadratic equation are complex, and will always be a
complex conjugate pair;
In the form: (x ­ )(x ­ )
Imaginary numbers can be expressed as points on a Cartesian graph on
an Argand Diagram.
The modulus of a complex number z = x + iy is given by x2 + y2
The argument of a complex number is the angle between the positive
real axis and the vector representing z on the Argand diagram
(SohCahToa). The argument is always -180 x 180.
The modulus-argument form of a complex number z = x + iy is given as
Z = modulus x (cos(arg) + i sin(arg))
You can compare real and imaginary parts of an equations to find a and b
e.g. 3+5i = (a+ib)(1+i)
= (a-b) + i(a+b)
i) a­b = 3
ii) a+b = 5 so... a=4 and b=1
To find the Square roots of a complex number, make the complex
number equal to (a+bi)2 and solve to find a and b (answer given as
complex num)
For a cubic equation, either all three roots are real or one root is real
and the other two roots form a complex conjugate pair. Use the formula:
X3 ­ ( + )x +
To find unknowns in a cubic, if a root is given, use the other complex
conjugate and form a quadratic then use the grid method
Tom Fenton

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Chapter 2 ­ Numerical Solutions of Equations
If you find an interval in which f(x) changes sign, the interval must
contain a root of f(x) = 0
Use the interval bisection method to find an approximation of the root
(trial and improvement to find when the sign changes)
Newton-Raphson formula can also be used:
Where X0 is the first approximated value, f(x) is the function and f `(x) is the
differentiated function
Chapter 3 ­ Coordinate Systems
Parametric equations are given in the form x…read more

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For a rectangular hyperbola, the equation is given in the form xy = c2
Where general points are given as x = ct2 and y= c/t
Chapter 4 ­ Matrix Algebra
corresponding elements of the two matrices
Multiply matrices by stacking one on top of the other
The Identity Matrix is:
A linear matrix is a matrix where the matrix = the origin when x,y=0
A rotation of 45° clockwise will give (1/2)

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Step 5 ­ Write conclusion: It's true for n=1, if it's true for n=k, then it's true for