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Grade 10 Mathematics Formula Sheet

CAPS / NSC & IEB · Paper 1 & Paper 2

^ Exponent Laws

Multiplying $a^{m} \cdot a^{n} = a^{m+n}$

Dividing $\dfrac{a^{m}}{a^{n}} = a^{m-n}$

Power of a power $(a^{m})^{n} = a^{mn}$

Negative exponent $a^{-n} = \dfrac{1}{a^{n}}$

Zero exponent $a^{0} = 1,\quad a \neq 0$

Root as a power $\sqrt[n]{a} = a^{\frac{1}{n}}$

These only combine powers of the same base. $2^{3}\cdot 3^{2}$ does not simplify — rewriting both sides to a common base is usually the first step the question is testing.

× Factorising

Difference of squares $a^{2}-b^{2} = (a-b)(a+b)$

Trinomial $x^{2}+(p+q)x+pq = (x+p)(x+q)$

Sum of cubes $a^{3}+b^{3} = (a+b)(a^{2}-ab+b^{2})$

Difference of cubes $a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})$

Grouping $ax+ay+bx+by = (a+b)(x+y)$

Take out the common factor first, every time. Half the "hard" factorising questions become one of the standard forms the moment the common factor is gone.

= Equations & Inequalities

Quadratic (factorised) $(x-p)(x-q)=0 \Rightarrow x=p \text{ or } x=q$

Simultaneous substitute one equation into the other, or add/subtract to eliminate

Inequality — divide by a negative reverse the sign

Interval notation $x \in [\,a;\ b\,)$

A quadratic must equal zero before you factorise it. Moving everything to one side is not tidying up — it is the step that makes the method valid.

∑ Linear Number Patterns

General term $T_n = a + (n-1)d$

Common difference $d = T_2 - T_1 = T_3 - T_2$

Check the difference between at least two pairs before trusting it. If it is not constant the pattern is not linear, and $T_n = a+(n-1)d$ does not apply.

ƒ Functions & Graphs

Function	Shape	Key features
$y = ax + q$	straight line	gradient $a$ · $y$-intercept $q$
$y = ax^{2} + q$	parabola	turning point $(0;\ q)$ · $a > 0$ opens up
$y = \dfrac{a}{x} + q$	hyperbola	asymptotes $x = 0$ and $y = q$
$y = ab^{x} + q$	exponential	asymptote $y = q$ · $y$-intercept $a + q$

Domain is the $x$-values the graph uses; range is the $y$-values it reaches. For the hyperbola and the exponential the asymptote is exactly the value the range excludes.

θ Trigonometry

In a right triangle $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$

$\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$

$\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

$\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin$	$0$	$\tfrac{1}{2}$	$\tfrac{\sqrt{2}}{2}$	$\tfrac{\sqrt{3}}{2}$	$1$
$\cos$	$1$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{\sqrt{2}}{2}$	$\tfrac{1}{2}$	$0$
$\tan$	$0$	$\tfrac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	undefined

SOH-CAH-TOA only works in a right-angled triangle. Without a right angle you need the sine or cosine rule, which arrive in Grade 11.

⊙ 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 $m = \dfrac{y_2-y_1}{x_2-x_1}$

Distance uses squares, so the order of the points never matters. Gradient does not — swap the points in the top and the bottom together, or the sign comes out wrong.

R Finance

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

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

$P$ the amount at the start

$i$ rate as a decimal ($8\% \rightarrow 0{,}08$)

$n$ number of periods

Interest compounded monthly at $12\%$ a year means $i = 0{,}01$ and $n$ counts months. Divide the rate and multiply the periods by the same number.

⬔ Measurement

Rectangular prism $V = l \times b \times h$

Any right prism $V = \text{base area} \times h$

Cylinder — volume $\pi r^{2} h$

Cylinder — surface area $2\pi r^{2} + 2\pi r h$

Circle $A = \pi r^{2},\quad C = 2\pi r$

Multiplying every dimension by $k$ multiplies the area by $k^{2}$ and the volume by $k^{3}$ — worth learning now, because Grade 11 examines it directly.

σ Statistics

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

Range $\text{max} - \text{min}$

Interquartile range $Q_3 - Q_1$

Five-number summary min · $Q_1$ · median · $Q_3$ · max

Order the data before finding a median or a quartile. Every mark lost on this topic is lost to an unsorted list.

P Probability

Probability $P(A) = \dfrac{\text{favourable}}{\text{total}}$

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

Relative frequency $\dfrac{\text{times it happened}}{\text{times you tried}}$

Subtracting $P(A \text{ and } B)$ is not optional — without it the overlap is counted twice, which is the single most common error in this section.

Grade 10 is where the gap opens, and where it is cheapest to close.

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