Key mathematical principles - itnett/FTD02H-N GitHub Wiki

A Level Mathematics Formula Sheet

Shapes

• Area of Triangle: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

• Area of Parallelogram: $$\text{Area} = \text{base} \times \text{height}$$

• Area of Rectangle: $$\text{Area} = \text{length} \times \text{width}$$

• Area of Trapezoid: $$\text{Area} = \frac{1}{2} (\text{sum of parallel sides}) \times \text{height}$$

• Circumference & Area of Circle: $$C = 2\pi r, \quad A = \pi r^2$$

• Surface Area of Cuboid: $$SA = 2lw + 2lh + 2wh \quad \text{where } l, w, h \text{ are side lengths}$$

• Volume of Cuboid: $$V = lwh \quad \text{where } l, w, h \text{ are side lengths}$$

• Surface Area of Cylinder: $$SA = 2\pi rh + 2\pi r^2 \quad \text{(Note: Curved part: } 2\pi rh \text{)}$$

• Volume of Cylinder: $$V = \pi r^2 h$$

• Surface Area of Cone: $$SA = \pi r l + \pi r^2 \quad \text{(Note: Curved part: } \pi r l \text{)}$$

• Volume of Cone: $$V = \frac{1}{3} \pi r^2 h$$

• Surface Area of Sphere: $$SA = 4\pi r^2 \quad \text{(Note: Hemisphere = } 2\pi r^2 + \pi r^2 = 3\pi r^2 \text{)}$$

• Volume of Sphere: $$V = \frac{4}{3} \pi r^3 \quad \text{(Note: Hemisphere = } \frac{2}{3} \pi r^3 \text{)}$$

• Volume of Prism: $$V = \text{Area of cross section} \times \text{height}$$

• Volume of Pyramid: $$V = \frac{1}{3} \times \text{Base Area} \times \text{height}$$

Indices

• Multiplication: $$a^m \times a^n = a^{m+n}$$ $$(a^m)^n = a^{mn}$$ $$(ab)^m = a^m b^m$$

• Division: $$\frac{a^m}{a^n} = a^{m-n}$$

• Negative Powers: $$a^{-m} = \frac{1}{a^m}$$

• Fractions: $$\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$$ $$\left( \frac{a}{b} \right)^{-m} = \frac{b^m}{a^m}$$

• Rational Powers: $$a^{m/n} = (a^m)^{1/n} = \sqrt[n]{a^m}$$

Series

• Arithmetic Sequence (nth term): $$u_n = a + (n-1)d \quad \text{where } a \text{ is the first term, } d \text{ is the common difference}$$

• Sum of n terms (Arithmetic): $$S_n = \frac{n}{2} [2a + (n-1)d] = \frac{n}{2} (a + l) \quad \text{where } l \text{ is the last term}$$

• Geometric Sequence (nth term): $$u_n = ar^{n-1} \quad \text{where } a \text{ is the first term, } r \text{ is the common ratio}$$

• Sum of n terms (Geometric): $$S_n = a \frac{r^n - 1}{r-1} = \frac{a(1-r^n)}{1-r} \quad (r \neq 1)$$

• Sum to Infinity (Geometric): $$S_{\infty} = \frac{a}{1-r} \quad (|r| < 1)$$

• Compound Interest: $$FV = PV \left(1 + \frac{r}{n}\right)^{nt}$$

  • $( FV )$: future value
  • $( PV )$: present value
  • $( t )$: number of years
  • $( r )$: nominal annual interest rate
  • $( n )$: number of compounding periods per year

• Binomial Theorem (integer powers): $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$

• Binomial Theorem (Fractional & Negative powers): $$(a + b)^n = a^n \left(1 + \frac{b}{a}\right)^n$$

  • For small ( x ): $$(1+x)^n \approx 1 + nx + \frac{n(n-1)}{2!} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \cdots$$

Geometry

• Straight Line (Equation):

  • Slope intercept form: $$y = mx + c$$
  • General form: $$Ax + By + C = 0$$
  • Point slope form: $$y - y_1 = m(x - x_1)$$

• Gradient: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• Distance between 2 points: $$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

• Midpoint: $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

• Equation of Circle: $$(x - a)^2 + (y - b)^2 = r^2 \quad \text{(centre } (a,b), \text{ radius } r)$$

Quadratics

• Quadratic Function (Solutions): $$ax^2 + bx + c = 0$$ $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad a \neq 0$$

• Axis of Symmetry: $$y = ax^2 + bx + c \implies x = -\frac{b}{2a}$$

• Discriminant: $$\Delta = b^2 - 4ac$$

  • $( \Delta > 0 )$: 2 real distinct roots
  • $( \Delta = 0 )$: 2 real repeated/double roots
  • $( \Delta < 0 )$: no real roots

