Core A level

Question 1

What is the exponential form of a complex number?

Answer 1

z = re^iθ where
r = |z|
and θ = argz

Question 2

Describe the proof for writing a complex number in exponential form

Answer 2

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

Question 3

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

Answer 3

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

Question 4

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

Answer 4

z₁z₂ = r₁r₂e^i(θ₁+θ₂)

Question 5

What is de Moivre’s theorem?

Answer 5

zⁿ = rⁿ(cosnθ + isinnθ)

Question 6

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

Answer 6

Exponential form

Question 7

z + 1/z =

Answer 7

2cosθ

Question 8

z - 1/z =

Answer 8

2isinθ

Question 9

zⁿ + 1/zⁿ =

Answer 9

2cosnθ

Question 10

zⁿ - 1/zⁿ =

Answer 10

2isin(nθ)

Question 11

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

Answer 11

z = r(cos(θ+2kπ) + isin(θ+2kπ))

Question 12

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

Answer 12

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

Question 13

When is the Maclaurin series valid?

Answer 13

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

Question 14

When is an integral improper?

Answer 14

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]

Question 15

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

If it doesn’t exist?

Answer 15

Exists: Convergent

Doesn’t exist: Divergent

Question 16

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

Answer 16

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

Question 17

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

Answer 17

= 1/(b-a) ∫ f(x) dx

where the integral limits are b and a

Question 18

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?

Answer 18

f + k over the interval [a,b]

Question 19

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)?

Answer 19

-f over the interval [a,b]

Question 20

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)?

Answer 20

kf over the interval [a,b]

Question 21

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?

Answer 21

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)

Question 22

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?

Answer 22

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)

Question 23

Polar co-ordinates:

Write x in terms of θ

Answer 23

rcosθ = x

Question 24

Polar co-ordinates:

Write y in terms of θ

Answer 24

rsinθ = y

Question 25

Polar co-ordinates: | Write r in terms of x and y

Answer 25

x² + y² = r

Question 26

Polar co-ordinates: | Write θ in terms of x and y

Answer 26

θ = arctan (y/x)

Question 27

What is the origin called in polar coordinates?

Answer 27

The pole

Question 28

Polar co-ordinates: | What is the initial line?

Answer 28

Usually the positive x-axis

Question 29

Polar co-ordinates: | What is the form of the coordinates? Eg, what are a and b in (a, b)

Answer 29

(r, θ)

Question 30

Polar co-ordinates: | r = a

Answer 30

circle with centre O and radius a

Question 31

Polar co-ordinates: | What shape is α = θ

Answer 31

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

Question 32

Polar co-ordinates: | What shape is r = aθ

Answer 32

Spiral starting at O

Question 33

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

Answer 33

p ≥ 2q

Question 34

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

Answer 34

q ≤ p < 2q

Question 35

Polar co-ordinates: | How do you find a tangent parallel to the initial line?

Answer 35

dy/dθ = 0

Question 36

Polar co-ordinates: | How do you find a tangent perpendicular to the initial line?

Answer 36

dx/dθ = 0

Question 37

sinh x =

Answer 37

(eˣ - e⁻ˣ)/2

Question 38

cosh x =

Answer 38

(eˣ + e⁻ˣ)/2

Question 39

tanh x =

Answer 39

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

Question 40

sinh (-a) =

Answer 40

-sinh a

Question 41

cosh (-a) =

Answer 41

cosh a

Question 42

arsinh x =

Answer 42

ln( x + √(x² + 1) )

Question 43

arcosh x =

Answer 43

ln( x + √(x² - 1) ), x ≥ 1

Question 44

artanh x =

Answer 44

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

Question 45

What is the hyperbolic identity that equals 1?

Answer 45

cosh² A - sinh² A = 1

Question 46

What is the sine addition formula for hyperbolic functions?

Answer 46

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

Question 47

What is the cosine addition formula for hyperbolic functions?

Answer 47

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

Question 48

What is Osborne's rule?

Answer 48

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

Question 49

What does sinh x differentiate to?

Answer 49

cosh x

Question 50

What does cosh x differentiate to?

Answer 50

sinh x

Question 51

What does tanh x differentiate to?

Answer 51

sech² x

Question 52

What does arsinh x differentiate to?

Answer 52

1/ √(x² + 1)

Question 53

What does arcosh x differentiate to?

Answer 53

1/ √(x² - 1), x > 1

Question 54

What does artanh x differentiate to?

Answer 54

1 / (1 - x²), |x| < 1

Question 55

Differential equations: | What is separation of variables?

Answer 55

If dy/dx = f(x) * g(y) | then, ∫ 1/g(y) dy = ∫ f(x) dx

Question 56

Differential equations: | How do you solve first order differential equations?

Answer 56

Write in the form dy/dx + P(x)y = Q(x) | then multiply by the integrating factor; e^ ∫ P(x) dx

Question 57

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

Answer 57

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

Question 58

Differential equations: | What is the auxiliary equation?

Answer 58

am² + bm + c = 0 | where a, b, and c are the coefficients of the derivatives (eg, a is the coefficient of the second derivative)

Question 59

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

Answer 59

y = Aeᵃˣ + Beᵇˣ | where a, and b are the roots of your auxiliary equation

Question 60

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

Answer 60

y = (A + Bx) eᵃˣ | where a is the root of your auxiliary equation

Question 61

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

Answer 61

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

Question 62

Differential equations: | What is the complementary function?

Answer 62

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

Question 63

Differential equations: | What is the particular interval?

Answer 63

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

Question 64

``` Differential equations: What are the particular intervals for these functions? p p + qx p + qx + rx² peᵏˣ pcosωx + qsinωx ```

Answer 64

``` p = λ p + qx = λ + μx p + qx + rx² = λ + μx + νx² peᵏˣ = λeᵏˣ pcosωx + qsinωx = λcosωx + μsinωx ```

Question 65

Differential equations: | How do solve second-order non-homogeneous differential equations?

Answer 65

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

Question 66

What is simple harmonic motion?

Answer 66

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

Question 67

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

Answer 67

x'' = -ω²x

Question 68

Simple harmonic motion: | write acceleration in terms of dv/dx

Answer 68

x'' = v dv/dx

Question 69

Simple harmonic motion: what is ω?

Answer 69

Angular velocity

Question 70

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

Answer 70

d²x/dt² + k dx/dt + ω²x = 0 | where k and ω² are positive constants

Question 71

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

Answer 71

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

Question 72

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

Answer 72

d²x/dt² + k dx/dt + ω²x = f(t) | where k and ω² are positive constants

Question 73

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

Answer 73

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

Question 74

What are coupled first-order linear differential equations?

Answer 74

``` dx/dt = ax + by + f(t) dy/dt = cx + dy + g(t) ```

Question 75

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?

Answer 75

Homogeneous