FP1

Question 0

Discriminant (b²-4ac) < 0

Answer 0

There are no real roots

Two imaginary roots

Question 1

i

Answer 1

i = √-1

Question 2

Complex Numbers

Definition

Answer 2

Sums of real and imaginary numbers

Written in the form a + bi or x + iy where x and y are real numbers

Question 3

Complex Conjugate Pairs

Answer 3

z = a + bi
z* = a - bi

Question 4

Imaginary Roots of Quadratic Equations

Answer 4

(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

Question 5

Argand Diagrams

Answer 5

X axis - real

Y axis - imaginary

Question 6

Argument

Answer 6

arg z = θ = tan(y/x) usually
Angle measured from right to left from the x axis
-π<θ≤π

Question 7

Modulus

Answer 7

|z| = r = √(x²+y²)

The modulus of any non zero complex number is positive

Question 8

Modulus-Argument Form

Answer 8

z = r (cosθ + i sinθ)

Where r = modulus and θ = argument

Question 9

Cubic Equations

Roots

Answer 9

• 3 real roots

- 1 real root and one complex conjugate pair

Question 10

Quadratic Equations

Roots

Answer 10

• Two real roots
• Two identical roots
• one complex conjugate pair

Question 11

Quarter Equations

Roots

Answer 11

• 4 real roots
• one complex conjugate pair and 2 real roots
• two complex conjugate pairs

Question 12

f(x) = 0

Answer 12

If there is an interval [a,b] in which f(x) changes sign

Then [a.b] must contain a root of f(x)=0

Question 13

Interval Bisection

Answer 13

Table

Question 14

Linear Interpolation

Answer 14

Line

Similar Triangles

Question 15

The Newton-Raphson Process

Answer 15

xn+1 = xn - f(xn)/f’(xn)

Question 16

Parametric Equations

Definition

Answer 16

x and y coordinates are expressed in the for of an independent variable, a parameter

Question 17

Cartesian Equation

Definition

Answer 17

An equation containing both x and y, this can be found by eliminating the parameter from the parametric equations

Question 18

Parabola

Definition

Answer 18

The loch of point which are equidistant from a fixed point, the focus, and a fixed line, the directrix.

Question 19

Parabola

Parametric Equations

Answer 19

x = at²
y = 2at

Question 20

Parabola

Cartesian Equation

Answer 20

y² = 4ax

Question 21

Rectangular Hyperbola

Answer 21

A hyperbola, two smooth curves the mirror image of each other, with rectangular / perpendicular asymptotes

Question 22

Rectangular Hyperbola

Parametric Equations

Answer 22

x = ct
y = c/t

Question 23

Rectangular Hyperbola

Cartesian Equation

Answer 23

xy = c²

Question 24

Identity Matrix

Answer 24

I = (1 0) | (0 1)

Question 25

Matrices | Rows and Columns

Answer 25

Rows x Columns | Dimension x Order

Question 26

Adding Matrices

Answer 26

A + B = B + A Add the corresponding elements of each matrix

Question 27

Subtracting Matrices

Answer 27

A - B = -B + A Subtract the corresponding elements of each matrix

Question 28

Multiplying Matrices

Answer 28

AB ≠ BA ABC = (AB)C = A(BC) Dimensions: (n x m) x (m x k) = (n x k)

Question 29

Representing Transformations as Matrices

Answer 29

(a b) (c d) Most linear transformations can be represented y a 2x2 matrix

Question 30

Unit Vectors

Answer 30

i = (1) (0) j = (0) (1)

Question 31

Enlargement

Answer 31

``` (k 0) (0 k) Enlargement Scale factor k Centre (0,0) ```

Question 32

Rotation

Answer 32

Anti-clockwise (cosθ -sinθ) (sinθ cosθ) Clockwise (cosθ sinθ) (-sinθ cosθ) Centre (0,0)

Question 33

Reflection | In x=0

Answer 33

(-1 0) | 0 1

Question 34

Reflection | In y=0

Answer 34

(1 0) | 0 -1

Question 35

Reflection | In y=x

Answer 35

(0 1) | 1 0

Question 36

Reflection | In y=-x

Answer 36

(0 -1) | -1 0

Question 37

Matrices To The Power -1

Answer 37

A = (a b) Then A^-1 = 1/(ad-bc) (d -b) (c d) (-c a) A(A^-1) = (A^-1)A = Identity Matrix

Question 38

Determinant

Answer 38

Det (A) = ad -nc

Question 39

Singular Matrix

Answer 39

``` If det(A) = 0 then A is singular This means that A^-1 does not exist ```

Question 40

Non Singular Matrix

Answer 40

If Det(A) ≠ 0 then A is a non singular matrix and A^-1 does exist

Question 41

Matrices | Area Scale Factor

Answer 41

Area of Image = Area of Object x det(M)

Question 42

Matrix Products and Transformations

Answer 42

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

Question 43

∑ r^0 r=1 to n

Answer 43

n

Question 44

∑ r for r = 1 to n

Answer 44

n/2 (n+1)

Question 45

∑ r² for r = 1 to n

Answer 45

n/6 (n+1) (2n+1)

Question 46

∑ r³ for r = 1 to n

Answer 46

n²/4 (n+1)²

Question 47

Proof By Mathematical Induction

Answer 47

1. Basis - prove that the general statement is true for n=1 2. Assumption - assume that the general statement is true for n=k 3. Inductive - show that the general statement is true for n=k+1 4. Conclusion - the general statement is true for all positive integers