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Grade 12 Mathematics: Key Formulas

The formulas most often needed in South African NSC Mathematics: algebra, sequences, finance, calculus, trigonometry, geometry and probability.

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Quadratic formula

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Quadratic formula

For

a x^{2} + b x + c = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Nature of the roots from the discriminant

Δ = b^{2} - 4 a c

Δ > 0

: two real, unequal roots;

Δ = 0

: equal roots;

Δ < 0

: no real roots.

Sum and product of the roots

x_{1} + x_{2} = - \frac{b}{a}

and

x_{1} x_{2} = \frac{c}{a}

General term of an arithmetic sequence

T_{n} = a + (n - 1) d

Sum of an arithmetic series

S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)

General term of a geometric sequence

T_{n} = a r^{n - 1}

Sum of a geometric series

S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}

r \neq = 1

Sum to infinity of a geometric series

S_{\infty} = \frac{a}{1 - r}

, valid when

- 1 < r < 1

Compound interest

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

Simple interest

A = P (1 + in)

Reducing-balance depreciation

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

Effective interest rate from a nominal rate

i^{(m)}

i_{e f f} = (1 + \frac{i ^{(m)}}{m})^{m} - 1

Derivative of

x^{n}

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Derivative from first principles

f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

Gradient of the tangent at

x = a

m = f^{'} (a)

; the tangent is

y - f (a) = f^{'} (a) (x - a)

Fundamental trigonometric identities

sin^{2} θ + cos^{2} θ = 1

and

tan θ = \frac{s i n θ}{c o s θ}

Compound angle:

sin (A + B)

sin A cos B + cos A sin B

Compound angle:

cos (A - B)

cos A cos B + sin A sin B

Double angle:

sin 2 θ

2 sin θ cos θ

Double angle:

cos 2 θ

cos^{2} θ - sin^{2} θ = 1 - 2 sin^{2} θ = 2 cos^{2} θ - 1

Sine rule

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}

Cosine rule

a^{2} = b^{2} + c^{2} - 2 b c cos A

Area rule for a triangle

Area = \frac{1}{2} ab sin C

Distance between two points

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Midpoint of a line segment

M (\frac{x _{1} + x _{2}}{2}; \frac{y _{1} + y _{2}}{2})

Gradient of a line through two points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Equation of a circle with centre

(a; b)

and radius

r

(x - a)^{2} + (y - b)^{2} = r^{2}

Gradients of perpendicular lines

m_{1} \times m_{2} = - 1

Probability of A or B

P (A or B) = P (A) + P (B) - P (A and B)

Probability of independent events A and B

P (A and B) = P (A) \times P (B)

Number of arrangements of n different objects

n!

Population standard deviation

σ = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}

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