Core Pure Flashcards

(45 cards)

1
Q

What is |z1z2| equivalent to?

A

|z1||z2|
(Same logic for division)

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2
Q

What is arg(z1z2) equivalent to?

A

arg(z1) + arg(z2)
(Same logic for division)

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3
Q

What can the determinant of a matrix tell you?

A

The determinant of a matrix is the scale factor for the area or volume of shapes before and after the transformation.

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4
Q

What 2x2 matrix rotates a point anticlockwise by θº about the origin?

A

[cosθ -sinθ]
[sinθ cosθ]

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5
Q

What 3x3 matrix rotates a point θº anticlockwise around the positive x axis?

A

[1 0 0]
[0 c -s]
[0 s c]

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6
Q

What 3x3 matrix rotates a point θº anticlockwise around the positive y axis?

A

[c 0 s]
[0 1 0]
[-s 0 c]

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7
Q

What 3x3 matrix rotates a point θº anticlockwise around the positive z axis?

A

[c -s 0]
[s c 0]
[0 0 1]

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8
Q

How can you convert from vector line equation to cartesian line equation?

A

r = (x1, y1, z1) + λ(a, b, c)
(x - x1)/a = (y - y1)/b = (z - z1)/c

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9
Q

How can you simplify arctan(a) + arctan (b)?

A

arctan(a) + arctan (b) = arctan( (a + b) / (1- ab) )

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10
Q

What is simple harmonic motion?

A

When the acceleration of a particle is always towards a fixed point, on a line of motion, where the acceleration is proportional to the displacement of the particle.

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11
Q

What is the equation for simple harmonic motion?

A

ẍ = -ω²x
- ω is the angular velocity of the particle.
- ω is the coefficient of t in the sin and cos function of the solved equation
- The minus sign meaning the acceleration is towards the origin.

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12
Q

What are the different types of simple harmonic motion?

A

ẍ + kẋ + ω²x = f(t)

f(t) = 0:
- Light damping (k² < 4ω²): a force is reducing the amplitude of the oscillations
- Heavy damping (k² > 4ω²): a force is resisting the motion so much that no oscillations are experienced
- Critical damping (k² = 4ω²): the minimum resistive force that results in heavy dampening

f(t) ≠ 0:
- Forced harmonic motion: other than the other two forces, the particle experiences another external force

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13
Q

How can you use complex numbers to solve this problem?
A particle moves one unit then turns an angle of α to the right. It does this four times in total. What is the distance from original to final position?

A
  • Original position is origin.
  • First moves by 1: distance is 1
  • Then moves by complex number e^αi
  • Then moves by complex number e^2αi, and so on
  • Sum these numbers to find the final displacement
  • Calculate this sum by doing a sum of a geometric series.
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14
Q

How do you solve an equation in the form:
cosnθ = …
sinnθ = …

A
  • let z = cosθ + isinθ
  • zⁿ = cosnθ + isinnθ
  • cosnθ = Re(zⁿ) = Re((cosθ + isinθ)ⁿ)
  • sinnθ = Im(…)
  • Expand binomially
  • Use sin²θ + cos²θ = 1 where appropriate

Remember: If finding real components, write imaginary terms of the expansion as +i(…), and if finding imaginary, write real as +(…) to save time.

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15
Q

How do you solve an equation in the form:
cosⁿθ = …
sinⁿθ = …

A

E.g. for cosⁿθ
- let z = cosθ + isinθ
- z + z⁻¹ = 2cosθ
- cosθ = (1/2)(z + z⁻¹)
- cosⁿθ = (1/2)(z + z⁻¹)ⁿ
- Expand binomially
- Collect terms using de Moivre’s theorem

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16
Q

What’s the formula for arc length, sector area and segment area?

A

s = rθ
Asec = (1/2)r²θ
Aseg = (1/2)r²(θ - sinθ)

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17
Q

How can you quickly multiply a matrix by a vector where the matrix is in terms of linear k?

A

Result =
(Matrix of constants)(Vector) + k(Matrix of coefficients)(Vector)

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18
Q

What must you remember when finding the area between two polar curves?

A

You MUST work out the different areas separately and subtract them. Subtracting the polar equations before integrating is incorrect because the radius is scaled down.

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19
Q

When finding the intersection between a plane and a line, what method should you use?

A

If line is in vector form and plane is in cartesian, substitute x, y and z into the cartesian in terms of λ from the vector equation.
Solve for λ and find coordinate.

20
Q

What are tips for tricky integrations?

A
  • Consider working both ways with the double angle formulae: a double angle can reduce the power of a trig expression, or a single angle could allow for a reverse chain rule to be used.
  • For modelling questions with decimals, keep in terms of decimals (e.g. keep 1/3.6² factorised out or whatever) so that numbers are as simple to work with as possible. Decimals can be less clunky than fractions.
21
Q

What equations relate cartesian to polar coordinates?

