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A Level Further Maths > Core A level > Flashcards

Core A level Flashcards

1
Q

What is the exponential form of a complex number?

A

z = re^iθ where
r = |z|
and θ = argz

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2
Q

Describe the proof for writing a complex number in exponential form

A

z = r(cosθ + isinθ)
Find the Maclaurin series expansion of cosθ, sinθ, and eˣ
subsitute x = iθ into the e-series, separate out the complex and real parts
you will get e^iθ = cosθ + isinθ

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3
Q

How do you go from the proof of exponential form of complex numbers to Euler’s identity?

A

e^iθ = cosθ + isinθ
θ = π
e^iθ = -1 + 0
e^iθ + 1 = 0

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4
Q

If z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, what does z₁*z₂ =

A

z₁z₂ = r₁r₂e^i(θ₁+θ₂)

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5
Q

What is de Moivre’s theorem?

A

zⁿ = rⁿ(cosnθ + isinnθ)

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6
Q

What can you quickly use, to prove de Moivre’s theorem for all n?

A

Exponential form

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7
Q

z + 1/z =

A

2cosθ

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8
Q

z - 1/z =

A

2isinθ

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9
Q

zⁿ + 1/zⁿ =

A

2cosnθ

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10
Q

zⁿ - 1/zⁿ =

A

2isin(nθ)

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11
Q

If zⁿ = w, what is the general solution to z, in modulus-argument form?

A

z = r(cos(θ+2kπ) + isin(θ+2kπ))

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12
Q

Describe what the roots of a complex number look like on an argand diagram

A

The roots lie at the vertices of a regular n-gon with its centre at the origin
n = no. of roots

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13
Q

When is the Maclaurin series valid?

A

When all the f(0), f’(0), f’‘(0), …, fʳ(0) all have finite values

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14
Q

When is an integral improper?

A

When:
one or both of the limits is infinite
f(x) is undefined at x = a, x = b, or any other point in the interval [a ,b]

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15
Q

If an improper integral exits, what is it described as?

If it doesn’t exist?

A

Exists: Convergent

Doesn’t exist: Divergent

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16
Q

If you have an integral, where the limits are ± ∞, how can you tell whether the integral is convergent or divergent?

A

Split the integral into two with limits (∞, c) and (c, -∞) where c is a number
If both integrals converge, then so does the original.
If either diverges, then the original is divergent

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17
Q

How do you calculate the mean value of a function? (In the interval [a, b])

A

= 1/(b-a) ∫ f(x) dx

where the integral limits are b and a

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18
Q

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…
What is the mean value of f(x) + k?

A

f + k over the interval [a,b]

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19
Q

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…
What is the mean value of -f(x)?

A

-f over the interval [a,b]

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20
Q

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…
What is the mean value of kf(x)?

A

kf over the interval [a,b]

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21
Q

x = f(t)
y = g(t)
What is the volume of the solid that is generated when the parametric curve is rotated about the x-axis, between x = a and x = b, through 2π radians?

A

Volume = π ∫ y² dx (between x=b, and x = a)
= π ∫ y² dx/dt dt (between t=p and t=q)

where a = f(p) and b = f(q)

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22
Q

x = f(t)
y = g(t)
What is the volume of the solid that is generated when the parametric curve is rotated about the y-axis, between y = a and y = b, through 2π radians?

A

Volume = π ∫ x² dy (between y = b, and y = a)
= π ∫ x² dy/dt dt (between t = p and t = q)

where a = f(p) and b = f(q)

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23
Q

Polar co-ordinates:

Write x in terms of θ

A

rcosθ = x

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24
Q

Polar co-ordinates:

Write y in terms of θ

A

rsinθ = y

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25
Q

Polar co-ordinates:

Write r in terms of x and y

A

x² + y² = r

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26
Q

Polar co-ordinates:

Write θ in terms of x and y

A

θ = arctan (y/x)

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27
Q

What is the origin called in polar coordinates?

A

The pole

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28
Q

Polar co-ordinates:

What is the initial line?

A

Usually the positive x-axis

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29
Q

Polar co-ordinates:

What is the form of the coordinates? Eg, what are a and b in (a, b)

A

(r, θ)

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30
Q

Polar co-ordinates:

r = a

A

circle with centre O and radius a

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31
Q

Polar co-ordinates:

What shape is α = θ

A

Half-line through O, making an angle α with the initial line

32
Q

Polar co-ordinates:

What shape is r = aθ

A

Spiral starting at O

33
Q

Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve convex (eg, egg shaped)?

A

p ≥ 2q

34
Q

Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve concave (at θ = π) (eg, dimple shaped)?

A

q ≤ p < 2q

35
Q

Polar co-ordinates:

How do you find a tangent parallel to the initial line?

A

dy/dθ = 0

36
Q

Polar co-ordinates:

How do you find a tangent perpendicular to the initial line?

A

dx/dθ = 0

37
Q

sinh x =

A

(eˣ - e⁻ˣ)/2

38
Q

cosh x =

A

(eˣ + e⁻ˣ)/2

39
Q

tanh x =

A

(eˣ - e⁻ˣ)/(eˣ + e⁻ˣ)
or
( e²ˣ - 1 )/ ( e²ˣ + 1 )

40
Q

sinh (-a) =

A

-sinh a

41
Q

cosh (-a) =

A

cosh a

42
Q

arsinh x =

A

ln( x + √(x² + 1) )

43
Q

arcosh x =

A

ln( x + √(x² - 1) ), x ≥ 1

44
Q

artanh x =

A

0.5 ln( (1 + x)/(1 - x), |x|<1

45
Q

What is the hyperbolic identity that equals 1?

