Core As

Question 1

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

Answer 1

z* = a - bi

Question 2

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

Answer 2

z = r(cosθ + isinθ)

where |z| = r and arg z = θ

Question 3

|z₁z₂| =

Answer 3

|z₁| * |z₂|

Question 4

arg(z₁z₂) =

Answer 4

arg(z₁) + arg(z₂)

Question 5

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

Answer 5

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

Question 6

If = x₁ + iy₁ and z₂ = x₂ + iy, what does |z - z₁| = |z - z₂| represent on an argand diagram?

Answer 6

A perpendicular bisector of the line segment joining z₁ and z₂

Question 7

If z₁ = x₁ + iy₁, what does arg (z - z₁) = θ represent on an argand diagram?

Answer 7

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

Question 8

What does 1/α + 1/β equal?

Answer 8

(α + β) / αβ

Question 9

What does 1/α + 1/β + 1/γ equal?

Answer 9

(αγ + βγ + αβ) / αβγ

Question 10

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

Answer 10

(αγδ + βγδ + αβδ + αβγ) / αβγδ

Question 11

What does α² + β² equal?

Answer 11

(α + β)² -2αβ

Question 12

What does α² + β² + γ² equal?

Answer 12

(α + β+ γ)² - 2(αγ + βγ + αβ)

Question 13

What does α² + β² + γ² + δ² equal?

Answer 13

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

Question 14

What does α³ + β³ equal?

Answer 14

(α + β)³ -3αβ(α + β)

Question 15

What does α³ + β³ + γ³ equal?

Answer 15

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

Question 16

What is the volume of a cone?

Answer 16

πr²h/3

Question 17

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

Answer 17

ad - bc

Question 18

When is a matrix singular?

Answer 18

When the determinant = 0

Question 19

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

Answer 19

Take away the second minor determinant!!

Question 20

When is a system of linear equations consistent?

Answer 20

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

Question 21

When is a system of linear equations inconsistent?

Answer 21

If there are no values that satisfies all the equations simultaneously

Question 22

What are the requirements for a linear transformation?

Answer 22

They always map the origin onto itself

They can be represented by a matrix

Question 23

What is the transformation represented by the matrix (a 0)
(0 b)

Answer 23

A stretch of scale factor a parallel to the x-axis and a stretch of scale factor b parallel to the y-axis

Question 24

For a stretch parallel to the x-axis, which points/lines are invariant?

Answer 24

Points on the y-axis are invariant

The line x=0 is the invariant line

Question 25

For a stretch parallel to the y-axis, which points/lines are invariant?

Answer 25

Points on the x-axis are invariant

The line y=0 is the invariant line

Question 26

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

Answer 26

Point: only the origin

No lines are invariant

Question 27

What does the determinant of a matrix represent?

Answer 27

The area scale factor

Question 28

What is the scalar product equation for vectors?

Answer 28

𝗮.𝗯 = |𝗮| |𝗯| cos θ

Question 29

When are two lines skew?

Answer 29

When they are not parallel and they do not intersect