Further Pure 1 FP1 Complete Revision Notes AQA

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FP1 Key Notes
Chapter 1 ­ Complex Numbers
i2 = -1
When adding/subtracting imaginary numbers, just add/subtract real
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
To add or subtract matrices you simply add or subtract the
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)
Determinant…read more

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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
n=k+1.…read more


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