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Unit P2: Pure Mathematics 2

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  1. P2.1 - Proof

    1. P2.1.1Structure of mathematical proof

      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:.

    2. P2.1.2Proof by exhaustion

      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.

    3. P2.1.3Disproof by counter example

      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.

  2. P2.2 - Algebra and functions

    1. P2.2.1

      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).

  3. P2.3 - Coordinate geometry in the (x, y) plane

    1. P2.3.1

      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.

  4. P2.4 - Sequences and series

    1. P2.4.1Sequences,

      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.

    2. P2.4.2Arithmetic sequences and series

      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.

    3. P2.4.3Increasing sequences, decreasing sequences and periodic

      Increasing sequences, decreasing sequences and periodic sequences.

    4. P2.4.4Geometric sequences and series

      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.

    5. P2.4.5Binomial expansion for positive integer powers

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

  5. P2.5 - Exponentials and logarithms

    1. P2.5.1y = ax and its graph

      y = ax and its graph. a > 0, a ≠ 1.

    2. P2.5.2Laws of logarithms

      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.

    3. P2.5.3Solving equations using logarithms

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

  6. P2.6 - Trigonometry

    1. P2.6.1Core trigonometric identities

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

    2. P2.6.2Solving trigonometric equations

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

  7. P2.7 - Differentiation

    1. P2.7.1

      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.

  8. P2.8 - Integration

    1. P2.8.1Evaluation of definite integrals

      Evaluation of definite integrals.

    2. P2.8.2Interpretation of the definite

      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.

    3. P2.8.3Trapezium rule for area approximation

      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.