2. The Formula Sheet & Mastery Ideas

You get the formula sheet, but at this level success is about combining tools and recalling a few facts the sheet leaves out.

Geometry

• Volumes: cylinder \(\pi r^2 h\), cone \(\tfrac13\pi r^2 h\), sphere \(\tfrac43\pi r^3\) (a hemisphere is half of that).
• Annulus (ring) area: \(\pi R^2 - \pi r^2\).
• Sector: arc \(=\frac{\theta}{360}(2\pi r)\), area \(=\frac{\theta}{360}(\pi r^2)\).

Algebra

• Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\).
• Fractional exponent: \(a^{m/n}=\left(\sqrt[n]{a}\right)^m\); negative exponent: \(a^{-n}=\dfrac{1}{a^n}\).
• Arithmetic sequence: \(a_n=a_1+(n-1)d\).

Mastery facts the sheet won't give you

• Scaling: lengths by \(k\), areas by \(k^2\), volumes by \(k^3\).
• Inverse variation: if more workers means fewer days, the product (worker-days) is constant.
• Average speed over a whole trip is total distance \(\div\) total time -- never the plain average of the speeds.
• Conditional probability: \(P(B\mid A)=\dfrac{P(A\text{ and }B)}{P(A)}\).
• Combinations: choosing \(r\) from \(n\) (order doesn't matter) is \({}_nC_r=\dfrac{n!}{r!\,(n-r)!}\).