Secondary 6 Mathematics Extended Part

A technique for integrals of products of two function types, using ∫u dv = uv − ∫v du.

Substituting x = a sin θ and the like to remove surds such as √(a²−x²) from an integrand.

The theorem that a definite integral equals the antiderivative at the upper limit minus its value at the lower limit.

An integral with limits whose value is a number, geometrically the signed area under a curve.

The variable of a definite integral, freely renamable since the value does not depend on it.

A function with f(−x) = −f(x), integrating to zero over a symmetric interval [−a, a].

A function with f(−x) = f(x), integrating over a symmetric interval to twice the half-interval.

Slicing a solid of revolution into thin discs, giving V = π∫[f(x)]² dx.

The solid swept out when a plane curve revolves about a coordinate axis or a line parallel to one.

The prescribed range of an inverse trigonometric function, e.g. sin⁻¹x lies in [−90°, 90°].

A rectangular array of numbers arranged in rows and columns, operated on as a whole.

A matrix with equal numbers of rows and columns; only square matrices have determinants and inverses.

The square matrix I with ones on the main diagonal and zeros elsewhere, the identity for multiplication.

The single number associated with a square matrix, written |A| or det A; zero means the matrix is singular.

A square matrix with zero determinant, hence no inverse; the corresponding system has no or infinitely many solutions.

A matrix B with AB = BA = I, written A⁻¹; unique when it exists, and A must be non-singular.

The rule solving a linear system as ratios of determinants, each obtained by replacing one column of the coefficient matrix.

Applying row operations to the augmented matrix until it is in reduced echelon form, then reading off the solution.

A system AX = 0 with all constants zero, always possessing the all-zero trivial solution.

A solution of a homogeneous system not all zero, existing if and only if the coefficient matrix is singular.

A quantity with both magnitude and direction, such as displacement, force or velocity.

A quantity with size only and no direction, such as temperature, mass or time.

The length of a vector, written |a|, the square root of the sum of squared components.

A vector of magnitude one; dividing a non-zero vector by its magnitude gives the unit vector of the same direction.

The vector from the origin to a point P, with components equal to the coordinates of P.

a·b = |a||b|cos θ = the sum of products of corresponding components; the result is a number.

In three dimensions, a×b is a vector perpendicular to both, with magnitude |a||b|sin θ.

The signed length (scalar projection) or the component vector (vector projection) of one vector along another.

Two non-zero vectors at right angles, exactly when their dot product is zero.

The formula locating a point dividing a segment in a given ratio; the midpoint formula is a special case.

A numerical method approximating a definite integral by trapezium areas over equal subintervals: T = (h2)[first + last + 2×(middle sum)].

Each of the small equal pieces of the integration interval; more pieces give better estimates.

The gap h = (b−a)/n between neighbouring points, the h in the trapezoidal formula.

An estimate above the true value; obtained by the trapezoidal rule when the curve opens upward (positive second derivative).

An estimate below the true value; obtained when the curve opens downward (negative second derivative).

The bending direction of a graph, decided by the sign of the second derivative and used to classify trapezoidal estimates.

The theorem computing a definite integral as the antiderivative at the upper limit minus its value at the lower.

Introducing a new variable u to simplify a composite integrand; in definite integrals the limits change too.

Defined as the limit of sums of rectangular strips under a curve; its value is a number.

The freely renamable variable of a definite integral: x may become t without changing the value.

Questions requiring written working, as opposed to multiple choice; the entire M1 paper is of this type.

The official weighting and timing of the examination components; for M1, one conventional paper of two and a half hours.

Reporting grades against preset standards and boundary scores, on a scale of one to five.

Official descriptions of what a typical candidate at each grade can do, phrased positively.

Converting a normal variable to the standard normal Z = (X − μ)/σ — the compulsory step of normal questions.

The distribution of sample means over repeated sampling, with mean μ and variance σ²/n.

Estimating a population parameter with a single value — the closing topic of M1 statistics.

P(A|B) = P(A∩B)/P(B), the probability of A given that B has occurred.

The theorem relating conditional probabilities to each other, applied to simple problems in M1.

A timed full-paper practice run in examination format, tuning pace and stamina.

The official weighting and timing of the examination; for M2, one conventional paper of two and a half hours.

A logical derivation of a conclusion from the given, admitting neither circularity nor special cases in its place.

Proof by verifying the initial value then deducing n = k + 1 from n = k; both steps and the conclusion are required.

An equality holding for all admissible values; proved by transforming one side into the other.

The structure of a long question whose parts pave the way, earlier results feeding later ones.

The signed minor determinant attached to each entry, used in third-order determinants and inverses.

The transpose of the cofactor matrix; a third-order inverse is the adjoint divided by the determinant.

Drawing a function's graph from intercepts, turning points, inflexion points and asymptotes.

The abilities long questions target — synthesis, argument and multi-step reasoning.

A revision record of mistakes and their causes — conceptual, computational, presentational — for targeted improvement.