Title: InkMaths / Further Pure 1: results and forms
On every note of this deck, unless a note says otherwise:
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Front: STANDARD RESULT · LEARN IT
Parabola:
y^2 = 4ax: parametric point? tangent there?\(y^{2} = 4ax: \text{parametric point?} \text{tangent there?}\)y^2 = 4ax: parametric point? tangent there?
Back: (at^2, 2at), tangent ty = x + at^2\((at^{2}, \, 2at), \text{tangent } ty = x + at^{2}\)(at^2, 2at), tangent ty = x + at^2

Front: DEFINE
Define: L'Hospital's rule.
Back: For an indeterminate form 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g', provided that second limit exists.

Front: STANDARD RESULT · LEARN IT
t-formula:
t = tan θ2: cos θ = ?\(t = \tan\frac{θ}{2}: \cos θ = ?\)t = tan (θ/2): cos θ = ?
Back: cos θ = 1 - t^21 + t^2\(\cos θ = \frac{1 - t^{2}}{1 + t^{2}}\)cos θ = (1 - t^2)/(1 + t^2)

Front: STANDARD RESULT · LEARN IT
Vector product:
|a × b| = ?\(|\text{a} × \text{b}| = ?\)| a × b| = ?
Back: |a||b| sin θ, the parallelogram's area\(|\text{a}||\text{b}| \sin θ, \text{the parallelogram's area}\)| a|| b| sin θ, the parallelogram's area

Front: RECALL
State how to build a series solution to a differential equation.
Back: The booklet prints the Taylor form, so the marks are in getting the numbers into it. Differentiate the equation repeatedly, substitute the given values at x = a each time to turn every derivative into a number, then feed those numbers into the series. Products need the product rule at every stage, which is where the working usually dies.

Front: RECALL
State how to get the tangent to a conic at a parametric point.
Back: Differentiate both parametric coordinates with respect to the parameter and divide, so dy/dx is (dy/dt) over (dx/dt). Then use y − y₁ = m(x − x₁) at the point. Implicit differentiation of the cartesian form gives the same gradient when the parameter is ugly.

Front: STANDARD RESULT · LEARN IT
Scalar triple product:
parallelepiped volume = ?, tetrahedron = ?\(\text{parallelepiped volume} = ?, \text{tetrahedron} = ?\)parallelepiped volume = ?, tetrahedron = ?
Back: |a · (b × c)|, 16|a · (b × c)|\(|\text{a} · (\text{b} × \text{c})|, \frac{1}{6}|\text{a} · (\text{b} × \text{c})|\)| a · (b × c)|, (1/6)| a · (b × c)|

Front: SPOT THE ERROR
Say what is wrong with this claim:
To solve x/(x − 2) < 3, multiply both sides by x − 2 and solve the result.
Back: x − 2 is negative when x < 2, and multiplying by a negative reverses the inequality. Subtract to one side, combine over a common denominator, and read a sign diagram; the shortcut loses the whole interval x < 2.

Front: STANDARD RESULT · LEARN IT
Weierstrass substitution:
t = tan x2: dx = ?\(t = \tan\frac{x}{2}: dx = ?\)t = tan (x/2): dx = ?
Back: dx = 2 dt1 + t^2\(dx = \frac{2 \, dt}{1 + t^{2}}\)dx = (2 dt)/(1 + t^2)

Front: STANDARD RESULT · LEARN IT
t-formula:
t = tan θ2: sin θ = ?\(t = \tan\frac{θ}{2}: \sin θ = ?\)t = tan (θ/2): sin θ = ?
Back: sin θ = 2t1 + t^2\(\sin θ = \frac{2t}{1 + t^{2}}\)sin θ = 2t/(1 + t^2)

Front: RECALL
State the condition Simpson's rule needs.
Back: The booklet prints the formula and stops there. The number of strips has to be even, so the ordinates come out odd in number and the coefficients run 1, 4, 2, 4, ..., 4, 1, starting and finishing on a 1. An odd strip count and the rule simply does not apply.

Front: STANDARD RESULT · LEARN IT
Ellipse against hyperbola: which way round?:
ellipse b^2 = ?, hyperbola b^2 = ?\(\text{ellipse } b^{2} = ?, \text{hyperbola } b^{2} = ?\)ellipse b^2 = ?, hyperbola b^2 = ?
Back: a^2(1 - e^2) where e is below 1, a^2(e^2 - 1) where e is above 1\(a^{2}(1 - e^{2}) \text{ where e is below 1}, a^{2}(e^{2} - 1) \text{ where e is above 1}\)a^2(1 - e^2) where e is below 1, a^2(e^2 - 1) where e is above 1