Core Pure

Question 1

Modulus and Argument Form?

Answer 1

r(cosθ + isinθ)

Question 2

Combining Modulus

Answer 2

⎮z₁z₂⎮=⎮z₁⎮⎮z₂⎮
⎮z₁/z₂⎮=⎮z₁⎮/⎮z₂⎮

Question 3

Combining Arguments

Answer 3

arg(z₁z₂) = arg(z₁) + arg(z₂)
arg(z₁/z₂) = arg(z₁) - arg(z₂)

Question 4

Loci on an Argand diagram?

Answer 4

Single Point: ⎮z - x - iy⎮= r
Produces circle centre (x,y) with radius r
2 points: ⎮z-z₁⎮=⎮z-z₂⎮
Produces perpendicular bisector of z₁→z₂

Question 5

What are the Series formulas?

Answer 5

Σ1 = n
Σr = ½n(n+1)
Σr² = ⅙n(n+1)(2n+1)
Σr³ = ¼n²(n+1)²

Question 6

Sum of roots rules?

Answer 6

Σ⍺ = -b/a
Σ⍺β = c/a
Σ⍺βɣ = -d/a
Σ⍺βɣδ = e/a

Question 7

Reciprocals:
❶ 1/⍺ + 1/β
❷ 1/⍺ + 1/β + 1/ɣ
❸ 1/⍺ + 1/β + 1/ɣ + 1/δ

Answer 7

❶ Σ⍺ / Σ⍺β
❷ Σ⍺β / Σ⍺βɣ
❸ Σ⍺βɣ / Σ⍺βɣδ

Question 8

Products of Powers:
❶ ⍺ⁿ x βⁿ
❷ ⍺ⁿ x βⁿ x ɣⁿ
❸ ⍺ⁿ x βⁿ x ɣⁿ x δⁿ

Answer 8

❶ (⍺β)ⁿ
❷ (⍺βɣ)ⁿ
❸ (⍺βɣδ)ⁿ

Question 9

Sum of Squares:
❶ ⍺² + β²
❷ ⍺² + β² + ɣ²
❸ ⍺² + β² + ɣ² + δ²

Answer 9

❶ (⍺ + β)² - 2⍺β
❷ (⍺ + β + ɣ)² -2(⍺β + βɣ + ⍺ɣ)
❸ (⍺ + β + ɣ + δ)² - 2(Σ⍺β)

Question 10

Sum of Cubes:
❶ ⍺³ + β³
❷ ⍺³ + β³ + ɣ³

Answer 10

❶ (⍺ + β)³ - 3⍺β(⍺ + β)
❷ (⍺ + β + ɣ)³ - 3(Σ⍺)(Σ⍺β) + 3⍺βɣ

Question 11

How do you Linearly transform roots?

Answer 11

If equation has roots ⍺,β,ɣ,δ
The equation with roots (g⍺ + h) can be found by
W = g⍺ + h → ⍺ = (W - h)/g

Question 12

How to you rotate areas around the x axis?

Answer 12

π ∫ y² dx
dx for around x

Question 13

How to you rotate areas around the y axis?

Answer 13

π ∫ x² dy
dy for around y

Question 14

Matrix Multiplication?

Answer 14

⎮a b⎮ x ⎮e f⎮ =
⎮c d⎮ ⎮g h⎮
⎮(ae + bg) (af + bh)⎮
⎮(ce + dg) (cf + dh)⎮

Question 15

Identity Matrix

Answer 15

⎮1 0 0⎮
⎮0 1 0⎮
⎮0 0 1⎮

Question 16

The Determinant 2x2

Answer 16

⎮a b⎮ = ab - cd
⎮c d⎮

Question 17

The Determinant 3x3

Answer 17

⎮a b c⎮
⎮e f g⎮
⎮h i j ⎮
= a(fj - ig) - b(ej - gh) + c(ei + fh)

Question 18

Inverting a 3x3 matrix

Answer 18

Form Cofactor Matrix
Apply Matrix of minors and transpose
Multiply by 1/determinant

Question 19

Using Matrices to solve Simultaneous Equations

Answer 19

. ⎮x⎮ ⎮a⎮
MatM X ⎮y⎮ = ⎮b⎮
⎮z⎮ ⎮c⎮
Where Mat M is formed by the coefficients of the simultaneous equations and a, b and c are what the equations equal.

Question 20

What is a sheaf?

Answer 20

Singular Matrix with infinitely many solutions. Plane intersect on a line.

Question 21

What is a prism?