• Completing the Square: $$ax^2 + bx + c = a \left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$$

• Max/Min Value: $$c - \frac{b^2}{4a}$$

Exponential and Logarithmic Functions

• Exponential Function: $$a^x = y \implies \log_a y = x$$

• Logarithm Rules:

  • $( \log_a (xy) = \log_a x + \log_a y )$
  • $( \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y )$
  • $( \log_a (x^b) = b \log_a x )$
  • $( \log_a x = \frac{\log_b x}{\log_b a} )$

Trigonometry

• Sine Rule: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

• Cosine Rule: $$c^2 = a^2 + b^2 - 2ab \cos C$$

• Area of Triangle: $$\text{Area} = \frac{1}{2}ab \sin C$$

• Degrees to Radians: $$\text{Degrees to Radians: } \times \frac{\pi}{180}$$ $$\text{Radians to Degrees: } \times \frac{180}{\pi}$$

• Length of an Arc: $$\text{Length} = \frac{\theta}{360} \times 2\pi r \quad (\text{degrees})$$ $$\text{Length} = r\theta \quad (\text{radians})$$

• Area of a Sector: $$\text{Area} = \frac{\theta}{360} \times \pi r^2 \quad (\text{degrees})$$ $$\text{Area} = \frac{1}{2} r^2 \theta \quad (\text{radians})$$

• Small Angle Approximations: $$\sin \theta \approx \theta, \quad \cos \theta \approx 1 - \frac{\theta^2}{2}, \quad \tan \theta \approx \theta$$

• Pythagorean Identities: $$\sin^2 \theta + \cos^2 \theta = 1$$ $$1 + \tan^2 \theta = \sec^2 \theta$$ $$1 + \cot^2 \theta = \csc^2 \theta$$

• Double Angle Formulas: $$\sin 2\theta = 2 \sin \theta \cos \theta$$ $$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta$$ $$\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$$

• Half Angle Formulas: $$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$ $$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$$ $$\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$$

Vectors

• Vector Form: $$\mathbf{a} = ai + bj + ck$$

• Magnitude of a Vector: $$|\mathbf{a}| = \sqrt{a^2 + b^2 + c^2}$$

• Unit Vector: $$\mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$$

• Midpoint: $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$$

• Scalar Product: $$\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}| |\mathbf{b}| \cos \theta$$

• Angle Between Two Vectors: $$\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$$

• Vector Equation of a Line: $$\mathbf{r} = \mathbf{a} + t\mathbf{b}$$

Probability and Statistics

• Mean: $$\bar{x} = \frac{\sum x_i}{n} \quad \text{(no frequency)}$$ $$\bar{x} = \frac{\sum f x_i}{\sum f} \quad \text{(with frequency)}$$

• Variance: $$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \quad \text{(no frequency)}$$ $$\sigma^2 = \frac{\sum f (x_i - \bar{x})^2}{\sum f} \quad \text{(with frequency)}$$

• Standard Deviation: $$\sigma = \sqrt{\sigma^2}$$

• Probability of Event A: $$P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$

• Complementary Events: $$P(A') = 1 - P(A)$$

• Combined Events (Addition Rule): $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

• Mutually Exclusive Events: $$P(A \cap B) = 0$$ $$P(A \cup B) = P(A) + P(B)$$

• Independent Events: $$P(A \cap B) = P(A)P(B)$$ $$P(A \cup B) = P(A) + P(B) - P(A)P(B)$$

• Conditional Probability: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

• Bayes' Theorem: $$P(A|B) = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A')P(A')}$$

Binomial Distribution

• Probability Mass Function (pmf): $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

• Mean: $$E(X) = np$$

• Variance: $$\text{Var}(X) = np(1-p)$$

Normal Distribution

• Probability Density Function (pdf): $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

• Standardisation: $$Z = \frac{X - \mu}{\sigma}$$

• Interquartile Range (IQR): $$\text{IQR} = Q_3 - Q_1$$

• Outliers:

  • Values greater than: $$Q_3 + 1.5 \times \text{IQR}$$
  • Values less than: $$Q_1 - 1.5 \times \text{IQR}$$

Mechanics

• SUVAT Equations: $$v = u + at$$ $$s = ut + \frac{1}{2}at^2$$ $$s = vt - \frac{1}{2}at^2$$ $$s = \frac{u+v}{2}t$$ $$v^2 = u^2 + 2as$$

Calculus (Differentiation and Integration)

• Turning/Stationary Points:

  • Solve: $$\frac{dy}{dx} = 0$$

• Max/Min:

