Cameroon GCE A-Level Pure Mathematics with Statistics key terms

The quantity b² − 4ac for the quadratic ax² + bx + c. Its sign tells whether the roots are real and distinct, real and equal, or a complex conjugate pair.

Rewriting ax² + bx + c as a(x + p)² + q by adding and subtracting a constant. It reveals the vertex of the parabola and the range of the function.

A function formed by applying one function to the output of another, written fg(x) or f(g(x)). It exists only where the inner output lies in the outer domain.

The function f⁻¹ that reverses f, so f⁻¹(f(x)) = x. It exists as a function only when f is one-to-one on its stated domain.

The domain is the set of input values a function accepts; the range is the set of output values it actually produces. Both must be stated when a function is defined.

Dividing polynomial f(x) by (x − a) leaves remainder f(a); (x − a) is a factor exactly when f(a) = 0. The theorem also fixes unknown coefficients.

Splitting a rational expression into a sum of simpler fractions with linear or repeated linear denominators. The result integrates and expands more easily than the original.

A sequence in which each term differs from the previous one by a fixed constant d. The nth term is a + (n − 1)d and the sum of n terms is n/2[2a + (n − 1)d].

A sequence in which each term is a fixed multiple r of the previous one. The nth term is ar^(n−1) and the sum of n terms is a(1 − rⁿ)/(1 − r) for r ≠ 1.

The limiting value a/(1 − r) of a geometric series, valid only when the common ratio satisfies |r| < 1. Otherwise the series diverges and no finite sum exists.

The property that the partial sums of a series approach a finite limit as more terms are added. For geometric series this happens precisely when the common ratio has modulus below one.

Writing (a + b)ⁿ as a sum of terms using binomial coefficients, with general term C(n, r)a^(n−r)b^r. For non-positive or fractional n it gives an infinite series valid for small |x|.

Summing a series by writing each term as a difference of consecutive values of a related expression, so intermediate terms cancel and only a few survive.

The change in y divided by the change in x between two points on a line. It measures steepness and appears as m in y = mx + c.

Two lines meeting at right angles. Their gradients m₁ and m₂ satisfy m₁m₂ = −1, so each gradient is the negative reciprocal of the other.

The form (x − a)² + (y − b)² = r², describing all points at distance r from centre (a, b). Expanding it gives x² + y² + 2gx + 2fy + c = 0.

A pair x = f(t), y = g(t) describing a curve through a third variable t. Differentiation then uses dy/dx = (dy/dt) ÷ (dx/dt).

The angle measure defined so that a full turn is 2π. Arc length is rθ and sector area is ½r²θ when θ is in radians.

Identities such as sin(A ± B) = sinA cosB ± cosA sinB and the matching cosine and tangent forms. They expand trigonometric expressions involving sums or differences of angles.

Identities expressing trigonometric ratios of 2A through A, for example sin2A = 2sinA cosA and cos2A = cos²A − sin²A. They are used to simplify and to integrate.

For small θ in radians, sinθ ≈ θ, tanθ ≈ θ and cosθ ≈ 1 − θ²/2. These replace trigonometric terms by polynomials in limiting and modelling problems.

The complete set of angles satisfying an equation, obtained by adding whole periods to a principal value, for example x = 180n° + (−1)ⁿα for sinx = sinα.

The rule dy/dx = dy/du × du/dx for differentiating a function of a function. It underlies related rates, parametric differentiation and integration by substitution.

Rules for differentiating products and quotients: (uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v². Both are needed for rational and mixed trigonometric expressions.

Differentiating an equation relating x and y term by term, treating y as a function of x and applying the chain rule, then solving for dy/dx.

The line through a point on a curve perpendicular to the tangent there. Its gradient is the negative reciprocal of the curve's gradient at that point.

A point where the first derivative is zero. The second derivative decides whether it is a maximum, a minimum, or a point of inflection.

Replacing a variable in an integral by a new one so the integrand becomes standard. Limits must be converted when the integral is definite.

