Pure Mathematics 1 - Paper 1

Worked examples (14)extended set

• 1Completing the square when the coefficient of x² is not 1
• 2Negative coefficient of x²: greatest value in context
• 3Intersection points, then
• 4Finding a quadratic from its vertex
• 5Line y = x + k tangent to a parabola
• 6Line through a fixed point that does not meet a curve
• 7Repeated root with exact surd values
• 8Roots given in terms of a parameter
• 9Simultaneous equations with an xy term
• 10Disguised quadratics in x², √x and 2ˣ
• 11Cube roots and a trigonometric quadratic
• 12Two distinct roots and positive for all x
• 13Completed square with a parameter, range and inverse
• 14Tangents to a circle with a given gradient

Worked examples (14)extended set

• 1Range from a completed square, with and without a restricted domain
• 2Inverse of a rational function and a self-inverse value
• 3Least value of a for an inverse, with a parameter
• 4Inverse on a left-hand domain, then (gf)⁻¹
• 5Sketch a function and its inverse, then find where they meet
• 6ff(x) with embedded fractions, then hence solve
• 7When can gf be formed? A parameter in the domain
• 8Working back from a given inverse
• 9Reflection and translation of y = x²
• 10Describe the transformations between two graphs on a grid
• 11Transforming a trigonometric graph step by step
• 12Constants from the range of a transformed quadratic
• 13Increasing function, inverse and an odd-function condition
• 14Composite equations with a parameter and the discriminant

Worked examples (13)extended set

• 1Midpoint, exact length and a parallel line
• 2Centre and radius by completing the square
• 3Circle from the ends of a diameter
• 4Centre on a given line through a perpendicular bisector
• 5Line meets circle: exact chord and triangle area
• 6Tangents at two symmetric points and where they meet
• 7Circle through three points, then a tangent
• 8Line and quadratic curve: set of values of k
• 9Line tangent to a circle via the discriminant
• 10Chord, angle at the centre and the larger segment
• 11Centre from distances to the axes; furthest point
• 12Right angle with a parameter, then the circle
• 13Unstructured: two tangents and the axes

Worked examples (12)extended set

• 1Radians, and a sector found from its perimeter and area
• 2Segment perimeter and area — the radian-mode trap
• 3Region between two arcs — exact answers in terms of π
• 4Triangle minus a sector — cosine rule for the angle
• 5Two tangents from an external point
• 6
• 7Exact major segment in terms of π and √3
• 8Circle and line: angle at the centre and the larger segment
• 9Fixed perimeter: the largest possible sector
• 10Three pipes held by a band — exact answers in context
• 11Circles meeting at right angles — the common region
• 12A growing segment — circular measure with rates of change

Worked examples (14)extended set

• 1Exact values of related angles and a ratio in a given quadrant
• 2Simple equations on a negative interval and in radians
• 3Equations involving inverse trigonometric notation
• 4Greatest and least values, sketch, and number of solutions
• 5Sketching a tan graph and solving with x + 30°
• 6Multiple and half angles: transform the interval first
• 7Quadratic in cos θ with a root to reject
• 8Do not divide by sin x: the lost solutions
• 9Prove an identity, then hence solve with ± roots
• 10Trigonometric model of an observation wheel
• 11Range and inverse of a trigonometric function
• 12Reading a, b and c from a graph, then the set of k
• 13Show that an equation is a quadratic in tan x, then hence twice
• 14A geometric progression with ratio sin θ

Worked examples (14)extended set

• 1Complete expansion with a negative 1/x term
• 2First three terms, then an unknown in a product
• 3Term independent of x with an unknown constant
• 4Product with a linear factor, then an approximation
• 5Coefficient in a product of two expansions gives a quadratic
• 6Surd substitution and the integer part of a power
• 7Arithmetic progression from a term and a sum
• 8Training plan: largest n with the total below a limit
• 9Sum of the terms lying between two values
• 10Second term and sum to infinity: rejecting a ratio
• 11Three terms in k: geometric, then arithmetic instead
• 12Bouncing ball: a geometric progression in context
• 13Geometric progression with a trigonometric ratio
• 14Terms of a GP that are also terms of an AP

Worked examples (13)extended set

• 1Fractional powers, f″(x) and the rate of change of the gradient
• 2Chain rule with negative and fractional powers
• 3Gradient as the limit of chord gradients
• 4Tangent and normal cutting the x-axis
• 5Normal parallel to a given line
• 6Increasing, decreasing and an inverse function
• 7Stationary points of a curve with an asymptote
• 8Stationary point to 3 significant figures
• 9Inflating a spherical balloon
• 10A point moving on a curve, with y given
• 11Minimising the surface area of an open box
• 12Line through two stationary points, then a sketch
• 13Finding constants, and when the second-derivative test fails

Worked examples (14)extended set

• 1Reverse chain rule, then a definite integral
• 2Improper integrals
• 3Curve from a quotient gradient, then a translation
• 4Definite integrals with an unknown constant or limit
• 5Finding k from a normal, then the curve
• 6Integrating twice from the second derivative
• 7When a definite integral is not the area
• 8Exact area from a maximum to a root
• 9Area enclosed by a curve and a line
• 10Area between two curves
• 11Volume of a bead: squaring y correctly
• 12Area and volume about the y-axis
• 13Volume of a region between a line and a curve
• 14Tangent, composite region and an infinite region