  • If: $$\frac{d^2y}{dx^2} > 0 \text{ (min)}$$ $$\frac{d^2y}{dx^2} < 0 \text{ (max)}$$

• Points of Inflection:

  • Solve: $$\frac{d^2y}{dx^2} = 0$$

• Increasing/Decreasing:

  • Increasing: $$\frac{dy}{dx} > 0$$
  • Decreasing: $$\frac{dy}{dx} < 0$$

• Convex/Concave:

  • Convex: $$\frac{d^2y}{dx^2} > 0$$
  • Concave: $$\frac{d^2y}{dx^2} < 0$$

• Tangents and Normals:

  • Tangent: $$y - y_1 = m(x - x_1)$$
  • Normal: $$y - y_1 = -\frac{1}{m}(x - x_1)$$

• Implicit Differentiation:

  • Differentiate both sides, applying chain rule where necessary.

Integration

• Area under a Curve:

  • Between curve and x-axis: $$\int_a^b y , dx$$
  • Between two curves: $$\int_a^b (y_{\text{top}} - y_{\text{bottom}}) , dx$$

• Kinematics:

  • Distance: $$\int |v(t)| , dt$$
  • Displacement: $$\int v(t) , dt$$

• Differentiation from First Principles: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

• Chain Rule: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

• Product Rule: $$\frac{d}{dx} [uv] = u'v + uv'$$

• Quotient Rule: $$\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$$

Derivatives

• Common Derivatives:
  • $x^n \rightarrow nx^{n-1}$
  • $e^{ax} \rightarrow ae^{ax}$
  • $\ln(ax) \rightarrow \frac{1}{x}$
  • $\sin(ax) \rightarrow a\cos(ax)$
  • $\cos(ax) \rightarrow -a\sin(ax)$
  • $\tan(ax) \rightarrow a\sec^2(ax)$
  • $\sec(ax) \rightarrow a\sec(ax)\tan(ax)$
  • $\csc(ax) \rightarrow -a\csc(ax)\cot(ax)$
  • $\cot(ax) \rightarrow -a\csc^2(ax)$
  • $\sin^{-1}(x) \rightarrow \frac{1}{\sqrt{1-x^2}}$
  • $\cos^{-1}(x) \rightarrow -\frac{1}{\sqrt{1-x^2}}$
  • $\tan^{-1}(x) \rightarrow \frac{1}{1+x^2}$

Integrals

• Common Integrals:
  • $ int x^n , dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$
  • $\int \frac{1}{x} , dx = \ln|x| + C$
  • $\int \sin(ax) , dx = -\frac{1}{a} \cos(ax) + C$
  • $\int \cos(ax) , dx = \frac{1}{a} \sin(ax) + C$
  • $\int e^{ax} , dx = \frac{1}{a} e^{ax} + C$
  • $\int a^x , dx = \frac{a^x}{\ln(a)} + C$
  • $\int \sec^2(ax) , dx = \frac{1}{a} \tan(ax) + C$
  • $\int \csc^2(ax) , dx = -\frac{1}{a} \cot(ax) + C$
  • $\int \sec(ax)\tan(ax) , dx = \frac{1}{a} \sec(ax) + C$
  • $\int \csc(ax)\cot(ax) , dx = -\frac{1}{a} \csc(ax) + C$
  • $\int \sinh(ax) , dx = \frac{1}{a} \cosh(ax) + C$
  • $\int \cosh(ax) , dx = \frac{1}{a} \sinh(ax) + C$
  • $\int \tanh(ax) , dx = \frac{1}{a} \ln|\cosh(ax)| + C$
  • $\int \text{sech}^2(ax) , dx = \frac{1}{a} \tanh(ax) + C$
  • $\int \text{csch}^2(ax) , dx = -\frac{1}{a} \coth(ax) + C$

Integration by Parts

• Formula: $$\int u , dv = uv - \int v , du$$

Trapezium Rule

• Formula: $$\text{Area} \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n]$$

Newton-Raphson Method

• Formula: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Functions

• Inverse Functions:

  • Replace ( f(x) ) with ( y ), swap ( x ) and ( y ), solve for ( y ).

• Composite Functions:

  • $$( (f \circ g)(x) = f(g(x)) )$$

• Odd and Even Functions:

  • Even: $( f(-x) = f(x) )$
  • Odd: $( f(-x) = -f(x) )$

• Transformations:

  • Vertical stretch: $( af(x) )$
  • Horizontal stretch: ( f(bx) )$
  • Translation: $( f(x - c) + d )$
  • Reflection: $( f(-x) ) (y-axis), ( -f(x) ) (x-axis)$

This cheat sheet covers the essential formulas and concepts in A Level Mathematics. Use this as a reference guide for quick recall of key mathematical principles.