Unit P1: Pure Mathematics 1
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P1.1 - Algebra and functions
P1.1.1Laws of indices
Laws of indices for all rational am × an = am + n, am ÷ an = am − n, (am)n = amn exponents. m The equivalence of a n and n am should be known.
P1.1.2Use and manipulate surds
Use and manipulation of surds.; Students should be able to rationalise denominators.
P1.1.3Quadratic functions and their graphs
Quadratic functions and their graphs.
P1.1.4Discriminant of a quadratic
The discriminant of a quadratic function.; Know and use b² − 4ac > 0, b² − 4ac = 0 and b² − 4ac < 0.
P1.1.5Completing the square
Completing the square and solving quadratic equations.; Solve quadratic equations by factorisation, formula, calculator methods and completing the square.
P1.1.6Solve simultaneous equations
Solve simultaneous equations; analytical solution by substitution.
P1.1.7Interpreting linear and quadratic inequalities
Interpret linear and quadratic inequalities graphically and algebraically, including inequalities with brackets or fractions reducible to linear or quadratic form; identify solution ranges from intersections of curves and lines.
P1.1.8Graphing linear and quadratic inequalities
Represent linear and quadratic inequalities graphically, such as y > x + r and y > ax² + bx + c.; Use shading and dotted/solid line conventions.
P1.1.9Solving linear and quadratic inequalities
Solve linear and quadratic inequalities, including comparisons between a quadratic expression and a linear expression.
P1.1.10Algebraic manipulation
Algebraic manipulation of polynomials.; Expand brackets, collect like terms and factorise polynomials of degree n, n <= 3.; The notation f(x) may be used.
P1.1.11Graphs of functions
Graphs of functions; sketching Functions to include simple cubic functions and the curves defined by simple equations. reciprocal functions Geometrical interpretation of k k algebraic solution of equations.; Use y = and y = with x ≠ 0. of intersection points of graphs of x x2 functions to solve equations.; Knowledge of the term asymptote is expected.; Also, trigonometric graphs.
P1.1.12Effect of simple
Knowledge of the effect of simple Students should be able to apply one of these transformations on the graph of transformations to any of the above functions (quadratics, cubics, reciprocals, sine, cosine, and tangent) and sketch the y = f(x) as represented by y = af(x), resulting graphs. y = f(x) + a, y = f(x + a), y = f(ax).; Given the graph of any function y = f(x), students should be able to sketch the graph resulting from one of these transformations.
P1.2 - Coordinate geometry in the (x, y) plane
P1.2.1Equation of a straight line
Equation of a straight line, To include: including the forms (i) the equation of a line through two given points y − y = m(x − x) and 1 1 (ii) the equation of a line parallel (or perpendicular) to a ax + by + c = 0. given line through a given point.; For example, the line perpendicular to the line 3x + 4y = 18 through the point (2, 3) has equation y − 3 = 4 (x − 2).
P1.2.2Parallel and perpendicular lines
Conditions for two straight lines to be parallel or perpendicular to each other.
P1.3 - Trigonometry
P1.3.1Sine rule, cosine rule and triangle area
The sine and cosine rules, and the Including the ambiguous case of the sine rule. area of a triangle in the form 1 ab sin C.
P1.3.2Radian measure,
Radian measure, including use for Use of the formulae s = rθ and A = 1 r2θ. arc length and area of sector.
P1.3.3Trigonometric functions and graphs
Understand sine, cosine and tangent functions, including their graphs, symmetries and periodicity; sketch transformations such as y = 3 sin x, y = sin(x + π/6) and y = sin 2x.
P1.4 - Differentiation
P1.4.1Derivative as gradient and rate of change
The derivative of f(x) as the dy For example, knowledge that is the rate of change of y gradient of the tangent to the graph dx of y = f(x) at a point; the gradient of with respect to x.; Knowledge of the chain rule is not the tangent as a limit; interpretation required. as a rate of change; second order derivatives.; The notation f ′(x) and f ′′(x) may be used.
P1.4.2Differentiating powers of x
Differentiation of xn, and related The ability to differentiate expressions such as sums, differences and constant x2 +5x −3 (2x + 5)(x − 1) and is expected. multiples. 3 x.
P1.4.3Tangents, normals and gradients
Applications of differentiation to Use of differentiation to find equations of tangents and gradients, tangents and normals. normals at specific points on a curve.
P1.5 - Integration
P1.5.1Indefinite integration as the reverse
Indefinite integration as the reverse Students should know that a constant of integration is of differentiation. required.
P1.5.2Integration of powers of x and related sums
Integration of xn and related sums, (Excluding n = −1 and related sums, differences and differences and constant multiples. multiples).; For example, the ability to integrate expressions such as 1 x2 −3x −1 2 and (x + 2)2 is expected. x Given f ′(x) and a point on the curve, students should be able to find an equation of the curve in the form y = f(x).