Year 1 - Core Pure Flashcards

1
Q

2.3 What does mod(z1z2) equal?

A

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

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2
Q

2.3 What does arg(z1z2) equal?

A

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

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3
Q

2.3 What does mod(z1/z2) equal?

A

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

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4
Q

2.3 What does arg(z1/z2) equal?

A

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

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5
Q

3.1 What is the formula of the first n integers?

A

Σ(r=1 -> n) r = ((1/2)n)(n+1)

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6
Q

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

A

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

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7
Q

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

A

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

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8
Q

3.1 Σ(r=1 -> n) (ar + b) = ?

A

aΣ(r=1 -> n)r + bΣ(r=1 -> n)1

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9
Q

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

A

α + β = -b/a

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10
Q

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

A

αβ = c/a

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11
Q

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

A

α + β + γ = -b/a

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12
Q

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

A

αβ + αγ + βγ = c/a

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13
Q

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

A

αβγ = -d/a

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14
Q

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

A

α + β + γ + δ = -b/a

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15
Q

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

A

αβ + αγ + αδ + βγ + βδ + γδ = c/a

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16
Q

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

A

αβγ + αβδ + αγδ + βγδ = -d/a

17
Q

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

A

αβγδ = e/a

18
Q

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

A

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

19
Q

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

A

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

20
Q

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

A

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)

21
Q

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

A

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

22
Q

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

A

Let w = 2x
x = w/2

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

23
Q

8.1 Describe the process for proof by induction

A
  1. Basis - Prove the general statement is true for n=1
  2. Assumption - Assume the general statement is true for n=k
  3. Inductive - Show the general statement is then true for n=k+1
  4. Conclusion - General statement is then true for all positive integers n
24
Q

8.1 What are the sentences to remember for proof by induction?

A
  • When n=1
  • therefore true for n=1 as RHS=LHS
  • Assume true for n=k
  • When n=k+1
  • Hence true for n=k+1
  • Since true for n=1, if true for n=k, then true for n=k+1, therefore true for all n
25
Q

8.2 How do you use proof by induction to prove divisibility results?

A
  1. Show n=1 is true
  2. Assume n=k is true: show f(k)
  3. Show n=k+1 is true: show f(k+1)
  4. Manipulate f(k+1) to = f(k) + a(x) where d is the multiple you are trying to show
26
Q

8.3 How do you use proof by induction with matrices?

A
  1. Show n=1 is true
  2. Assume n=k is true: show M^k
  3. Show n=k+1 is true: multiply M^k by M