The limit of a sum of areas, written ∫ from a to b of f(x) dx, evaluated as F(b) − F(a) for an antiderivative F. It gives signed area under a curve.

The rule ∫u dv = uv − ∫v du, used for products such as x·eˣ or x·cosx. Choice of u follows the usual priority of functions.

The expansion f(x) = f(0) + xf′(0) + (x²/2!)f″(0) + …, giving polynomial approximations such as eˣ, sinx and ln(1 + x).

The number a·b = |a||b|cosθ formed from two vectors. It tests perpendicularity, since a·b = 0 exactly when the vectors meet at right angles.

The form r = a + tb, where a is a fixed position vector, b a direction vector and t a scalar parameter. It describes every point on the line.

A number computed from a square matrix; for a 2×2 matrix with entries a, b, c, d it is ad − bc. A zero determinant means the matrix has no inverse.

The matrix A⁻¹ satisfying AA⁻¹ = I. For 2×2 matrices it is (1/det A) times the matrix with a and d swapped and b and c negated.

A coefficient matrix written alongside the constant column and reduced by row operations to solve simultaneous linear equations.

For a complex number z = x + iy, the modulus |z| is √(x² + y²) and the argument is the angle its vector makes with the positive real axis, usually taken in (−π, π].

Writing a complex number as r(cosθ + isinθ), where r is the modulus and θ the argument. It makes multiplication, division and powers straightforward.

The result that [r(cosθ + isinθ)]ⁿ = rⁿ(cos nθ + isin nθ). It gives powers and roots of complex numbers and derives trigonometric identities.

Two roots of the form a ± ib that occur together when a polynomial with real coefficients has non-real roots. Their sum 2a is real and their product a² + b² is real.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. It underlies tree diagrams and Bayes-type reasoning.

Events whose occurrence does not affect each other's probability, so P(A ∩ B) = P(A)P(B). Equivalently P(A|B) = P(A).

Events that cannot happen together, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B). They are not independent unless one has probability zero.

The distribution of the number of successes in n independent trials each with the same success probability p. Its mean is np and variance np(1 − p).

A model for counts of rare events in a fixed interval, with P(X = r) = e^(−λ)λʳ/r!. Its mean and variance both equal λ.

A symmetric bell-shaped continuous distribution fixed by mean μ and variance σ². Standardising with z = (x − μ)/σ lets tables of the standard normal be used.

The variable Z with mean 0 and variance 1 obtained by z = (x − μ)/σ. Its cumulative probabilities are read from printed tables.

The adjustment of a discrete value by 0.5 when a discrete distribution is approximated by a continuous one, for example P(X ≥ 7) becomes P(Y > 6.5).

Three measures of central tendency: the arithmetic average, the middle value of ordered data, and the most frequent value. Each suits different data types and shapes.

The square root of the variance, measuring the typical spread of values about the mean in the original units. A small value means values cluster tightly.

A graph of running totals against upper class boundaries. Quartiles and the median are read off at 25%, 50% and 75% of the total frequency.

A bar chart for continuous data in which bar area, not height, represents frequency. Heights are therefore frequency densities when class widths differ.

Data transformed by y = (x − a)/b to simplify calculation. The mean and standard deviation transform by the same rule, while the variance scales by b².

A statistic whose expected value equals the population parameter it estimates. The sample mean is unbiased for μ, while the sample variance uses divisor n − 1.

The result that sample means from a large sample are approximately normally distributed about the population mean, with standard error σ/√n, whatever the population shape.

A range of plausible values for a population parameter, built as estimate ± z × standard error. The stated percentage gives the long-run capture rate.

The null hypothesis H₀ states no effect or no change and is assumed true during the test; the alternative H₁ states the claim being tested, one- or two-sided.

The probability, chosen before testing, of rejecting a true null hypothesis, commonly 5% or 1%. It fixes the critical region against which the test statistic is compared.

A Type I error rejects a true null hypothesis; a Type II error fails to reject a false one. Lowering the significance level reduces Type I risk but raises Type II risk.