Unit P1: Pure Mathematics 1

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.

Use laws of indices to connect the rule to the data and decision in the question.

This matters because laws of indices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply laws of indices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Laws of indices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use and manipulation of surds.; Students should be able to rationalise denominators.

Use use and manipulate surds to connect the rule to the data and decision in the question.

This matters because use and manipulate surds determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply use and manipulate surds to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Use and manipulate surds is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Quadratic functions and their graphs.

Use quadratic functions and their graphs to connect the rule to the data and decision in the question.

This matters because quadratic functions and their graphs determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply quadratic functions and their graphs to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Quadratic functions and their graphs is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

The discriminant of a quadratic function.; Know and use b² − 4ac > 0, b² − 4ac = 0 and b² − 4ac < 0.

Use discriminant of a quadratic to connect the rule to the data and decision in the question.

This matters because discriminant of a quadratic determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply discriminant of a quadratic to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Discriminant of a quadratic is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Completing the square and solving quadratic equations.; Solve quadratic equations by factorisation, formula, calculator methods and completing the square.

Use completing the square to connect the rule to the data and decision in the question.

This matters because completing the square determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply completing the square to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Completing the square is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Solve simultaneous equations; analytical solution by substitution.

Use solve simultaneous equations to connect the rule to the data and decision in the question.

This matters because solve simultaneous equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply solve simultaneous equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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.

Use interpreting linear and quadratic inequalities to connect the rule to the data and decision in the question.

This matters because interpreting linear and quadratic inequalities determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply interpreting linear and quadratic inequalities to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Interpreting linear and quadratic inequalities is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Represent linear and quadratic inequalities graphically, such as y > x + r and y > ax² + bx + c.; Use shading and dotted/solid line conventions.

Use graphing linear and quadratic inequalities to connect the rule to the data and decision in the question.

This matters because graphing linear and quadratic inequalities determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply graphing linear and quadratic inequalities to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Graphing linear and quadratic inequalities is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Solve linear and quadratic inequalities, including comparisons between a quadratic expression and a linear expression.

Use solving linear and quadratic inequalities to connect the rule to the data and decision in the question.

This matters because solving linear and quadratic inequalities determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply solving linear and quadratic inequalities to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Solving linear and quadratic inequalities is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Algebraic manipulation of polynomials.; Expand brackets, collect like terms and factorise polynomials of degree n, n <= 3.; The notation f(x) may be used.

Use algebraic manipulation to connect the rule to the data and decision in the question.

This matters because algebraic manipulation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply algebraic manipulation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Algebraic manipulation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use graphs of functions to connect the rule to the data and decision in the question.

This matters because graphs of functions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply graphs of functions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Graphs of functions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use effect of simple to connect the rule to the data and decision in the question.

This matters because effect of simple determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply effect of simple to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Effect of simple is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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).

Use equation of a straight line to connect the rule to the data and decision in the question.

This matters because equation of a straight line determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply equation of a straight line to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

Conditions for two straight lines to be parallel or perpendicular to each other.

Use parallel and perpendicular lines to connect the rule to the data and decision in the question.

This matters because parallel and perpendicular lines determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply parallel and perpendicular lines to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Parallel and perpendicular lines is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use sine rule, cosine rule and triangle area to connect the rule to the data and decision in the question.

This matters because sine rule, cosine rule and triangle area determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply sine rule, cosine rule and triangle area to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Sine rule, cosine rule and triangle area is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Radian measure, including use for Use of the formulae s = rθ and A = 1 r2θ. arc length and area of sector.

Use radian measure, to connect the rule to the data and decision in the question.

This matters because radian measure, determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply radian measure, to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Radian measure, is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use trigonometric functions and graphs to connect the rule to the data and decision in the question.

This matters because trigonometric functions and graphs determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply trigonometric functions and graphs to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Trigonometric functions and graphs is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use derivative as gradient and rate of change to connect the rule to the data and decision in the question.

This matters because derivative as gradient and rate of change determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply derivative as gradient and rate of change to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Derivative as gradient and rate of change is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use differentiating powers of x to connect the rule to the data and decision in the question.

This matters because differentiating powers of x determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply differentiating powers of x to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Differentiating powers of x is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Applications of differentiation to Use of differentiation to find equations of tangents and gradients, tangents and normals. normals at specific points on a curve.

Use tangents, normals and gradients to connect the rule to the data and decision in the question.

This matters because tangents, normals and gradients determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply tangents, normals and gradients to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Tangents, normals and gradients is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Indefinite integration as the reverse Students should know that a constant of integration is of differentiation. required.

Use indefinite integration as the reverse to connect the rule to the data and decision in the question.

This matters because indefinite integration as the reverse determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply indefinite integration as the reverse to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Indefinite integration as the reverse is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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).

Use integration of powers of x and related sums to connect the rule to the data and decision in the question.

This matters because integration of powers of x and related sums determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply integration of powers of x and related sums to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Integration of powers of x and related sums is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.