A

x = rcosθ
y = rsinθ

22
Q

What should you include when sketching the positions of complex numbers on an Argand diagram?

A

Just mark them as dots or xs and label the axis that they are in line with. If defined by mod arg, draw angle and line.

23
Q

How do you know if a polar curve has a dimple or not?

A

Put in form r = a(b+cosθ)
If 1<|b|<2, there is a dimple.
Similar for sinθ

24
Q

How can you transform a shape in the complex plane so that it’s around the centre and has a vertex of 1?

A
  • Take a point z1 on the shape.
  • Find α, the centre.
  • z1 - α centres the coordinate.
  • β(z1 - α) rotates and scales the coordinate so it lands on 1.
  • β = 1 / (z1 - α)
25
How can you simplify trig functions of inverse trig functions, e.g. sin(4arccos 1/√3)
if x = arccos 1/√3, cos x = 1/√3 - We want an expression in terms of cos x - Take sin4x and use the double angle formulae - use sin x = √( 1 - cos²x) for any remaining sin x
26
How would you go about finding the real and imaginary components of 1 / (1 - 2e^θi)
Multiply both sides of the fraction by (1 - 2e^-θi)
27
What is the strategy for converting polar to cartesian?
- If required, use double angle formula to get all trig functions in terms of θ. - Keep multiplying and dividing by r to make all trig functions either rcosθ or rsinθ, then replace them with x and y. - Keep multiplying and dividing by r to make each r term r², then replace with x² + y². - Simplify.
28
What do you always forget when finding the area of a region?
YOU FORGET TO SUBTRACT THE TRIANGLE!!!
29
When justifying a differential equation model describing the flow of two fluids, what are they looking for?
- Explain an equation for change in total volume over time, then integrate that to find volume in terms of time, solving for the constant using initial conditions. - Describe the input of one liquid. - Describe the output of the other liquid as a proportion of the total liquid.
30
When skim reading modelling questions, what's the most important information to check?
- Variable definitions. - Variable units. - Initial conditions. - Assumptions. - Rounding requirements.
31
What is a sheaf?
A common line of multiple planes. It's direction can be calculated by doing n1 x n2.
32
What's the strategy for finding an invariant line?
- M(x1, mx1+c) = (x2, mx2+c) - Sub (1) into (2) - Rearrange to equal zero - Factorise out x1 and c - Solve for valid combinations of m and c, where x1∈R
33
What shortcut can you take for solving f(x) = g(x) if f(x) and g(x) have a common root?
Divide both sides by the common root, AND include it as a solution.
34
What do you forget when finding the real or imaginary component of a complex fraction?
The denominator MUST be fully real when you choose your components.
35
How do you make the denominator of 1/(1-z) real?
- Easier if you convert to exponential. - Multiply both sides of the fraction by (1-z⁻¹). - Expand and use de Moivre's theorem on the denominator. - Convert to trigonometric form and select components.
36
How can you remove x from the numerator of (4 - 2x)/(5 - 4x)? This could be applied to an integration, or it could be applied to a trig function, such as for a component of a sum of a complex series.
(4 - 2x)/(5 - 4x) = (3 + 5 - 4x) / 2(5 - 4x) = 1/2 + 3/2(5-4x)
37
How can you tell if a matrix inverts an image?
The determinant will be negative.
38
A represents a reflection. What is Det(A)?
-1
39
B represents a rotation. What is Det(B)?
1
40
What do you always get wrong when solving second order differentials?
You substitute in y y' and y'' into the equation in the wrong order.
41
Show f(4) = 0
Calculate EACH TERM – don't just substitute 4 in as x.
42
If a question asks 'using de Moivres Theorem', what are they looking for in your working? What should you make sure not to forget?
zⁿ = rⁿ(cosnθ + isinnθ) Do not forget to raise r to the power of n.
43
If you are doing cos then sin when solving equations, what must you be extra careful for?
- DO NOT get A and B the wrong way around when writing solution. - DO NOT get λ and µ the wrong way around when finding the P.I.
44
How should you describe motion?
- (If true, ) simple harmonic. - Oscillations about the origin. - With time period/frequency... - Type of damping. - NOTE: simple harmonic motion has a constant amplitude, so damped harmonic motion is not a type of simple harmonic motion.
45
How would you find the locus of: |(3-z)/(z+1)| = 3
- z = x + iy. - The modulus rule means that the LHS is equal to |3-z| divided by |z+1|. - 3-z = (3-x) + (-y)i. - z+1 = (x+1) + (y)i. - Express each modulus as √... in terms of x and y. - Solve.