Year 1 - Core Pure

Question 1

2.3 What does mod(z1z2) equal?

Answer 1

mod(z1z2) = mod(z1) mod(z2)

Question 2

2.3 What does arg(z1z2) equal?

Answer 2

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

Question 3

2.3 What does mod(z1/z2) equal?

Answer 3

mod(z1/z2) = mod(z1)/mod(z2)

Question 4

2.3 What does arg(z1/z2) equal?

Answer 4

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

Question 5

3.1 What is the formula of the first n integers?

Answer 5

Σ(r=1 -> n) r = ((1/2)n)(n+1)

Question 6

3.1 What is the formula for the sum of the first n squares?

Answer 6

Σ (r=1 -> n) r^2 = ((1/6)n)(n+1)(2n+1)

Question 7

3.1 What is the formula for the sum of the first n cubes?

Answer 7

((1/4)n^2)(n+1)^2

Question 8

3.1 Σ(r=1 -> n) (ar + b) = ?

Answer 8

aΣ(r=1 -> n)r + bΣ(r=1 -> n)1

Question 9

4.1 What do the sum of 2 roots of a quadratic equal?

Answer 9

α + β = -b/a

Question 10

4.1 What do the product of 2 roots of a quadratic equal?

Answer 10

αβ = c/a

Question 11

4.2 What do the sum of 3 roots of a cubic equal?

Answer 11

α + β + γ = -b/a

Question 12

4.2 What do the sum of all 3 of the pairs of products of a cubic equal?

Answer 12

αβ + αγ + βγ = c/a

Question 13

4.2 What do the product of all 3 roots of a cubic equal?

Answer 13

αβγ = -d/a

Question 14

4.3 What do the sum of 4 roots of a quartic equal?

Answer 14

α + β + γ + δ = -b/a

Question 15

4.3 What do the sum of all 5 pairs of a quartic equal?

Answer 15

αβ + αγ + αδ + βγ + βδ + γδ = c/a

Question 16

4.3 What do the sum of all 4 triples of a quartic equal?

Answer 16

αβγ + αβδ + αγδ + βγδ = -d/a

Question 17

4.3 What does the product of all 4 roots of a quartic equal?

Answer 17

αβγδ = e/a

Question 18

4.4 Give the rules for the sums of the reciprocals of the roots of a polynomial (quadratic, cubic, quartic)

Answer 18

Quadratic: (1/α) + (1/β) = (α + β)/αβ

Cubic: (1/α) + (1/β) + (1/γ) = (α + β + γ)/αβγ

Quartic: (1/α) + (1/β) + (1/γ) + (1/δ) = (α + β + γ + δ)/αβγδ

Question 19

4.4 Give the rules for the products of powers of the roots of a polynomial (quadratic, cubic, quartic)

Answer 19

Quadratic: (α^n)(β^n) = (αβ)^n

Cubic: (α^n)(β^n)(γ^n) = (αβγ)^n

Quartic: (α^n)(β^n)(γ^n)(δ^n) = (αβγδ)^n

Question 20

4.4 Give the rules of the sums of squares of the roots of a polynomial (quadratic, cubic, quartic)

Answer 20

Quadratic: α^2 + β^2 = (α + β)^2 - 2αβ

Cubic: α^2 + β^2 + γ^2 = (α + β + γ)^2 - 2(αβ + αγ + βγ)

Quartic: α^2 + β^2 + γ^2 + δ^2 = (α + β + γ + δ)^2 -2(αβ + αγ + αδ + βγ + βδ + γδ)

Question 21

4.4 Give the rules of the sums of cubes of the roots of a polynomial (quadratic, cubic)

Answer 21

Quadratic: α^3 + β^3 = (α + β)^3 - 3αβ(α + β)

Cubic: α^3 + β^3 + γ^3 = (α + β + γ)^3 - 3(α + β + γ)(αβ + αγ + βγ) + 3αβγ

Question 22

4.5 e.g. The cubic x^3 - 2x^2 + 3x - 4 = 0 has roots α, β, & γ, find the equation with roots 2α, 2β, and 2γ

Answer 22

Let w = 2x
x = w/2

Substitute x = w/2 into the cubic, simplify until coefficients are integers