A

cosh² A - sinh² A = 1

46
Q

What is the sine addition formula for hyperbolic functions?

A

sinh ( A ± B ) = sinh A cosh B ± cosh A sinh B

47
Q

What is the cosine addition formula for hyperbolic functions?

A

cosh ( A ± B ) = cosh A cosh B ± sinh A sinh B

48
Q

What is Osborne’s rule?

A

Whenever converting between normal trig identites and hyperbolic ones:
replace cos A by cosh A
and replace sin B by sinh B
however:
replace any product of two sin terms by minus the product of the two sinh terms
eg, sin² A would go to - sinh² A

49
Q

What does sinh x differentiate to?

A

cosh x

50
Q

What does cosh x differentiate to?

A

sinh x

51
Q

What does tanh x differentiate to?

A

sech² x

52
Q

What does arsinh x differentiate to?

A

1/ √(x² + 1)

53
Q

What does arcosh x differentiate to?

A

1/ √(x² - 1), x > 1

54
Q

What does artanh x differentiate to?

A

1 / (1 - x²), |x| < 1

55
Q

Differential equations:

What is separation of variables?

A

If dy/dx = f(x) * g(y)

then, ∫ 1/g(y) dy = ∫ f(x) dx

56
Q

Differential equations:

How do you solve first order differential equations?

A

Write in the form dy/dx + P(x)y = Q(x)

then multiply by the integrating factor; e^ ∫ P(x) dx

57
Q

Differential equations:

How do you solve second-order, homogeneous differential equations?

A

Find the roots of the auxiliary equation, and write in the correct form, depending on whether there is one root, two or complex

58
Q

Differential equations:

What is the auxiliary equation?

A

am² + bm + c = 0

where a, b, and c are the coefficients of the derivatives (eg, a is the coefficient of the second derivative)

59
Q

Differential equations:

Auxiliary equation, if b²- 4ac > 0, what is the general solution to the differential equation?

A

y = Aeᵃˣ + Beᵇˣ

where a, and b are the roots of your auxiliary equation

60
Q

Differential equations:

Auxiliary equation, if b²- 4ac = 0, what is the general solution to the differential equation?

A

y = (A + Bx) eᵃˣ

where a is the root of your auxiliary equation

61
Q

Differential equations:

Auxiliary equation, if b²- 4ac < 0, what is the general solution to the differential equation?

A

y = eᵖˣ ( Acos qx + Bsin qx) where p ± qi is the solution to the auxiliary equation

62
Q

Differential equations:

What is the complementary function?

A

It is the general solution to the homogeneous bit of the second-order non-homogeneous equation.

63
Q

Differential equations:

What is the particular interval?

A

It is a function which satisfies the original differential equation
When solving second-order non-homogeneous differential equations, it’s the function of x on the other side of the equal sign to the differential equation

64
Q 
Differential equations:
What are the particular intervals for these functions?
p
p + qx
p + qx + rx²
peᵏˣ
pcosωx + qsinωx
A 
p = λ
p + qx = λ + μx
p + qx + rx² = λ + μx + νx²
peᵏˣ = λeᵏˣ
pcosωx + qsinωx = λcosωx + μsinωx
65
Q

Differential equations:

How do solve second-order non-homogeneous differential equations?

A

Solve the corresponding homo. equation to find the complementary function (C.F)
Choose a particular integral (P.I), and substitute into the original equation to the find the value of any coefficients in the P.I
The general solution = C.F + P.I

66
Q

What is simple harmonic motion?

A

Motion in which the acceleration of the particle P is always towards a fixed point O on the line of motion of P
The acceleration is proportional to the displacement of P from O
Where O = the centre of oscillation

67
Q

What is the algebraic version of the definition of simple harmonic motion?

A

x’’ = -ω²x

68
Q

Simple harmonic motion:

write acceleration in terms of dv/dx

A

x’’ = v dv/dx

69
Q

Simple harmonic motion: what is ω?

A

Angular velocity

70
Q

What is the equation for a particle moving with damped harmonic motion?

A

d²x/dt² + k dx/dt + ω²x = 0

where k and ω² are positive constants

71
Q

Describe (in terms of k and ω) when a particle is being heavily, critically or lightly damped

A

Heavily: k² > 4ω²
Critically: k² = 4ω²
Lightly: k² < 4ω²

72
Q

What is the equation for a particle moving with forced harmonic motion?

A

d²x/dt² + k dx/dt + ω²x = f(t)

where k and ω² are positive constants

73
Q

How do you solve coupled first-order linear differential equations?

A

By eliminating one of the dependent variables to form a second-order differential equation
Eg. dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)

differentiate the top equation, then substitute the second equation in for dy/dt

74
Q

What are coupled first-order linear differential equations?

A 
dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)
75
Q

Coupled first-order differential equations:
dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)

if f(t) and g(t) are both zero, what is the system said to be?

A

Homogeneous

A Level Further Maths (5 decks)

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