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Core Pure Year 2
A-Level Further Maths- Edexcel
| Question | Answer |
|---|---|
| What is the exponential form of a complex number? | z=re^iθ (r is modulus, θ is argument) |
| What is Euler's relation? | e^iθ = cosθ + i sinθ |
| re^iθ x ge^ix = | rge^i(θ+x) (multiply modulus, add argument) |
| re^iθ / ge^ix = | (r/g)e^i(θ-x) (divide modulus, minus argument) |
| What is De Moivre's theorem? | (r(cosθ+isinθ))^n = r^n (cos(nθ)+isin(nθ)) |
| How do you use De Moivre's to show that sin(nθ) or cos(nθ) = some expansion involving powers of sinθ or cosθ | Expand (cosθ+isinθ)^n using both the binomial expansion and De Moivre's. Equate real parts for cos or imaginary parts for sin. Use trig identities to finish the question |
| z + 1/z = | 2 cosθ |
| z - 1/z = | 2i sinθ |
| 2 cosθ = | z + 1/z |
| 2i sinθ = | z - 1/z |
| z^n + 1/z^n = | 2 cos(nθ) |
| z^n - 1/z^n = | 2i sin(nθ) |
| 2 cos(nθ) = | z^n + 1/z^n |
| 2i sin(nθ) = | z^n - 1/z^n |
| How do you make (cosθ)^n or (sinθ)^n into the sum of non-exponential cos/sin (i.e. 20cos5θ + 3cos3θ + ...) (De Moivre's) | Use the [2cosθ = z + 1/z] or [2isinθ = z - 1/z] identity and raise it to the relevant power. Expand both sides. Rewrite the 'z' side using the [z^n +- 1/z^n] identities and replace with 2cos/2isin. Re-arrange into the correct form |
| How do we work out the sum of a complex series? | Make everything into exponentials. Use reg. maths formulae. The fraction will be over something including e^iθ, so times the num. and den, by e^-i(0.5)θ. You should now have a [z +- 1/z] which you can replace with trig. Replace other e's with trig |
| For any sum of a complex series, what do the real and imaginary parts equal? | Re = cosθ + cos2θ + cos3θ + ... Im = sinθ + sin2θ + sin3θ + ... |
| When given two series (one involving cos, the other sin) how can you show what a specific series is equal to? | With series C (cos based) and S (sin based), work out C+iS. Use reverse binomial expansion to find an exact value. Use any trig identities needed, then C = real parts, S = im parts |
| How do we solve an equation z^n = a + bi ? | Rewrite a + bi into the mod-argument form. Replace [cosθ+isinθ] with [cos(θ+2kπ)+isin(θ+2kπ)]. Do root n of both sides. If n=4, sub in k=0,1,2,3 to find all your solutions (etc) |
| When you find the nth roots of a complex number what is special about the roots? | They form each vertex of a regular n-gon with centre (0,0). |
| If given one complex vertex of a shape, how can you find the rest? | Keep multiplying the exponential form of the vertex by 2π/(no. sides) |
| How can we use the method of differences to find the sum of a series? | Get it in a form where you have the f(r) - f(r+1). Expand the sum and you should be able to cancel all expect the first and last term |
| What will happen if a method of differences is applied to f(r) - f(r+2) ? | All will cancel except 2 terms at the start and 2 at the end |
| How can we show the series expansion of any function using the maclaurin series? | Differentiate the function and plug in the correct values into the formula. |
| How can we find the series expansion of compound functions? (Like e^sinx) | Apply both formulas from formula book |
| What is a polar co-ordinate? | Co-ordinate written (r,θ) where r is its distance from the pole (0,0) and θ is the angle made with the initial line (usually + x axis) |
| Polar Co-ordinates: x= | rcosθ |
| Polar Co-ordinates: y= | rsinθ |
| Polar Co-ordinates: r^2= | x^2 + y^2 |
| Polar Co-ordinates: θ= | arctan(y/x) |
| What form are polar equations typically? | r = f(θ) |
| r=a forms what graph | circle |
| θ = a forms what graph | half line |
| r= aθ forms what graph | spiral |
| r = asin(xθ) / acos(xθ) forms what graph | rose |
| For a curve with equation r = a(p+qcosθ) where p≥q, what are the 3 scenarios for a graph? | If p=q then cardioid. If p≥2q then egg. If q<p<2q then dimple |
| How do we work out the enclosed area of a polar curve between θ=a and θ=b? | 0.5 ∫ (between a and b) r^2 dθ |
| How do we work out enclosed area of the intersection of polar curves? | Find points of intersection. Split area into sections which can be found individually |
| How do you work out the points on a polar curve which have tangents parallel to the initial line? | dy/dθ = 0 [y=rsinθ] |
| How do you work out the points on a polar curve which have tangents perpendicular to the initial line? | dx/dθ = 0 [x=rcosθ] |
| How do you work out the volume of revolutions of parametrics? | Work out y^2, dx/dt and the new bounds then plug them into the integral. (Or x^2, dy/dt if around y-axis) |
| What makes an integral improper? | At least one limit is infinite, or (in the range [a,b]) a, b or any number between is undefined |
| How do we evaluate improper integrals? | Make infinity or the undefined value t. Do the limit as t->infinity/ undefined. Sub in t at the end |
