A-level Mathematics/OCR/C1/Appendix A: Formulae

By the end of this module you will be expected to have learned the following formulae:

The Laws of Indices

1. $x^{a}x^{b}=x^{a+b}\,$
2. ${\frac {x^{a}}{x^{b}}}=x^{a-b}$
3. $x^{-n}={\frac {1}{x^{n}}}$
4. $\left(x^{a}\right)^{b}=x^{ab}$
5. $\left(xy\right)^{n}=x^{n}y^{n}$
6. $\left({\frac {x}{y}}\right)^{n}={\frac {x^{n}}{y^{n}}}$
7. $x^{\frac {a}{b}}={\sqrt[{b}]{x^{a}}}$
8. $x^{0}=1\,$
9. $x^{1}=x\,$

The Laws of Surds

1. ${\sqrt {xy}}={\sqrt {x}}\times {\sqrt {y}}$
2. ${\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}$
3. ${\frac {a}{b+{\sqrt {c}}}}={\frac {a}{b+{\sqrt {c}}}}\times {\frac {b-{\sqrt {c}}}{b-{\sqrt {c}}}}={\frac {a(b-{\sqrt {c}})}{b^{2}-c}}$

Polynomials

Parabolas

If f(x) is in the form $a(x+b)^{2}+c$

1. -b is the axis of symmetry
2. c is the maximum or minimum y value

Axis of Symmetry = ${\frac {-b}{2a}}$

Completing the Square

$ax^{2}+bx+c=0\,$ becomes $a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c$

The Quadratic Formula

• The solutions of the quadratic $ax^{2}+bx+c=0$ are: $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$
• The discriminant of the quadratic $ax^{2}+bx+c=0$ is $b^{2}-4ac$

Errors

1. $Absolute\ error=value\ obtained-true\ value$
2. $Relative\ error={\frac {absolute\ error}{true\ value}}$
3. $Percentage\ error=relative\ error\times 100$

Coordinate Geometry

Gradient of a line

$m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$

Point-Gradient Form

The equation of a line passing through the point $\left(x_{1},y_{1}\right)$ and having a slope m is $y-y_{1}=m\left(x-x_{1}\right)$.

Perpendicular lines

Lines are perpendicular if $m_{1}\times m_{2}=-1$

Distance between two points

$d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}$

Mid-point of a line

$\left({\frac {{x_{1}}+{x_{2}}}{2}};{\frac {{y_{1}}+{y_{2}}}{2}}\right)$

General Circle Formulae

$Area=\pi r^{2}\,$

$Circumference=2\pi r\,$

Equation of a Circle

$\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}$, where (h,k) is the center and r is the radius.

Differentiation

Differentiation Rules

1. Derivative of a constant function:

${\frac {dy}{dx}}\left(c\right)=0$

1. The Power Rule:

${\frac {dy}{dx}}\left(x^{n}\right)=nx^{n-1}$

1. The Constant Multiple Rule:

${\frac {dy}{dx}}cf\left(x\right)=c{\frac {dy}{dx}}f\left(x\right)$

1. The Sum Rule:

${\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)+g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)+{\frac {dy}{dx}}g\left(x\right)$

1. The Difference Rule:

${\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)-g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)-{\frac {dy}{dx}}g\left(x\right)$

Rules of Stationary Points

• If $f'\left(c\right)=0$ and $f''\left(c\right)<0$, then c is a local maximum point of f(x). The graph of f(x) will be concave down on the interval.
• If $f'\left(c\right)=0$ and $f''\left(c\right)>0$, then c is a local minimum point of f(x). The graph of f(x) will be concave up on the interval.
• If $f'\left(c\right)=0$ and $f''\left(c\right)=0$ and $f'''\left(c\right)\neq 0$, then c is a local inflection point of f(x).

This is part of the C1 (Core Mathematics 1) module of the A-level Mathematics text.