FP1 Flashcards
Discriminant (b²-4ac) < 0
There are no real roots
Two imaginary roots
i
i = √-1
Complex Numbers
Definition
Sums of real and imaginary numbers
Written in the form a + bi or x + iy where x and y are real numbers
Complex Conjugate Pairs
z = a + bi z* = a - bi
Imaginary Roots of Quadratic Equations
(x-α)(x-β) = 0
x² - (α+β)x + αβ = 0
If the roots α and β of a quadratic equation are complex they will always form a complex conjugate pair
Argand Diagrams
X axis - real
Y axis - imaginary
Argument
arg z = θ = tan(y/x) usually
Angle measured from right to left from the x axis
-π<θ≤π
Modulus
|z| = r = √(x²+y²)
The modulus of any non zero complex number is positive
Modulus-Argument Form
z = r (cosθ + i sinθ)
Where r = modulus and θ = argument
Cubic Equations
Roots
- 3 real roots
- 1 real root and one complex conjugate pair
Quadratic Equations
Roots
- Two real roots
- Two identical roots
- one complex conjugate pair
Quarter Equations
Roots
- 4 real roots
- one complex conjugate pair and 2 real roots
- two complex conjugate pairs
f(x) = 0
If there is an interval [a,b] in which f(x) changes sign
Then [a.b] must contain a root of f(x)=0
Interval Bisection
Table
Linear Interpolation
Line
Similar Triangles
The Newton-Raphson Process
xn+1 = xn - f(xn)/f’(xn)
Parametric Equations
Definition
x and y coordinates are expressed in the for of an independent variable, a parameter
Cartesian Equation
Definition
An equation containing both x and y, this can be found by eliminating the parameter from the parametric equations
Parabola
Definition
The loch of point which are equidistant from a fixed point, the focus, and a fixed line, the directrix.
Parabola
Parametric Equations
x = at² y = 2at
Parabola
Cartesian Equation
y² = 4ax
Rectangular Hyperbola
A hyperbola, two smooth curves the mirror image of each other, with rectangular / perpendicular asymptotes
Rectangular Hyperbola
Parametric Equations
x = ct y = c/t
Rectangular Hyperbola
Cartesian Equation
xy = c²
Identity Matrix
I = (1 0)
(0 1)
Matrices
Rows and Columns
Rows x Columns
Dimension x Order
Adding Matrices
A + B = B + A
Add the corresponding elements of each matrix
Subtracting Matrices
A - B = -B + A
Subtract the corresponding elements of each matrix
Multiplying Matrices
AB ≠ BA
ABC = (AB)C = A(BC)
Dimensions: (n x m) x (m x k) = (n x k)
Representing Transformations as Matrices
(a b)
(c d)
Most linear transformations can be represented y a 2x2 matrix
Unit Vectors
i = (1)
(0)
j = (0)
(1)
Enlargement
(k 0) (0 k) Enlargement Scale factor k Centre (0,0)
Rotation
Anti-clockwise
(cosθ -sinθ)
(sinθ cosθ)
Clockwise
(cosθ sinθ)
(-sinθ cosθ)
Centre (0,0)
Reflection
In x=0
(-1 0)
0 1
Reflection
In y=0
(1 0)
0 -1
Reflection
In y=x
(0 1)
1 0
Reflection
In y=-x
(0 -1)
-1 0
Matrices To The Power -1
A = (a b) Then A^-1 = 1/(ad-bc) (d -b)
(c d) (-c a)
A(A^-1) = (A^-1)A = Identity Matrix
Determinant
Det (A) = ad -nc
Singular Matrix
If det(A) = 0 then A is singular This means that A^-1 does not exist
Non Singular Matrix
If Det(A) ≠ 0 then A is a non singular matrix and A^-1 does exist
Matrices
Area Scale Factor
Area of Image = Area of Object x det(M)
Matrix Products and Transformations
Matrix Product ABC means:
- first do the transformation represented by C
- second do the transformation represented by B
- third do the transformation represented by A
∑ r^0 r=1 to n
n
∑ r for r = 1 to n
n/2 (n+1)
∑ r² for r = 1 to n
n/6 (n+1) (2n+1)
∑ r³ for r = 1 to n
n²/4 (n+1)²
Proof By Mathematical Induction
- Basis - prove that the general statement is true for n=1
- Assumption - assume that the general statement is true for n=k
- Inductive - show that the general statement is true for n=k+1
- Conclusion - the general statement is true for all positive integers
