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TMUA AS Pure Maths Formulas and Rules

Thirty flashcards of the algebra, coordinate geometry, trigonometry, sequences, logarithms and calculus rules needed for TMUA-style Paper 1 practice, since no formula booklet is allowed. Study material written for practice.

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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}

Discriminant

b^{2} - 4 a c

: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Completing the square

x^{2} + b x + c = (x + \frac{b}{2})^{2} + c - \frac{b ^{2}}{4}

. The vertex is at

x = - \frac{b}{2}

Sum and product of roots

For

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

with roots

α, β

α + β = - b

and

α β = c

. Then

α^{2} + β^{2} = (α + β)^{2} - 2 α β

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

Factor and remainder theorem

When

f (x)

is divided by

(x - a)

the remainder is

f (a)

(x - a)

is a factor exactly when

f (a) = 0

Laws of indices

a^{m} a^{n} = a^{m + n}

\frac{a ^{m}}{a ^{n}} = a^{m - n}

(a^{m})^{n} = a^{mn}

a^{- n} = \frac{1}{a ^{n}}

a^{1/ n} = n a

Laws of logarithms

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y

lo g_{a} x^{k} = k lo g_{a} x

Change of base

lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}

. Also

lo g_{a} b = \frac{1}{lo g _{b} a}

Logarithm and exponent

lo g_{a} x = y

means

a^{y} = x

lo g_{a} 1 = 0

and

lo g_{a} a = 1

Binomial expansion

(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}

, where

(r n) = \frac{n !}{r ! ( n - r )!}

Arithmetic series

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

and

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

Geometric series

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

and

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

. For

∣ r ∣ < 1

the sum to infinity is

\frac{a}{1 - r}

Gradient, midpoint and distance

Gradient

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

; midpoint

= (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

; distance

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

Parallel and perpendicular lines

Parallel lines have equal gradients. Perpendicular gradients multiply to

- 1

Equation of a circle

Centre

(a, b)

, radius

r

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

. A tangent is perpendicular to the radius at the point of contact.

Trigonometric identities

sin^{2} θ + cos^{2} θ = 1

and

tan θ = \frac{sin θ}{cos θ}

Exact trigonometric values

sin 3 0^{\circ} = \frac{1}{2}

cos 3 0^{\circ} = \frac{3}{2}

tan 3 0^{\circ} = \frac{1}{3}

;

sin 4 5^{\circ} = cos 4 5^{\circ} = \frac{2}{2}

tan 4 5^{\circ} = 1

;

sin 6 0^{\circ} = \frac{3}{2}

cos 6 0^{\circ} = \frac{1}{2}

tan 6 0^{\circ} = 3

Sine rule and cosine rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

and

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

Area of a triangle

Area

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

where

C

is the angle between sides

a

and

b

Arc length and sector area

With

θ

in radians: arc length

= r θ

and sector area

= \frac{1}{2} r^{2} θ

π

radians

= 18 0^{\circ}

Differentiation of powers

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

. The gradient of a curve at a point is the value of

\frac{d y}{d x}

there.

Tangent and normal

Tangent at

(a, f (a))

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

. The normal has gradient

- \frac{1}{f ^{'} ( a )}

(when

f^{'} (a) \neq = 0

Stationary points

Where

f^{'} (x) = 0

. If

f^{''} (x) > 0

it is a minimum, if

f^{''} (x) < 0

a maximum; if

f^{''} (x) = 0

further checking is needed (it may be an inflexion).

Integration of powers

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

for

n \neq = - 1

. Definite integrals:

\int_{a}^{b} f (x) d x = F (b) - F (a)

Area between curves

f (x) \geq g (x)

[a, b]

the area between them is

\int_{a}^{b} [f (x) - g (x)] d x

. Areas below the axis count as positive area, so split the integral at the roots.

Exponential and natural logarithm

e^{l n x} = x

and

ln (e^{x}) = x

\frac{d}{d x} e^{x} = e^{x}

and

\frac{d}{d x} ln x = \frac{1}{x}

Modulus inequalities

∣ x ∣ < a

means

- a < x < a

;

∣ x ∣ > a

means

x < - a

x > a

. Equations such as

∣ f (x) ∣ = g (x)

need case checks.

Surds

ab = a b

for non-negative

a, b

. Rationalise

\frac{1}{p + q}

by multiplying top and bottom by

p - q

Inequalities with quadratics

Factorise and sketch. For an upward parabola with roots

α < β

f (x) < 0

between the roots and

f (x) > 0

outside them.

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