Unit P2: Pure Mathematics 2

Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof stated below:.

Use structure of mathematical proof to connect the rule to the data and decision in the question.

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

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

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

Proof by exhaustion Proof by exhaustion.; This involves trying all the options.; Suppose x and y are odd integers less than 7.; Prove that their sum is divisible by 2.

Use proof by exhaustion to connect the rule to the data and decision in the question.

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

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

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

Disproof by counter example.; Disproof by counter example – show that the statement “n2 − n + 1 is a prime number for all values of n” is untrue.

Use disproof by counter example to connect the rule to the data and decision in the question.

This matters because disproof by counter example determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply disproof by counter example to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Disproof by counter example is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Simple algebraic division; use of Only division by (ax + b) or (ax – b) will be required, e.g. the Factor Theorem and the b students should know that if f(x) = 0 when x =, Remainder Theorem. a then (ax – b) is a factor of f(x).; Students may be required to factorise cubic expressions such as x3 + 3x2 − 4 and 6x3 + 11x2 − x − 6.; Students should be familiar with the terms ‘quotient’ and ‘remainder’ and be able to determine the remainder when the polynomial f(x) is divided by (ax + b).

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

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

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

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

Coordinate geometry of the circle Students should be able to find the radius and the using the equation of a circle in the coordinates of the centre of the circle, given the equation of form (x − a)2 + (y − b)2 = r2 and the circle and vice versa. including use of the following circle properties: (i) the angle in a semicircle is a right angle.; (ii) the perpendicular from the centre to a chord bisects the chord.; (iii) the perpendicularity of radius and tangent.

Use coordinate geometry of the circle to connect the rule to the data and decision in the question.

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

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

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

Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form x = f(x). n + 1 n.

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

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

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

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

Understand and work with The proof of the sum formula should be known. arithmetic sequences and series, including the formula for the nth term and the sum of a finite Σ Understanding of notation will expected. arithmetic series.; the sum of the first n natural numbers.

Use arithmetic sequences and series to connect the rule to the data and decision in the question.

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

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

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

Increasing sequences, decreasing sequences and periodic sequences.

Use increasing sequences, decreasing sequences and periodic to connect the rule to the data and decision in the question.

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

Example: apply increasing sequences, decreasing sequences and periodic to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Increasing sequences, decreasing sequences and periodic is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Understand and work with For example, given the sum of a series students should be geometric sequences and series, able to use logs to find the value of n. including the formulae for the nth The proof of the sum formula for a finite series should be term and the sum of a finite known. geometric series.; the sum to infinity of a convergent geometric series, The sum to infinity may be expressed as S ∞. including the use of |r| < 1.

Use geometric sequences and series to connect the rule to the data and decision in the question.

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

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

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

Expand (a + bx)^n for positive integer n and use the notations n!, binomial coefficients and nCr.

Use binomial expansion for positive integer powers to connect the rule to the data and decision in the question.

This matters because binomial expansion for positive integer powers determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply binomial expansion for positive integer powers to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Binomial expansion for positive integer powers is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

y = ax and its graph. a > 0, a ≠ 1.

Use y = ax and its graph to connect the rule to the data and decision in the question.

This matters because y = ax and its graph determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply y = ax and its graph to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: y = ax and its graph is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x − log_a y, log_a(x^k) = k log_a x, log_a(1/x) = −log_a x and log_a a = 1, for valid positive arguments and a ≠ 1.

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

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

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

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

The solution of equations of the Students may use the change of base formula. form ax = b.

Use solving equations using logarithms to connect the rule to the data and decision in the question.

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

Example: apply solving equations using logarithms 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.

Knowledge and use of sinθ tan θ =, cosθ and sin2 θ + cos2 θ = 1.

Use core trigonometric identities to connect the rule to the data and decision in the question.

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

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

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

Solve simple trigonometric equations in specified degree or radian intervals, including equations requiring identities, transformations or reduction to a quadratic form.

Use solving trigonometric equations to connect the rule to the data and decision in the question.

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

Example: apply solving trigonometric 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.

Applications of differentiation to To include applications to curve sketching.; Maxima and maxima and minima and stationary minima problems may be set in the context of a practical points, increasing and decreasing problem. functions.

Use applications of differentiation to connect the rule to the data and decision in the question.

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

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

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

Evaluation of definite integrals.

Use evaluation of definite integrals to connect the rule to the data and decision in the question.

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

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

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

Interpretation of the definite Students will be expected to be able to evaluate the area of integral as the area under a curve. a region bounded by a curve and given straight lines.; For example, find the finite area bounded by the curve y = 6x − x2 and the line y = 2x. ∫ x dy will not be required.; Students will be expected to be able to evaluate the area of a region bounded by two curves.

Use interpretation of the definite to connect the rule to the data and decision in the question.

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

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

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

Approximation of area under a For example, curve using the trapezium rule. use the trapezium rule to approximate ∫ (2x +1) dx using four strips.; Use of increasing number of trapezia to improve accuracy and an estimate of the error may be required.

Use trapezium rule for area approximation to connect the rule to the data and decision in the question.

This matters because trapezium rule for area approximation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

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

Boundary: Trapezium rule for area approximation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.