Cameroon GCE A-Level Pure Mathematics with Statistics flashcards

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

Remainder Theorem

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). For divisor (ax − b) the remainder is f(b/a).

Factor Theorem

(x − a) is a factor of f(x) exactly when f(a) = 0. Used to find unknown coefficients and to factorise cubics fully.

Inverse Function Condition

A function has an inverse only if it is one-to-one. Restrict the domain first, then reflect the graph in the line y = x to obtain the inverse graph.

Repeated Linear Partial Fraction

A denominator factor (x − a)² requires two partial fractions: A/(x − a) plus B/(x − a)². Solve for A and B by substituting convenient values.

Sum of an Arithmetic Progression

Sₙ = (n/2)[2a + (n − 1)d], equivalently (n/2)(first term + last term). The nth term is a + (n − 1)d.

Sum to Infinity of a GP

Provided |r| < 1, the sum to infinity of a geometric progression is a/(1 − r). If |r| ≥ 1 the series diverges and no finite sum exists.

Binomial Expansion General Term

For positive integer n, the term containing bʳ in (a + b)ⁿ is C(n,r)aⁿ⁻ʳbʳ. Setting this equal to a target power finds unknown coefficients.

Method of Differences

Split the general term into partial fractions so that consecutive terms cancel, leaving only the first and last surviving terms of the sum.

Perpendicular Gradient Rule

Two straight lines are perpendicular exactly when the product of their gradients equals −1, that is m₁m₂ = −1. Parallel lines have equal gradients.

Circle Equation Complete Square

x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c), found by completing the square.

Tangent Radius Property

A tangent to a circle is perpendicular to the radius drawn to the point of contact. This gives the tangent gradient from the radius gradient.

Line Curve Discriminant Test

Substituting a line into a curve gives a quadratic. Its discriminant b² − 4ac is positive for two intersections, zero for a tangent, negative for none.

Double Angle Formulas

sin2A = 2sinA cosA and cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A. These convert products to sums in integration.

R-Form Transformation

a sinθ + b cosθ equals R sin(θ + α) where R = √(a² + b²) and tanα = b/a. The maximum value of the expression is R.

Cosine Rule

In any triangle, a² = b² + c² − 2bc cosA. Rearranged, cosA = (b² + c² − a²)/(2bc), used when three sides or two sides and the included angle are known.

Chain Rule

dy/dx = dy/du × du/dx. For parametric curves, dy/dx = (dy/dt) divided by (dx/dt), evaluated at the required parameter value.

Quotient Rule

For y = u/v, dy/dx = (u′v − uv′)/v². The numerator order matters: the derivative of the numerator comes first.

Second Derivative Test

At a stationary point where dy/dx = 0, a negative second derivative indicates a maximum and a positive second derivative a minimum.

Volume of Revolution

Rotating y = f(x) about the x-axis through 360° gives volume π∫y² dx between the stated limits. About the y-axis use π∫x² dy.

Scalar Product

a·b = |a||b|cosθ = x₁x₂ + y₁y₂ + z₁z₂. The dot product is zero exactly when the two vectors are perpendicular.

Inverse of a 2×2 Matrix

For M = [[a, b], [c, d]], the inverse is (1/(ad − bc))[[d, −b], [−c, a]]. It exists only when the determinant ad − bc is nonzero.

Singular Matrix Meaning

A matrix with zero determinant is singular and has no inverse. In a linear system Ax = b it signals no unique solution exists.

De Moivre's Theorem

For z = r(cosθ + i sinθ), zⁿ = rⁿ(cos nθ + i sin nθ). It is the standard tool for powers and roots of complex numbers.

Modulus and Argument

For z = a + bi, the modulus is √(a² + b²) and the argument is arctan(b/a), adjusted by quadrant. Argument is measured anticlockwise from the positive real axis.

nth Roots of Unity

Solutions of zⁿ = 1 lie on the unit circle at arguments 2πk/n for k = 0 to n − 1. They form a regular polygon and sum to zero.

Circle Locus in Argand Plane

The locus |z − a| = r is a circle with centre a and radius r. The locus |z − a| = |z − b| is the perpendicular bisector of ab.

Binomial Distribution Conditions

X ~ B(n, p) requires a fixed number n of independent trials, two outcomes per trial, and constant probability p. Mean np, variance np(1 − p).

Poisson Distribution Formula

X ~ Po(λ) has P(X = r) = e^{−λ}λʳ/r!. Mean and variance both equal λ. It models rare random events in time or space.

Bayes' Theorem

P(A|B) = P(B|A)P(A)/P(B). The denominator is often expanded as total probability over a partition of the sample space.

Standardising a Normal Variable

If X ~ N(μ, σ²), then z = (x − μ)/σ follows N(0, 1). Tables of the standard normal then give required probabilities.

Coding in Statistics

With coding y = (x − a)/b, the mean recovers as x̄ = a + bȳ and standard deviation as sₓ = |b|s_y. Adding a does not change spread.

Frequency Density

For histograms with unequal class widths, frequency density equals frequency divided by class width. The area of each bar equals the class frequency.

Confidence Interval for a Mean

A 95% interval for μ is x̄ ± 1.96(σ/√n) when σ is known. For 99% replace 1.96 with 2.576. Increasing n narrows the interval.

Type I and Type II Errors

Type I error rejects a true null hypothesis, with probability equal to the significance level α. Type II error fails to reject a false null hypothesis.