Pure Mathematics 1

Spec coverage

Every spec point, and what actually teaches it

The real numbered content list from the official Pearson specification, each point matched to the lesson that covers it. Click through to a lesson from its spec point below, or rate how confident you feel — saved in this browser so it’s there next time you come back. Any point with no matching lesson is flagged, not hidden.

22 spec points · 22 covered

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1 — Algebra and functions

  • 1.1

    Laws of indices for all rational exponents

    • Laws of indices for all rational exponents.
    • Guidance: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. The equivalence of a^(m/n) and ⁿ√aᵐ should be known.
    The hidden quadratic — substitution from an indices/exponential equationDifferentiating Negative and Fractional Indices, Then Finding an Exact GradientPaper Anatomy

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  • 1.2

    Use and manipulation of surds

    • Use and manipulation of surds.
    • Guidance: Students should be able to rationalise denominators.
    Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

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  • 1.3

    Quadratic functions and their graphs

    • Quadratic functions and their graphs.
    Quadratic Functions, the Discriminant, and Solving Quadratic Equations

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  • 1.4

    The discriminant of a quadratic function

    • The discriminant of a quadratic function.
    • Guidance: Need to know and to use b² − 4ac > 0, b² − 4ac = 0 and b² − 4ac < 0.
    The Discriminant with a Parameter — "No Real Roots" StyleQuadratic Functions, the Discriminant, and Solving Quadratic EquationsSolving simultaneous equations by substitution

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  • 1.5

    Completing the square; solution of quadratic equations

    • Completing the square. Solution of quadratic equations.
    • Guidance: Solution of quadratic equations by factorisation, use of the formula, use of a calculator and completing the square. ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)).
    The hidden quadratic — substitution from an indices/exponential equationQuadratic Functions, the Discriminant, and Solving Quadratic Equations

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  • 1.6

    Simultaneous equations by substitution

    • Solve simultaneous equations; analytical solution by substitution.
    Solving simultaneous equations by substitution

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  • 1.7

    Interpreting linear and quadratic inequalities graphically

    • Interpret linear and quadratic inequalities graphically.
    • Guidance: For example, ax + b > cx + d, px² + qx + r ≥ 0, px² + qx + r < ax + b. Interpreting the third inequality as the range of x for which the curve y = px² + qx + r is below the line with equation y = ax + b. Including inequalities with brackets and fractions. These would be reducible to linear or quadratic inequalities, e.g. a/x < b becomes ax < bx², x ≠ 0.
    Quadratic inequalities, interpreted and represented graphically

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  • 1.8

    Representing linear and quadratic inequalities graphically

    • Represent linear and quadratic inequalities graphically.
    • Guidance: Represent linear and quadratic inequalities such as y > x + r and y > ax² + bx + c graphically. Shading and use of dotted and solid line convention is required.
    Quadratic inequalities, interpreted and represented graphically

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  • 1.9

    Solutions of linear and quadratic inequalities

    • Solutions of linear and quadratic inequalities.
    • Guidance: For example, solving ax + b > cx + d, px² + qx + r ≥ 0, px² + qx + r < ax + b.
    Quadratic inequalities, interpreted and represented graphically

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  • 1.10

    Algebraic manipulation of polynomials

    • Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation.
    • Guidance: Students should be able to use brackets. Factorisation of polynomials of degree n, n ≤ 3, e.g. x³ + 4x² + 3x. The notation f(x) may be used.
    Sketching cubic graphs from factored form

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  • 1.11

    Graphs of functions; sketching curves and solving equations graphically

    • Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.
    • Guidance: Functions to include simple cubic functions and the reciprocal functions y = k/x and y = k/x² with x ≠ 0. Knowledge of the term asymptote is expected. Also, trigonometric graphs.
    Sketching cubic graphs from factored form

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  • 1.12

    Effect of simple transformations on a graph

    • Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
    • Guidance: Students should be able to apply one of these transformations to any of the above functions (quadratics, cubics, reciprocals, sine, cosine, and tangent) and sketch the resulting graphs. Given the graph of any function y = f(x), students should be able to sketch the graph resulting from one of these transformations.
    Graph Transformations Described in Words

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2 — Coordinate geometry in the (x, y) plane

3 — Trigonometry

  • 3.1

    The sine and cosine rules; area of a triangle

    • The sine and cosine rules, and the area of a triangle in the form ½ab sin C.
    • Guidance: Including the ambiguous case of the sine rule.
    The Sine Rule, the Cosine Rule, and the Ambiguous Case

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  • 3.2

    Radian measure, arc length and area of sector

    • Radian measure, including use for arc length and area of sector.
    • Guidance: Use of the formulae s = rθ and A = ½r²θ.
    Sector Area, Arc Length, and Composite Perimeter

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  • 3.3

    Sine, cosine and tangent functions and their graphs

    • Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.
    • Guidance: Knowledge of graphs of curves with equations such as y = 3 sin x, y = sin(x + π/6), y = sin 2x is expected.
    Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

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4 — Differentiation

5 — Integration