Pure Mathematics 3 - Paper 3

Worked examples (13)extended set

• 1Expansion with a negative fractional power, then an estimate
• 2Adapting (a + bx)ⁿ: take out the constant first
• 3Three distinct linear factors (factorise first)
• 4Recap: two-modulus inequality with a parameter
• 5Recap: repeated factor, found with the derivative
• 6Repeated linear factor
• 7Improper fraction: the constant term
• 8Expanding a quotient with a surd denominator
• 9Coefficient given: find the constant, then a new coefficient
• 10Partial fractions, then a repeated-factor expansion
• 11Partial fractions with a zero constant, then a tangent
• 12Expansion validity: the narrower range wins
• 13Improper fraction with a parameter a, then expand

Worked examples (14)extended set

• 1Logarithm of an expression containing an exponential
• 2Two different bases with an exact answer
• 3Disguised quadratic in 2ˣ with a negative index
• 4Exponential equation in fractional form
• 5Log equation with a constant and a surd answer
• 6Making y the subject of a log relation
• 7Modulus inequality with exponential terms
• 8Quadratic inequality in 5ˣ, then a base less than 1
• 9Linear form with ln y against x² and exact constants
• 10Two decaying substances
• 11Graph of y = |4 − e²ˣ| and number of roots
• 12Upper limit from a definite integral, then area
• 13Differential equation with logs of a non-trivial argument
• 14Mixing bases 2, 3 and 6 by factorising

Worked examples (3)

• 1Equation with reciprocal trig (8 marks)
• 2Proving a trig identity (6 marks)
• 3Inverse trig values (5 marks)

Worked examples (13)extended set

• 1Differentiating tan⁻¹ with the chain rule
• 2Product rule with tan⁻¹ and an exact tangent
• 3Proving a standard derivative with the quotient rule
• 4Rate of change of an angle using tan⁻¹
• 5Stationary point with tan⁻¹ and ln: exact coordinates
• 6Trig stationary point by double-angle, then an exact area
• 7Exponential of a trig function: quadratic in cos x
• 8Implicit curve: where the normal is parallel to the y-axis
• 9Two tangents on the y-axis and the angle between them
• 10Parametric spiral: dy/dx as tan(t + π/4)
• 11Eliminating a parameter k from dy/dx
• 12Exponential parameters: vertical tangent and a normal that never returns
• 13Stationary point that sets up an iteration

Worked examples (14)extended set

• 1An inverse tangent integral with a non-unit x² coefficient
• 2Recognising f′(x)/f(x), including tan x
• 3Exact area under y = ln 2x by parts
• 4By parts leading to ∫ tan x dx
• 5Integration by parts twice
• 6x tan⁻¹ by parts, using division
• 7Partial fractions giving ln and tan⁻¹ terms
• 8Partial fractions with a repeated factor
• 9Improper fraction with a parameter
• 10A given trig substitution: show the new integral, then evaluate
• 11Area enclosed between two curves
• 12Maximum point, then area by substitution
• 13Exponential integrand: substitution and recognition together
• 14Integral equation leading to an iteration

Worked examples (12)extended set

• 1Three roots from a sketch, then consecutive integers
• 2Stationary point of x² cos x: a sign change, and a false one
• 3Relating an inverse-cosine iteration to its equation
• 4Integral by parts equals a constant, then locate and iterate
• 5Maximum of e^(−x/4) tan⁻¹x
• 6Tangents to a circle: the region outside the sector
• 7Segment perimeter: a slowly oscillating iteration
• 8Bacteria model: a differential equation solution equal to a value
• 9Parametric stationary point and the π − tan⁻¹ rearrangement
• 10Implicit curve: stationary points by iteration
• 11Halving the area under y = x cos x
• 12Substitution integral: an iteration that fails, and one that works

Worked examples (12)extended set

• 1Displacement, magnitude, unit vector and midpoint
• 2Parallelogram: the fourth vertex, then show it is a square
• 3Line through two points; does a point lie on it?
• 4Two lines that intersect, and the acute angle between them
• 5Two tunnels: show the lines are skew
• 6Parallel lines with unknown components; a perpendicular direction
• 7Exact cosine of an angle, hence the exact area of a triangle
• 8Foot of the perpendicular, distance and reflection in a line
• 9Cuboid on axes: vectors, angle and a unit vector
• 10Square-based pyramid: exact angle and shortest distance to an edge
• 11A point on a line making a 60° angle: reject the false root
• 12Show a perpendicular distance, then points at a given distance

Worked examples (13)extended set

• 1Exponentials on both sides: split the power first
• 2f′(x)/f(x) and where the constant goes
• 3Forming a differential equation: a deflating balloon
• 4Partial fractions first, then solve
• 5Integration by parts inside a DE, long-term behaviour and a sketch
• 6Trig identity on one side, ln sec y on the other
• 7A substitution hidden inside a DE
• 8Splitting y/(y + 2) and leaving an implicit relation
• 9Modelling a drip: form the DE, solve and interpret
• 10Algebraic division, then ln and tan⁻¹ terms
• 11Two conditions: one for c, one for k
• 12Parts with ln x, sin 2y and the two-value trap
• 13Rate of reaction: formulate, partial fractions, limit and sketch

Worked examples (14)extended set

• 1Division by the conjugate, then modulus and argument
• 2Exact modulus-argument and exponential forms
• 3Multiplying and dividing in polar form
• 4Square roots by forming a quartic, then a complex quadratic
• 5An equation in z and z* with a complex fraction
• 6A quotient that is real, with a modulus condition
• 7Finding a parameter from arg z = π/4
• 8Two numbers on an Argand diagram: angle, length ratio and transformation
• 9A cubic with real coefficients and one complex root
• 10A quartic with a root in exponential form
• 11An arc locus, then the locus of z* − 2
• 12Shading a region bounded by a circle and two half-lines
• 13Perpendicular bisector region: least |z| and the greatest-argument trap
• 14Reading inequalities from a diagram; greatest |z| and least arg z