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Grade 12 Mathematics Formula Sheet
CAPS / NSC & IEB · Paper 1 & Paper 2 · Grades 10–12
α Algebra & Equations
Quadratic formula
$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta = b^2 - 4ac$
$\Delta > 0$
2 distinct real roots
$\Delta = 0$
1 repeated real root
$\Delta < 0$
no real roots
Exponent laws
$a^m \cdot a^n = a^{m+n}$
$\dfrac{a^m}{a^n} = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^{-n} = \dfrac{1}{a^n}$ $\sqrt[n]{a} = a^{1/n}$
Log laws
$\log(xy) = \log x + \log y$
$\log\!\left(\dfrac{x}{y}\right) = \log x - \log y$
$\log(x^n) = n\log x$
Change of base
$\log_a x = \dfrac{\log x}{\log a}$
∑ Sequences & Series
Arith. term
$T_n = a + (n-1)d$
Arith. sum
$S_n = \dfrac{n}{2}\bigl[2a + (n-1)d\bigr]$
$S_n = \dfrac{n}{2}(a + l)$ [$l$ = last term]
Geom. term
$T_n = a \cdot r^{\,n-1}$
Geom. sum
$S_n = \dfrac{a(r^n - 1)}{r - 1}$ $r \neq 1$
Infinite sum
$S_\infty = \dfrac{a}{1-r}$ $|r| < 1$
$a$ = first term $d$ = common difference $r$ = common ratio $n$ = number of terms
R Finance, Growth & Decay
Simple interest
$A = P(1 + in)$
Simple decay
$A = P(1 - in)$
Compound growth
$A = P(1 + i)^n$
Compound decay
$A = P(1 - i)^n$
Effective rate
$(1+i_{\text{eff}}) = \!\left(1 + \dfrac{i_{\text{nom}}}{m}\right)^{\!m}$
Future value
$F = \dfrac{x\bigl[(1+i)^n - 1\bigr]}{i}$
Present value
$P = \dfrac{x\bigl[1 - (1+i)^{-n}\bigr]}{i}$
$P$ = principal $A$ = accrued amount $i$ = rate per period $n$ = periods $x$ = payment $m$ = compoundings/year
∫ Differential Calculus
First principles
$f'(x) = \displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$
Power rule
$\dfrac{d}{dx}\bigl[x^n\bigr] = nx^{n-1}$
Constant
$\dfrac{d}{dx}\bigl[k\bigr] = 0$
Scalar multiple
$\dfrac{d}{dx}\bigl[k\,f(x)\bigr] = k\,f'(x)$
Sum / difference
$\dfrac{d}{dx}\bigl[f \pm g\bigr] = f'(x) \pm g'(x)$
$f'(x) = 0$
stationary point (max or min)
$f''(x) = 0$
point of inflection
$f'(x) > 0$
function increasing
$f'(x) < 0$
function decreasing
$f''(x) > 0$
concave up (local min)
$f''(x) < 0$
concave down (local max)
θ Trigonometry
SOH-CAH-TOA
$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$ $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$ $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$
Pythag. identity
$\sin^2\theta + \cos^2\theta = 1$
Quotient identity
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
Co-functions (90°)
$\sin(90°\!-\!\theta)=\cos\theta$ $\cos(90°\!-\!\theta)=\sin\theta$
Reduction (180°)
$\sin(180°\!-\!\theta)=\sin\theta$ $\cos(180°\!-\!\theta)=-\cos\theta$
$\sin(180°\!+\!\theta)=-\sin\theta$ $\cos(180°\!+\!\theta)=-\cos\theta$
Reduction (360°)
$\sin(360°\!-\!\theta)=-\sin\theta$ $\cos(360°\!-\!\theta)=\cos\theta$
Negative angles
$\sin(-\theta)=-\sin\theta$ $\cos(-\theta)=\cos\theta$
$\sin(\alpha \pm \beta)$
$\sin\alpha\cos\beta \pm \cos\alpha\sin\beta$
$\cos(\alpha \pm \beta)$
$\cos\alpha\cos\beta \mp \sin\alpha\sin\beta$
$\sin 2\alpha$
$2\sin\alpha\cos\alpha$
$\cos 2\alpha$
$\cos^2\!\alpha - \sin^2\!\alpha$
$1 - 2\sin^2\!\alpha$
$2\cos^2\!\alpha - 1$
Sine rule
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
Cosine rule
$a^2 = b^2 + c^2 - 2bc\cos A$
Area rule
$\text{Area} = \tfrac{1}{2}ab\sin C$
| $0°$ | $30°$ | $45°$ | $60°$ | $90°$ | |
|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos\theta$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ |
| $\tan\theta$ | $0$ | $\dfrac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | undef |
CAST rule — All positive (Q1) · Sin positive (Q2) · Tan positive (Q3) · Cos positive (Q4) | General solution adds $n\cdot360°$ (or $n\cdot180°$ for tan), $n\in\mathbb{Z}$
⊙ Analytical Geometry
Distance
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint
$M = \!\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right)$
Gradient
$m = \dfrac{y_2-y_1}{x_2-x_1} = \tan\theta$
Parallel
$m_1 = m_2$
Perpendicular
$m_1 \times m_2 = -1$
Line (slope-int.)
$y = mx + c$
Line (point-slope)
$y - y_1 = m(x - x_1)$
Circle at origin
$x^2 + y^2 = r^2$
Circle at $(a,b)$
$(x-a)^2 + (y-b)^2 = r^2$
σ Statistics
Mean
$\bar{x} = \dfrac{\sum x_i}{n}$
Variance
$\sigma^2 = \dfrac{\sum(x_i - \bar{x})^2}{n}$
Std deviation
$\sigma = \sqrt{\dfrac{\sum(x_i - \bar{x})^2}{n}}$
Regression line
$\hat{y} = a + bx$
Slope $b$
$b = \dfrac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2}$
Intercept $a$
$a = \bar{y} - b\bar{x}$
Correlation: $r \approx +1$ strong positive · $r \approx -1$ strong negative · $r \approx 0$ no correlation
P Probability & Counting
Complement
$P(A') = 1 - P(A)$
Addition rule
$P(A\cup B) = P(A) + P(B) - P(A\cap B)$
Mutually excl.
$P(A\cap B) = 0$
Independent
$P(A\cap B) = P(A)\cdot P(B)$
Factorial
$n! = n\times(n-1)\times\cdots\times 2\times 1$
Permutations
${}^{n}P_r = \dfrac{n!}{(n-r)!}$
Combinations
${}^{n}C_r = \dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$
Counting princ.
$n_1 \times n_2 \times \cdots \times n_k$
△ Euclidean Geometry — Key Circle Theorems
Chord & centre
Perpendicular from centre bisects chord (and conversely)
Angle at centre
Angle at centre $= 2\times$ angle at circumference (same arc)
Semicircle
Angle in a semicircle $= 90°$
Cyclic quad.
Opposite angles of a cyclic quad. are supplementary (sum $= 180°$)
Same chord
Angles in the same segment (subtended by the same chord) are equal
Tangent ⊥ radius
Tangent is perpendicular to radius at point of tangency
Tangent–chord
Tangent–chord angle = inscribed angle in alternate segment
Equal tangents
Two tangents from an external point are equal in length
Knowing the formula is not the same as knowing when to use it.
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