Maths - Misc

Flashcards

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

How can you simplify this?? Rewrite as

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

and complete the square.

2022-01-19

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

How can you simplify this?

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Rewrite as

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

and complete the square.

2022-02-02

Given a parametric curve

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

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

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

2022-02-03

How could you find the area of the shaded region?

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

Given a parametric curve

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

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

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

2022-04-12

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

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

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

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

2022-04-18

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

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\[4\sin x \cos x\]

2022-06-03

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

doesn’t look the same as

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

but they are. How?? 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 $\log$.

2022-06-07

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\[p^3 - q^3\]

How could you rewrite this??

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

2022-06-18

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

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