Core As Flashcards
If z = a + bi, what is the complex conjugate?
z* = a - bi
What is the modulus argument form of the complex number z?
z = r(cosθ + isinθ)
where |z| = r and arg z = θ
|z₁z₂| =
|z₁| * |z₂|
arg(z₁z₂) =
arg(z₁) + arg(z₂)
If z₁ = x₁ + iy₁, what does |z - z₁| = r represent on an argand diagram?
A circle with centre (x₁, y₁) and radius r
If = x₁ + iy₁ and z₂ = x₂ + iy, what does |z - z₁| = |z - z₂| represent on an argand diagram?
A perpendicular bisector of the line segment joining z₁ and z₂
If z₁ = x₁ + iy₁, what does arg (z - z₁) = θ represent on an argand diagram?
Half line from, but not including, the fixed point z₁ making an angle θ with a line from the fixed point z₁ parallel to the real axis
What does 1/α + 1/β equal?
(α + β) / αβ
What does 1/α + 1/β + 1/γ equal?
(αγ + βγ + αβ) / αβγ
What does 1/α + 1/β + 1/γ + 1/δ equal?
(αγδ + βγδ + αβδ + αβγ) / αβγδ
What does α² + β² equal?
(α + β)² -2αβ
What does α² + β² + γ² equal?
(α + β+ γ)² - 2(αγ + βγ + αβ)
What does α² + β² + γ² + δ² equal?
(α + β+ γ+ δ)² - 2(αγ + βγ + αβ + γδ + βδ + αδ)
What does α³ + β³ equal?
(α + β)³ -3αβ(α + β)
What does α³ + β³ + γ³ equal?
(α + β+ γ)³ - 3(α + β+ γ)(αγ + βγ + αβ) + 3αβγ
What is the volume of a cone?
πr²h/3
What is the determinant of a 2-by-2 matrix?
ad - bc
When is a matrix singular?
When the determinant = 0
What do you need to remember when calculating the determinant of a 3-by-3 matrix?
Take away the second minor determinant!!
When is a system of linear equations consistent?
When there is at least one set of values that satisfies all the equations simultaneously
When is a system of linear equations inconsistent?
If there are no values that satisfies all the equations simultaneously
What are the requirements for a linear transformation?
They always map the origin onto itself
They can be represented by a matrix
What is the transformation represented by the matrix (a 0)
(0 b)
A stretch of scale factor a parallel to the x-axis and a stretch of scale factor b parallel to the y-axis
For a stretch parallel to the x-axis, which points/lines are invariant?
Points on the y-axis are invariant
The line x=0 is the invariant line
For a stretch parallel to the y-axis, which points/lines are invariant?
Points on the x-axis are invariant
The line y=0 is the invariant line
For stretches in both the x- and the y-axis, which points/lines are invariant?
Point: only the origin
No lines are invariant
What does the determinant of a matrix represent?
The area scale factor
What is the scalar product equation for vectors?
𝗮.𝗯 = |𝗮| |𝗯| cos θ
When are two lines skew?
When they are not parallel and they do not intersect
