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

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

The Laws of Indices[edit]

  1. x^ax^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[edit]

  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[edit]

Parabolas[edit]

If f(x) is in the form a(x + b)^2 + e

  1. -d is the axis of symmetry
  2. e is the maximum or minimum value

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

Completing the Square[edit]

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

The Quadratic Formula[edit]

  • 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[edit]

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


Coordinate Geometry[edit]

Gradient of a line[edit]

m=\frac {y_2-y_1}{x_2-x_1}

Point-Gradient Form[edit]

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[edit]

Lines are perpendicular if m_1 \times m_2=-1

Distance between two points[edit]

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Mid-point of a line[edit]

\left(\frac {{x_1} + {x_2}}{2} ; \frac {{y_1} + {y_2}}{2}\right)

General Circle Formulae[edit]

Area = \pi r^2\,

Circumference = 2 \pi r\,

Equation of a Circle[edit]

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

Differentiation[edit]

Differentiation Rules[edit]

  1. Derivative of a constant function:

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

  2. The Power Rule:

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

  3. The Constant Multiple Rule:

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

  4. 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 )

  5. 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[edit]

  • 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 ) \ne 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.