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EnglandMaths

Pure mathematics: advanced

7 dot points across 7 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you work with exponential and logarithmic functions, the number e, and use logarithms to solve equations and linearise models?

Exponential functions and their graphs, the number e and the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, and using logarithms to estimate parameters in exponential and power-law models.
A focused answer to the OCR A-Level Mathematics A exponentials and logarithms content, covering exponential functions and their graphs, the number e and its derivative, the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, and using log-linear and log-log graphs to estimate parameters in growth and power-law models.
12 min answer →

How do you work with functions as mappings, compose and invert them, and handle the modulus function in graphs and equations?

The language of functions (domain, range, composite and inverse functions), the modulus function and its graph, and solving modulus equations and inequalities.
A focused answer to the OCR A-Level Mathematics A functions content, covering domain and range, composite and inverse functions, the conditions for an inverse to exist, the modulus function and its graph, and solving modulus equations and inequalities both algebraically and graphically.
11 min answer →

How do you split a rational expression into partial fractions, and why does that make integration and binomial expansion easier?

Decomposing a proper algebraic fraction into partial fractions, including denominators with distinct linear factors and a repeated linear factor, and using partial fractions in integration and binomial expansion.
A focused answer to the OCR A-Level Mathematics A partial fractions content, covering decomposing a proper algebraic fraction with distinct linear factors and with a repeated linear factor, dealing with an improper fraction by dividing first, and using partial fractions to integrate and to expand with the binomial series.
10 min answer →

How do you measure angles in radians, and how do you find arc lengths, sector areas and small-angle approximations?

Radian measure, the relationship between radians and degrees, arc length and the area of a sector and segment, and the small-angle approximations for sine, cosine and tangent.
A focused answer to the OCR A-Level Mathematics A radian content, covering the definition of a radian, converting between radians and degrees, exact values in radians, arc length and sector and segment area, and the small-angle approximations for sine, cosine and tangent.
10 min answer →

How do you use the trigonometric identities, including the compound and double angle formulae and the R form, to prove results and solve equations?

The Pythagorean and quotient identities, the reciprocal functions, the compound and double angle formulae, the R form for a sin theta plus b cos theta, and solving trigonometric equations over an interval.
A focused answer to the OCR A-Level Mathematics A trigonometric identities content, covering the Pythagorean and quotient identities, the reciprocal and inverse functions, the compound and double angle formulae, expressing a sine plus b cosine in R form, and solving trigonometric equations over a given interval.
12 min answer →

How do you work with the three trigonometric functions, the sine and cosine rules, and exact values to model and solve triangle problems?

The sine, cosine and tangent functions and their graphs, the sine and cosine rules, the area of a triangle, and exact values of trigonometric ratios for standard angles.
A focused answer to the OCR A-Level Mathematics A trigonometry content, covering the sine, cosine and tangent functions and their graphs, the sine and cosine rules, the area of a triangle, the ambiguous case, and the exact values of trigonometric ratios for standard angles.
11 min answer →

How do you represent quantities with both magnitude and direction, and use vectors in two and three dimensions for geometry and motion?

Vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, position vectors and unit vectors, and geometric applications including collinearity and the midpoint.
A focused answer to the OCR A-Level Mathematics A vectors content, covering vectors in two and three dimensions, component and i, j, k notation, magnitude and direction, addition and scalar multiplication, position vectors, unit vectors, and geometric applications such as proving collinearity and finding a midpoint.
10 min answer →