Maths

What is the quadratic formula?

x = (-b ± √(b² - 4ac)) / 2a, for an equation in the form ax² + bx + c = 0.

What is the discriminant of a quadratic?

b² - 4ac.

What does a positive discriminant mean?

The quadratic has two distinct real roots.

What does a discriminant of zero mean?

The quadratic has one repeated real root.

What does a negative discriminant mean?

The quadratic has no real roots.

How do you expand two brackets such as (x + a)(x + b)?

Multiply every term in the first bracket by every term in the second, then collect like terms.

How do you factorise x² + bx + c?

Find two numbers that multiply to c and add to b, then write the expression as two brackets.

What does completing the square do?

It rewrites a quadratic in the form (x + a)² + b, which can help find turning points and solve equations.

How do you solve simultaneous equations by elimination?

Multiply equations if necessary so one variable has matching coefficients, add or subtract the equations to eliminate it, then solve and substitute back.

What is an inequality?

A statement comparing values using symbols such as

What happens to an inequality sign when multiplying or dividing by a negative number?

The inequality sign reverses direction.

What is a function?

A rule that maps each input to an output.

What does f(x) mean?

The value of the function f when the input is x.

What is an inverse function?

A function that reverses the effect of another function.

What is a composite function?

A function formed by applying one function and then another, for example fg(x) = f(g(x)).

What is the nth term of an arithmetic sequence?

A formula that gives the value of any term. For a linear sequence it has the form dn + c.

How do you find the common difference of a linear sequence?

Subtract one term from the next.

What is a geometric sequence?

A sequence where each term is found by multiplying the previous term by the same value.

What is the equation of a straight line?

y = mx + c, where m is the gradient and c is the y-intercept.

How do you calculate gradient between two points?

(y₂ - y₁) / (x₂ - x₁).

What is true about the gradients of parallel lines?

Parallel lines have the same gradient.

What is true about gradients of perpendicular lines?

Their gradients multiply to -1, so each is the negative reciprocal of the other.

What does y = x² look like?

A U-shaped parabola with its minimum at the origin.

What does y = x³ look like?

A cubic curve passing through the origin with an S-like shape.

What does y = 1/x look like?

A reciprocal graph with two separate branches that approach but do not touch the axes.

What does a graph transformation y = f(x) + a do?

Translates the graph vertically by a units.

What does y = f(x + a) do to a graph?

Translates the graph horizontally by a units to the left.

What does y = af(x) do to a graph?

Stretches the graph vertically by scale factor a.

What does y = f(ax) do to a graph?

Stretches the graph horizontally by scale factor 1/a.

What is Pythagoras' theorem?

a² + b² = c² for a right-angled triangle, where c is the hypotenuse.

What does SOHCAHTOA stand for?

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

What is the sine rule?

a/sin A = b/sin B = c/sin C.

What is the cosine rule for finding a side?

a² = b² + c² - 2bc cos A.

What is the cosine rule for finding an angle?

cos A = (b² + c² - a²) / 2bc.

What is the formula for the area of a triangle using sine?

Area = ½ab sin C.

What is the sum of angles in a triangle?

180°.

What is the sum of interior angles of an n-sided polygon?

(n - 2) × 180°.

How do you find one exterior angle of a regular polygon?

360° divided by the number of sides.

What do exterior angles of any polygon sum to?

360°.

What is the area of a triangle?

½ × base × perpendicular height.

What is the area of a parallelogram?

base × perpendicular height.

What is the area of a trapezium?

½(a + b)h, where a and b are the parallel sides.

What is the area of a circle?

πr².

What is the circumference of a circle?

2πr or πd.

What is the area of a sector?

(angle/360) × πr².

What is the length of an arc?

(angle/360) × 2πr.

What is the volume of a prism?

Area of cross-section × length.

What is the volume of a cylinder?

πr²h.

What is the volume of a pyramid?

⅓ × area of base × perpendicular height.

What is the volume of a cone?

⅓πr²h.

What is the volume of a sphere?

(4/3)πr³.

What is the surface area of a sphere?

4πr².

What does congruent mean?

Exactly the same shape and size.

What does similar mean?

The same shape with corresponding lengths in the same ratio.

How do lengths change with a linear scale factor k?

Lengths are multiplied by k.

How do areas change with a linear scale factor k?

Areas are multiplied by k².

How do volumes change with a linear scale factor k?

Volumes are multiplied by k³.

What is a vector?

A quantity with both magnitude and direction.

How are vectors usually written in GCSE Maths?

As column vectors showing horizontal and vertical movement.

How do you add vectors?

Add their corresponding components.

How do you subtract vectors?

Subtract their corresponding components.

What is a bearing?

An angle measured clockwise from north and written using three digits.

What is standard form?

a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

What is the index law for multiplying the same base?

aᵐ × aⁿ = aᵐ⁺ⁿ.

What is the index law for dividing the same base?

aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

What is the index law for raising a power to another power?

(aᵐ)ⁿ = aᵐⁿ.

What does a⁰ equal?

1, provided a ≠ 0.

What does a negative power mean?

a⁻ⁿ = 1/aⁿ.

What does a fractional power of 1/n mean?

a^(1/n) = ⁿ√a.

What does a^(m/n) mean?

Take the nth root of a and raise the result to the power m.

What is a surd?

An irrational root left in exact form, such as √2.

How do you simplify √a × √b?

√(ab).

What does rationalising the denominator mean?

Rewriting a fraction so there is no surd in its denominator.

What is direct proportion?

Two quantities vary so that one is a constant multiple of the other, for example y = kx.

What is inverse proportion?

As one quantity increases the other decreases according to a constant relationship, for example y = k/x.

What does k represent in proportion equations?

The constant of proportionality.

How do you calculate percentage change?

(change ÷ original amount) × 100.

What multiplier represents a percentage increase?

1 + the percentage written as a decimal.

What multiplier represents a percentage decrease?

1 - the percentage written as a decimal.

How do you calculate repeated percentage change?

Multiply by the appropriate percentage multiplier once for each time period.

What is simple interest?

Interest calculated only on the original amount.

What is compound interest?

Interest calculated on the changing balance, so previous interest also earns interest.

What is probability written as?

A value between 0 and 1, where 0 means impossible and 1 means certain.

What do probabilities of all mutually exclusive possible outcomes sum to?

1.

What does mutually exclusive mean?

Two events cannot happen at the same time.

What does independent mean in probability?

The outcome of one event does not affect the probability of another.

How do you calculate the probability of two independent events both occurring?

Multiply their probabilities.

When can probabilities be added?

For mutually exclusive alternative outcomes.

What is relative frequency?

Number of times an event occurs ÷ total number of trials.

What is expected frequency?

Probability of the event × number of trials.

What is a sample space diagram?

A diagram or table showing all possible outcomes of an experiment.

What is a tree diagram used for?

Representing sequences of events and their probabilities.

What is the mean?

Sum of all values ÷ number of values.

What is the median?

The middle value when the data is arranged in order.

What is the mode?

The most frequently occurring value.

What is the range?

Highest value - lowest value.

How do you estimate the mean from a grouped frequency table?

Use each class midpoint × frequency, add the results, then divide by total frequency.

What is cumulative frequency?

A running total of frequencies up to each value or class.

What does the interquartile range measure?

The spread of the middle 50% of the data: upper quartile - lower quartile.

What does correlation mean?

A relationship or association between two variables.