LGCSE Mathematics (0178) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Standard form

A number written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Used to express very large or very small values compactly.

Highest common factor

The largest positive integer dividing two or more numbers exactly. Found by multiplying the shared prime factors of the numbers.

Lowest common multiple

The smallest positive integer divisible by each of two or more numbers. Found by taking the highest power of every prime factor present.

Compound interest formula

Amount = P(1 + r/100)ⁿ, where P is the principal, r the rate per period and n the number of periods. Interest is added each period.

Upper and lower bounds

For a value rounded to a given unit, the lower bound is half a unit below and the upper bound half a unit above the stated value.

Quadratic formula

x = [−b ± √(b² − 4ac)]/2a solves ax² + bx + c = 0. The discriminant b² − 4ac determines whether two, one or no real roots exist.

Perpendicular gradient rule

Two lines are perpendicular exactly when the product of their gradients equals −1, so one gradient is the negative reciprocal of the other.

Completing the square

Rewriting ax² + bx + c as a(x + p)² + q. This reveals the turning point (−p, q) and the minimum or maximum value of the expression.

Inverse function

The function f⁻¹ that reverses f, so f⁻¹(f(x)) = x. Its graph is the reflection of the graph of f in the line y = x.

Composite function

The function fg(x) means apply g first, then apply f to the result. Order matters: fg(x) is generally not equal to gf(x).

Simultaneous equations

Two or more equations solved together to find values satisfying all of them. Eliminate one variable by combining scaled equations.

Alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment, that is, the angle subtended by the chord on the opposite arc.

Angle in a semicircle

Any angle subtended by a diameter at a point on the circumference is a right angle, measuring exactly 90 degrees.

Cosine rule

a² = b² + c² − 2bc cos A relates the three sides of a triangle to one angle. Use it when two sides and the included angle are known.

Sine rule

a/sin A = b/sin B = c/sin C links sides and opposite angles in any triangle. Use it with a matching side–angle pair.

Exterior angle of polygon

For a regular polygon with n sides, each exterior angle equals 360°/n. The exterior angles of any convex polygon sum to 360°.

Triangle area using sine

Area = ½ab sin C, where a and b are two sides and C is the angle between them. Useful when the perpendicular height is unknown.

Volume of a prism

Volume = cross-sectional area × length. The cross-section must be uniform along the length; a cylinder is a prism with circular cross-section.

Arc length

Arc length = (θ/360) × 2πr, where θ is the central angle in degrees. It is the fraction of the circumference cut off by the angle.

Sector area

Area = (θ/360) × πr², where θ is the central angle in degrees. It is that fraction of the whole circle's area.

Cone slant height

For a right cone, slant height l = √(r² + h²), from Pythagoras. Curved surface area is πrl and volume is ⅓πr²h.

Similar solids scaling

If corresponding lengths of similar solids are in ratio k, surface areas are in ratio k² and volumes in ratio k³.

Density formula

Density = mass ÷ volume. Rearranged, mass = density × volume and volume = mass ÷ density, with consistent units throughout.

Distance between two points

d = √[(x₂ − x₁)² + (y₂ − y₁)²], obtained by applying Pythagoras to the horizontal and vertical differences between the points.

Point–gradient form

y − y₁ = m(x − x₁) is the equation of the line through point (x₁, y₁) with gradient m. Expand to reach y = mx + c.

Line of symmetry of parabola

The curve y = ax² + bx + c is symmetric about the vertical line x = −b/2a, which passes through its turning point.

Interquartile range

IQR = upper quartile − lower quartile. It measures the spread of the middle half of the data and ignores extreme values.

Estimated mean for grouped data

Multiply each class midpoint by its frequency, sum these products, then divide by the total frequency. This estimates the mean of grouped data.

Mutually exclusive events

Events that cannot occur together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).

Independent events

Events where one outcome does not affect the other, so P(A and B) = P(A) × P(B). Selection without replacement creates dependence.

Tree diagram rules

Multiply probabilities along a path of branches to find combined outcomes, then add the probabilities of paths giving the required result.

Cumulative frequency

A running total of frequencies. Plotting it against the upper class boundary allows the median and quartiles to be read from the curve.