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Grade 11 Mathematics Formula Sheet

CAPS / NSC & IEB · Paper 1 & Paper 2

√ Exponents & Surds

Rational exponent $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$

Surd laws $\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}$

$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Rationalising $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$

with a binomial $\dfrac{1}{\sqrt{a}+\sqrt{b}} \times \dfrac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}$

Multiply by the conjugate, not by the same expression — that is what clears the surd, because $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a - b$.

α Equations & Nature of Roots

Quadratic formula $x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

Discriminant $\Delta = b^{2} - 4ac$

$\Delta > 0$ two unequal real roots

$\Delta = 0$ two equal real roots

$\Delta < 0$ non-real roots

$\Delta$ a perfect square roots are rational

"Real and unequal" and "rational" are different questions. $\Delta > 0$ answers the first; whether $\Delta$ is a perfect square answers the second, and the exam asks for both.

∑ Quadratic Number Patterns

General term $T_n = an^{2} + bn + c$

Second difference $2a$

First difference $3a + b$

First term $a + b + c$

Build the three equations from the top of the pattern and solve in order — $a$ from the second difference, then $b$, then $c$. Guessing $T_n$ and checking is slower and loses method marks.

⊙ 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 $m = \dfrac{y_2-y_1}{x_2-x_1}$

Parallel $m_1 = m_2$

Perpendicular $m_1 \cdot m_2 = -1$

Line through a point $y - y_1 = m(x - x_1)$

Angle of inclination $m = \tan\theta$

If $\tan\theta$ is negative the calculator returns a negative angle; add $180^\circ$ to get the inclination, which is measured anticlockwise from the positive $x$-axis and is never negative.

ƒ Functions & Graphs

Function	Shape	Key features
$y = a(x+p)^{2} + q$	parabola	turning point $(-p,\ q)$ · axis of symmetry $x = -p$ · $a > 0$ opens up
$y = \dfrac{a}{x+p} + q$	hyperbola	asymptotes $x = -p$ and $y = q$ · no turning point
$y = ab^{x+p} + q$	exponential	asymptote $y = q$ · $b > 1$ grows, $0 < b < 1$ decays

The sign inside the bracket is the opposite of the shift: $y = (x+3)^2$ moves left three. Losing that costs a mark on almost every transformation question.

θ Trigonometry

Square identity $\sin^{2}\theta + \cos^{2}\theta = 1$

Quotient identity $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

Reduction $\sin(180^\circ - \theta) = \sin\theta$

$\cos(180^\circ - \theta) = -\cos\theta$

Co-function $\sin(90^\circ - \theta) = \cos\theta$

General solution $\sin\theta = k \Rightarrow \theta = \sin^{-1}k + n\cdot360^\circ$

or $\theta = 180^\circ - \sin^{-1}k + n\cdot360^\circ$

$\cos\theta = k$ $\theta = \pm\cos^{-1}k + n\cdot360^\circ$

$\tan\theta = k$ $\theta = \tan^{-1}k + n\cdot180^\circ$

The full reduction table and the CAST diagram are on the trigonometry sheet. Writing $n \in \mathbb{Z}$ is worth a mark on its own.

◺ Sine, Cosine & Area Rules

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 $\text{Area} = \tfrac{1}{2}ab\sin C$

Sine rule when an angle is opposite a known side

Cosine rule when three sides, or two sides and the angle between them

In a 3-D problem, name the triangle you are working in before writing anything down. Most lost marks here are a correct rule applied to the wrong triangle.

⬔ Measurement

Right pyramid — V $\tfrac{1}{3} \times \text{base area} \times h$

Cone — volume $\tfrac{1}{3}\pi r^{2} h$

Cone — surface area $\pi r^{2} + \pi r \ell$

Sphere — volume $\tfrac{4}{3}\pi r^{3}$

Sphere — surface area $4\pi r^{2}$

Multiplying every dimension by $k$ multiplies area by $k^{2}$ and volume by $k^{3}$. That is the whole of the "effect of change" question.

◯ Circle Geometry — the theorems you must name

Centre to chord A line from the centre perpendicular to a chord bisects it (and conversely)

Angle at centre The angle at the centre is twice the angle at the circumference on the same arc

Semicircle The angle in a semicircle is $90^\circ$

Same segment Angles in the same segment are equal

Cyclic quadrilateral Opposite angles are supplementary; an exterior angle equals the interior opposite angle

Tangent A tangent is perpendicular to the radius at the point of contact; two tangents from a point are equal

Every statement in a geometry proof needs a reason, and the reason must be the theorem's name. A correct chain of statements with no reasons scores close to nothing.

R Finance, Growth & Decay

Simple interest $A = P(1 + in)$

Compound interest $A = P(1 + i)^{n}$

Straight-line depreciation $A = P(1 - in)$

Reducing balance $A = P(1 - i)^{n}$

Nominal to effective $1 + i_{\text{eff}} = \left(1 + \dfrac{i^{(m)}}{m}\right)^{m}$

"Depreciation on a reducing balance" is compound interest with a minus. Straight-line depreciation is simple interest with a minus. Reading which one the question means is most of the work.

σ Statistics

Mean $\bar{x} = \dfrac{\sum x}{n}$

Variance $\sigma^{2} = \dfrac{\sum (x - \bar{x})^{2}}{n}$

Standard deviation $\sigma = \sqrt{\dfrac{\sum (x-\bar{x})^{2}}{n}}$

Interquartile range $Q_3 - Q_1$

Outlier below $Q_1 - 1{,}5(\text{IQR})$ or above $Q_3 + 1{,}5(\text{IQR})$

Skewed right mean $>$ median

Skewed left mean $<$ median

Use the calculator's statistics mode for $\sigma$ — the formula is here so you can explain it, not so you compute it by hand under time pressure.

P Probability

Complement $P(\text{not } A) = 1 - P(A)$

Addition rule $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Mutually exclusive $P(A \text{ and } B) = 0$

Independent $P(A \text{ and } B) = P(A) \times P(B)$

Independent and mutually exclusive are not the same thing and cannot both be true for events with non-zero probability. The exam sets that trap most years.

Grade 11 is where maths marks quietly slip.

The content doubles and nothing is examined again until it matters. I tutor Grade 10–12 maths one-to-one in Gqeberha, at your own table, in English or Afrikaans. The first consultation is free.

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An independent revision aid aligned to the CAPS/NSC Mathematics syllabus. Not an official Department of Basic Education document.