Answer 21

Singular matrix with zero solutions. There is no point where all 3 planes meet.

Question 22

What is a matrix that has 1 solution?

Answer 22

Invertible matrices have one solution. The three planes meet at a single point.

Question 23

Reflection in the Y axis

Answer 23

(-1 0)
( 0 1)

Question 24

Reflection in the X axis

Answer 24

(1 0 )
(0 -1)

Question 25

Reflection in the line y = x

Answer 25

(0 1)
(1 0)

Question 26

Reflection in the line y = -x

Answer 26

( 0 -1)
(-1 0)

Question 27

Rotation of θ degrees anticlockwise

Answer 27

(cosθ -sinθ)
(sinθ cosθ)

Question 28

Enlargement

Answer 28

(a 0)
(0 b)
a and b are the scale factors

Question 29

Stretch

Answer 29

x: (a 0)
. (0 1)

y: (1 0)
. (0 b)

Question 30

What does the determinant of a transformation represent?

Answer 30

The modulus of the determinant represents the scale factor

Question 31

Applying successive transformations

Answer 31

To apply MatA then MatB
MatB x MatA x Point

Question 32

Reflection in the plane x=0

Answer 32

⎮-1 0 0⎮
⎮ 0 1 0⎮
⎮ 0 0 1⎮

Question 33

Reflection in the plane y=0

Answer 33

⎮1 0 0⎮
⎮0 -1 0⎮
⎮0 0 1⎮

Question 34

Reflection in the plane z=0

Answer 34

⎮1 0 0 ⎮
⎮0 1 0 ⎮
⎮0 0 -1⎮

Question 35

Rotation about the X-axis

Answer 35

⎮1 0 0 ⎮
⎮0 cosθ -sinθ⎮
⎮0 sinθ cosθ⎮

Question 36

Rotation about the Y-axis

Answer 36

⎮ cosθ 0 sinθ⎮
⎮ 0 1 0 ⎮
⎮-sinθ 0 cosθ⎮

Question 37

Rotation about the Z-axis

Answer 37

⎮cosθ -sinθ 0⎮
⎮sinθ cosθ 0⎮
⎮ 0 0 1⎮

Question 38

How can you undo a tranformation?

Answer 38

If you inverse the transformation matrix and apply it, then you can undo the transformation.

Question 39

How can you find invariant points and lines

Answer 39

⎮a b⎮ x ⎮x⎮ = ⎮x⎮
⎮c d⎮ ⎮y⎮ ⎮y⎮

⎮a b⎮ x ⎮ x ⎮ = ⎮ x’ ⎮
⎮c d⎮ ⎮mx+c⎮ ⎮mx’+c⎮

Question 40

Proof by induction?

Answer 40

Base: Prove true for n=1
Assume: True for n=k
Induction: Using the fact that n=k is true prove true for n=k+1
If true for n=1 and if true for n=k then true for n=k+1 you can assume true for all values n.

Question 41

Vector equation for a line

Answer 41

r = a + λb
where a and b are vectors

Question 42

Cartesian equation of a line

Answer 42

λ = (x-a₁)/b₁ = (y-a₂)/b₂ = (z-a₃)/b₃

Question 43

Scalar Product

Answer 43

⎮a₁⎮ ⎮b₁⎮
⎮a₂⎮ · ⎮b₂⎮ =
⎮a₃⎮ ⎮b₃⎮
a₁b₁ + a₂b₂ + a₃b₃

Question 44

How to find the angle between two vectors?

Answer 44

cosθ = a·b / ⎮a⎮⎮b⎮
a and b are perpendicular when a·b = 0

Question 45

Vector Equation of a Plane

Answer 45

r = a + λb + μc

Question 46

Cartesian Equation of a Plane

Answer 46

n₁x + n₂y + n₃z = d
where d = a·n
We can simplify this to r·n = d
where n perpendicular to the plane

Question 47

Angle between 2 intersecting lines

Answer 47

cosθ = ⎮(a·b / ⎮a⎮⎮b⎮)⎮

Question 48

Angle between line r=a+λb and plane r·n=k

Answer 48

sinθ = ⎮(b·n / ⎮b⎮⎮n⎮)⎮
sin because we want angle to plane not normal

Question 49

Angle between planes r·n₁=k₁ and r·n₂=k₂

Answer 49

cosθ = ⎮(n₁·n₂ / ⎮n₁⎮⎮n₂⎮)⎮

Question 50

How to find shortest distances

Answer 50

Shortest distances occur when things are perpendicular.