Maths - Misc | Olly Britton

Flashcards

Surds and completing the square

\[\sqrt{3 - 2\sqrt{2}}\]

How can you simplify this?

simplify-nested-surd-by-completing-square

Rewrite as

\[\sqrt{2}^2 - 2\cdot\sqrt{2}\cdot + 1^2\]

and complete the square.

Area of a shaded circle region

WACKY CIRCLE How could you find the area of the shaded region?

shaded-region-sector-plus-triangle

Split it up into a sector and a right-angled triangle.

WACKY CIRCLE ANSWER

Rotating a parametric curve

Given a parametric curve

\[(x(t), y(t))\]

how can you work out the parametric curve for rotating it by an angle $\theta$?

rotate-parametric-curve-by-rotation-matrix

\[\left(\begin{matrix} \cos \theta & -\sin \theta \\\\ \sin \theta & \cos \theta \end{matrix}\right)\]

Area of a triangle from coordinates

What is the formula for the area of a triangle with coordinates $(0, 0)$, $(x _ 1, y _ 1)$ and $(x _ 2, y _ 2)$?

triangle-area-two-points-determinant

\[\frac{1}{2} \left \vert \begin{matrix} x _ 1 & y _ 1 \\\\ x _ 2 & y _ 2 \end{matrix}\right \vert \]

What is the formula for the area of a triangle with coordinates $(x _ 1, y _ 1)$, $(x _ 2, y _ 2)$ and $(x _ 3, y _ 3)$?

triangle-area-three-points-determinant

\[\frac{1}{2} \left \vert \begin{matrix} x _ 1 & y _ 1 & 1 \\\\ x _ 2 & y _ 2 & 1 \\\\ x _ 3 & y _ 3 & 1 \end{matrix}\right \vert \]

Trigonometric and exponential identities

Careful: What is an alternative way of writing $2\sin (2x)$?

double-angle-2sin2x-rewrite

\[4\sin x \cos x\]

Careful: What is $e^{-\ln x}$?

e-to-minus-ln-x-is-reciprocal

\[\frac{1}{x}\]

Writing set-builder answers

When giving a set as an answer, like $x$ is a real number above $4$, where does the $x \in \mathbb{R}$ go?

set-builder-membership-at-start

At the start

\[\{ x \in \mathbb{R}\ \vert \ x > 4 \}\]

Equivalent logarithm and factorisation forms

\[\frac{1}{5}\ln\left \vert \frac{t + 2}{1 - 2t} \right \vert + c\]

doesn’t look the same as

\[\frac{1}{5}\ln\left \vert \frac{2t + 4}{2t - 1} \right \vert \]

but they are. How?

equivalent-log-expressions-via-modulus

You can flip the order of the minuses thanks to the modulus sign and the factor of 2 and top can become part of the constant when you take it out of the $\log$.

\[p^3 - q^3\]

How could you rewrite this?

difference-of-cubes-factorisation

\[(p-q)(p^2 + pq + q^2)\]

Choosing summary statistics for skewed data

If data is skew, is it better to use the median and IQR or the mean and s.d.?

skewed-data-prefer-median-iqr

The median and IQR.