Pure Mathematics 2 - Paper 2

Worked examples (12)extended set

• 1Modulus equation with two modulus signs, then a sec θ follow-on
• 2Modulus inequality in context: a manufacturing tolerance
• 3Sketch a V-graph and a line, then reject the false root
• 4Linear expression less than a modulus: when squaring fails
• 5Modulus inequality, then smallest and largest integers N via ln
• 6Modulus inequality, then an exponential with a decreasing base
• 7Factor theorem with the unknown in two terms, then a remainder
• 8Divide by a linear factor, then integrate the remainder term
• 9Divide a quartic by a quadratic: show the remainder, then exact roots
• 10A given remainder, factorising p(x) + c, and a disguised cos θ cubic
• 11Two unknowns from a factor and a remainder, then a cot 2θ equation
• 12From a quartic with a known factor to a fixed-point iteration

Worked examples (13)extended set

• 1Laws of logarithms and the link with indices
• 2Graphs of e^(kx) for positive and negative k
• 3Unknown in the index on both sides
• 4Exponential and logarithm as inverse functions
• 5Log equations: domain check and exact answers
• 6Equations quadratic in eˣ or 2ˣ
• 7Exponential inequalities and the sign of the logarithm
• 8Modulus equation, hence an exponential and a log equation
• 9Exponential decay model from a graph of ln m against t
• 10Power law y = kxⁿ from a graph of ln y against ln x
• 11Linear form with a shifted exponent: y = k·bˣ⁺²
• 12Using natural logarithms to reach an exact linear form
• 13Stationary points that lead to a quadratic in eˣ

Worked examples (13)extended set

• 1Exact values from compound-angle formulae
• 2A quadratic in cosec θ, with an impossible root
• 3Sketching y = cot x to count roots
• 4The graph of y = 2 cosec x: asymptotes, range and solutions
• 5Double-angle equations: tan 2θ and cos 2x
• 6Factor theorem, then a cubic in sec θ
• 7tan θ + cot θ = k via a double angle
• 8R sin(θ − α) with a negative right-hand side
• 9tan(x + 45°) = k tan x: show, solve, then the set of k
• 10A tide model: R sin(x + α) in context
• 11Identity, range, then an exact integral
• 12R cos(θ + α): a reciprocal expression and a doubled angle
• 13From a product to R sin(2θ − α) + c

Worked examples (14)extended set

• 1Standard derivatives and the chain rule
• 2Product rule with a logarithm: exact stationary point
• 3Quotient with a squared denominator, then a normal
• 4Stationary points of a rational function and where it increases
• 5Exact maximum of an exponential times a root
• 6Drug concentration: exact time of the maximum
• 7Maxima of a trigonometric curve
• 8Tangent and normal to a logarithmic quotient
• 9Exact tangent to an implicit trigonometric curve
• 10Implicit curve: missing points, gradient where y = 0, no vertical tangent
• 11Parametric curve: normal at the x-axis and a given gradient
• 12Trigonometric quotient: an impossible factor and the range
• 13Implicit curve: no stationary point, but a vertical tangent
• 14Maximum point of a trigonometric parametric curve

Worked examples (13)extended set

• 1Standard integrals with a linear inside
• 2Equation of a curve from its gradient
• 3Integrating sin² and cos² with double-angle identities
• 4Exact log answer, a phase-shifted cosine and a set of values
• 5tan² using the sec² identity
• 6Prove an identity, then hence integrate
• 7Divide first, then integrate an improper fraction
• 8Area between two exponential curves
• 9Region between a curve and a chord
• 10Trapezium rule and whether it under-estimates
• 11An integral equal to a constant, leading to an iteration
• 12Exact area between the maxima of a trig curve and its chord
• 13Trapezium estimate in context compared with the exact value

Worked examples (12)extended set

• 1Finding the consecutive integers either side of a root
• 2When a sign change does not mean a root
• 3Using a given iterative formula to 4 significant figures
• 4Sketching two curves to show there is exactly one root
• 5A root from an integral equal to a constant, in context
• 6Exponential meets sec x: iterating in radians
• 7Quartic with a known factor: cube-root iteration for a negative root
• 8Finding the exact limit of a convergent sequence
• 9Iterating for a limit, then using the root in an area
• 10Modulus and log curves: two roots, two branches
• 11Two rearrangements: one fails, one converges
• 12Stationary point found by iteration, then an exact shortcut