| Can all improper integrals be solved? | No, some are not convergent |
| How do you work out the mean value of a function between [a,b] ? | 1/(b-a) ∫ (between a and b) f(x) dx |
| How do we differentiate inverse trig functions (2 methods)? | Use implicit differentiation after removing the arc, OR use chain rule alongside formula in the book |
| How do we integrate inverse trig functions? | Get it into the form of the formulas and apply them. (SPLIT INTO TWO FRACTIONS IF NEEDED) |
| How do we integrate fractions which can't be made into ln ? | Use partial fractions |
| If the mean value of a function f(x) between a,b is M, what is the mean value of kf(x), f(x) + k and -f(x) ? | kf(x) = kM, f(x) + k = M+k, -f(x) = -M |
| How do we solve first order differential equations? | Get it in the form dy/dx + P(x)y = Q(x), multiply everything by the IF (e^∫P(x)dx), simplify LHS into d/dx(y * IF) ,solve the equation to get y |
| How do we solve second order homogenous differentials? | Make aux equations (am^2 + bm + c = 0) using co-efficients of question. Solve for m. Plug into solutions for y= depending on amount of m values |
| SOHD: Two real m values. y= | Ae^(m1)x + Be^(m2)x |
| SOHD: One real m values. y= | (A+Bx)e^mx |
| SOHD: Zero real m values. y= | [m=a+bi] y=e^ax(Acosbx + Bsinbx) |
| How to non-homogeneous differentials differ? | After solving it as if it were homogeneous, posit a suitable form for y, differentiate twice and plug in to find the value. Add on at the end |
| if f(x) is in the form k, what is the form of the particular integral | λ |
| if f(x) is in the form ax+b, what is the form of the particular integral | λx + μ |
| if f(x) is in the form ax^2, what is the form of the particular integral | λx^2 + μx + ν |
| if f(x) is in the form ke^px, what is the form of the particular integral | λe^px |
| if f(x) is in the form mcos(wx), what is the form of the particular integral | λcos(wx)+μsin(wx) |
| if f(x) is in the form mcos(wx) + msin(wx), what is the form of the particular integral | λcos(wx)+μsin(wx) |
| What is the issue which can arise when solving second-order differentials and what is the solution | Issue is if the thing we posit is already involved in the C.F, so we multiply what we are positting by x |
| How do we use boundary conditions? | After finding y, plug in anything stated (differentiate if needed) to find A and B |
| Boundary conditions: what does it mean if y is bounded? | A or B needs to be a certain number so that y stops shooting off. So if y=Ae^x + Be^-x, A=0 so that the graph doesn't shoot off |
| If the denominator of a partial fraction includes (x^2 + c) [c>0] as a factor, what is the expanded form? | (A+Bx) / (x^2 + c) |
| What is sinh(x) defined as? | (e^x - e^-x) / 2 |
| What is cosh(x) defined as? | (e^x + e^-x) / 2 |
| What is tanh(x) defined as? | (e^x - e^-x) / (e^x + e^-x) OR (e^2x - 1) / (e^2x + 1) |
| What is the range of y=cosh(x) | y ≥ 1 |
| What is arsinh(x) defined as? | ln(x+√(x^2 + 1)) |
| What is arcosh(x) defined as? | ln(x+√(x^2 - 1)) WHERE x≥1 |
| What is artanh(x) defined as? | 1/2 ln((1+x)/(1-x)) WHERE |x |<1 |
| What is the identity involving cosh and sinh | (coshA)^2 - (sinhA)^2 = 1 |
| sinh(A±B) | sinhAcoshB ± coshAsinhB |
| cosh(A±B) | coshAcoshB ± sinhAsinhB |
| Differentiate sinh x | cosh x |
| Differentiate cosh x | sinh x |
| Differentiate tanh x | (sech x)^2 |
| Differentiate arsinh x | 1 / √(x^2 + 1) |
| Differentiate arcosh x | 1 / √(x^2 - 1) |
| Differentiate artanh x | 1 / (1 - x^2) |
| Integrate sinh x | cosh x + c |
| Integrate cosh x | sinh x + c |
| Integrate tanh x | ln(cosh x) + c |
| Integrate 1 / √(a^2 + x^2) | arsinh (x/a) + c |
| Integrate 1 / √(x^2 - a^2) | arcosh (x/a) + c WHERE x>a |
| What is the equation for simple harmonic motion | ẍ = -ω^2 (x) |
| What is ω equal to in terms of harmonic motion | 2π / T (T is time period) |
| What is the equation for damped harmonic motion? | ẍ + kẋ + ω^2 x = 0 |
| How do we know we have CRITICAL damping? | AUX equation has 1 real solution |
| How do we know we have HEAVY damping? | AUX equation has 2 real solutions |
| How do we know we have LIGHT damping? | AUX equation has 0 real solutions |
| How do we combine differentials? | Differentiate one and use the other one to get only x's (or only y's) |
| Once we solve one differential of a pair, how can we solve the other? | Differentiate it once and then plug it into the formula for the other |
| Reservoir has 10,000L unpolluted water. Leaking 200L /day. Contaminated fluid IN at 300L /day, containing 4g contaminant in every L. x grams after t days. Form differential equation. | rate of contaminant = grams in each day - concentration x litres lost dx/dt = 1200 - 200(x/10000-200t+300t) dx/dt = 1200 - 2x/(100+t) |
Created by:
XanderMoore
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