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Northern IrelandMaths

A2 1 Pure Mathematics

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do the chain, product and quotient rules let us differentiate any combination of functions, including implicitly?

Differentiating exponential, logarithmic and trigonometric functions, the chain, product and quotient rules, implicit differentiation, and connected rates of change.
A CCEA A-Level Mathematics answer on differentiating exponential, logarithmic and trigonometric functions, the chain, product and quotient rules, implicit differentiation, and using the chain rule for connected rates of change.
13 min answer →

How do we describe, combine and invert functions, and split a rational expression into partial fractions?

Functions, domain and range, composite and inverse functions, the modulus function and modulus equations, and expressing a rational function as partial fractions.
A CCEA A-Level Mathematics answer on functions with their domain and range, composite and inverse functions, the modulus function and solving modulus equations, and expressing a rational function as partial fractions.
13 min answer →

How do substitution, integration by parts and partial fractions let us integrate harder functions and solve differential equations?

Integrating standard functions, integration by substitution and by parts, integrating with partial fractions, and forming and solving differential equations by separating the variables.
A CCEA A-Level Mathematics answer on integrating standard exponential, logarithmic and trigonometric functions, integration by substitution and by parts, integrating with partial fractions, and solving first-order differential equations by separating the variables.
13 min answer →

How do we locate roots and estimate areas when exact methods fail?

Locating roots by a sign change, iterative methods including the Newton-Raphson method and fixed-point iteration, and the trapezium rule for estimating a definite integral.
A CCEA A-Level Mathematics answer on locating roots by a change of sign, iterative methods including fixed-point iteration and the Newton-Raphson method, and using the trapezium rule to estimate the value of a definite integral.
12 min answer →

How do parametric equations describe a curve through a parameter, and how do we convert and differentiate them?

Curves defined parametrically, converting between parametric and Cartesian form, and finding the gradient of a parametric curve using the chain rule.
A CCEA A-Level Mathematics answer on curves defined parametrically, converting between parametric and Cartesian form by eliminating the parameter, and finding the gradient of a parametric curve using dy by dx equals dy by dt divided by dx by dt.
12 min answer →

How do we prove a mathematical statement is always true, or show it is false?

Methods of proof, including proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction.
A CCEA A-Level Mathematics answer on the methods of proof - direct proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction - with worked examples including the irrationality of root 2.
12 min answer →

How do arithmetic and geometric series sum, and how does the binomial expansion extend to any power?

Arithmetic and geometric sequences and series and their sums, the sum to infinity of a convergent geometric series, and the binomial expansion for any rational power with its validity condition.
A CCEA A-Level Mathematics answer on arithmetic and geometric sequences and series, the formulae for their sums, the sum to infinity of a convergent geometric series, and the binomial expansion for any rational power with its range of validity.
13 min answer →

How do radians, the compound and double-angle formulae and the harmonic form extend trigonometry?

Radian measure with arc length and sector area, the reciprocal and inverse trigonometric functions, the compound-angle and double-angle identities, and the form $R\sin(\theta \pm \alpha)$ for solving equations.
A CCEA A-Level Mathematics answer on radian measure with arc length and sector area, the reciprocal and inverse trigonometric functions, the compound-angle and double-angle identities, and writing a sin plus b cos in the form R sin of theta plus alpha to solve equations.
13 min answer →

How do vectors extend to three dimensions, and how do we measure distances and angles between them?

Three-dimensional vectors in component form, the magnitude and distance between points in space, the scalar (dot) product and the angle between two vectors, and the condition for perpendicular vectors.
A CCEA A-Level Mathematics answer on three-dimensional vectors in component form, the magnitude of a vector and the distance between points in space, the scalar product, the angle between two vectors, and the condition for two vectors to be perpendicular.
12 min answer →