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Flashcards in Core As Deck (29)
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1

If z = a + bi, what is the complex conjugate?

z* = a - bi

2

What is the modulus argument form of the complex number z?

z = r(cosθ + isinθ)
where |z| = r and arg z = θ

3

|z₁z₂| =

|z₁| * |z₂|

4

arg(z₁z₂) =

arg(z₁) + arg(z₂)

5

If z₁ = x₁ + iy₁, what does |z - z₁| = r represent on an argand diagram?

A circle with centre (x₁, y₁) and radius r

6

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

7

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

8

What does 1/α + 1/β equal?

(α + β) / αβ

9

What does 1/α + 1/β + 1/γ equal?

(αγ + βγ + αβ) / αβγ

10

What does 1/α + 1/β + 1/γ + 1/δ equal?

(αγδ + βγδ + αβδ + αβγ) / αβγδ

11

What does α² + β² equal?

(α + β)² -2αβ

12

What does α² + β² + γ² equal?

(α + β+ γ)² - 2(αγ + βγ + αβ)

13

What does α² + β² + γ² + δ² equal?

(α + β+ γ+ δ)² - 2(αγ + βγ + αβ + γδ + βδ + αδ)

14

What does α³ + β³ equal?

(α + β)³ -3αβ(α + β)

15

What does α³ + β³ + γ³ equal?

(α + β+ γ)³ - 3(α + β+ γ)(αγ + βγ + αβ) + 3αβγ

16

What is the volume of a cone?

πr²h/3

17

What is the determinant of a 2-by-2 matrix?

ad - bc

18

When is a matrix singular?

When the determinant = 0

19

What do you need to remember when calculating the determinant of a 3-by-3 matrix?

Take away the second minor determinant!!

20

When is a system of linear equations consistent?

When there is at least one set of values that satisfies all the equations simultaneously

21

When is a system of linear equations inconsistent?

If there are no values that satisfies all the equations simultaneously

22

What are the requirements for a linear transformation?

They always map the origin onto itself
They can be represented by a matrix

23

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

24

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

25

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

26

For stretches in both the x- and the y-axis, which points/lines are invariant?

Point: only the origin
No lines are invariant

27

What does the determinant of a matrix represent?

The area scale factor

28

What is the scalar product equation for vectors?

𝗮.𝗯 = |𝗮| |𝗯| cos θ

29

When are two lines skew?

When they are not parallel and they do not intersect