36-Week WAEC Revision Programme — Further Mathematics
September to the May WAEC sitting. SS3 is not a year of new material so much as a year of pressure on material already met, so this is not a course to work through from the beginning — it is a testing programme. Each week you answer cold, work the ones you got wrong, and have the AI explain why you got them wrong. Anything you turn out never to have been taught, you learn there and then, and meet again three weeks later.
A week is one block of the syllabus and the prompts that cover it. You are not meant to do all of them: answer the questions first, and let what you get wrong decide which of the rest you need. Fifteen or twenty minutes a subject is a real week's work — nine subjects do not fit any other way.
Every week, in this order
Start with the questions, not the explanation. The first two steps are in every week; a purpose-written mistake or pitfalls prompt appears in about half of them, and where it does not, your own wrong answers are the material for step three.
- 1. Retrieve — Answer cold, with nothing in front of you. Wrong answers are the point — they show you where the week's work is.
- 2. Drill — Work the ones you missed properly, showing every step, so you and the AI can both see where it went wrong.
- 3. Explain my error — Ask why your answer was wrong, not just what the right one is. Where the week offers a Spot My Mistake or WAEC Pitfalls prompt, use it.
- 4. Teach me — Only if the error turns out to be something you were never taught. Then come back and retrieve it again.
Sets, Logic & Algebra
↑ Jump to week- Week 1
- Quiz me rapid-fire on evaluating functions
- Inequality word problem with a real-life constraint
- WAEC-style past question on functions
- Show all working: full function exam question with three parts
- WAEC exam question: remainder theorem with two conditions
- WAEC-style question on evaluating composite functions
- Structured question: solve an exponential equation using indices
- Structured question: rationalise and simplify a surd expression
- WAEC exam question: solving an equation involving surds
- Structured question: solve a logarithmic equation with two conditions
- Common Mistakes: Solving an equation in which the unknown sits under a root
- Simplifying an expression before solving an equation
- Factorising an expression with a common algebraic bracket factor
- Simplifying algebraic fractions that need factorising a quadratic first
- Solving an equation formed by equating two algebraic fractions
- What a function is and function notation
- Evaluating a function at a given value
- Evaluating a function at a negative value
- Evaluating a function at an algebraic input
- Domain of a simple algebraic function
- Domain of a function involving a square root
- Range of a simple function from its graph or values
- Mapping diagrams and function types
- Testing whether a relation is a function
- The vertical line test
- Composite functions fg(x)
- Composite functions gf(x)
- Showing that fg(x) is not the same as gf(x)
- Finding the inverse of a simple linear function
- Finding the making x the subject of a fractional function
- Verifying an inverse function is correct
- Domain and range of an inverse function
- Graphs of a function and its inverse as reflections
- Self-inverse functions
- Composite function leading to solving an equation
- Multi-part function exam question
- Piecewise-style simple function evaluation
- Function notation applied to a real-world formula
- One-to-one versus many-to-one functions
- Why not every function has an inverse function
- Restricting the domain so an inverse exists
- Finding f(x) given a description of the function rule
- Solving f(x) = a given value
- Composite function with three functions applied in order
- Explaining function notation to a classmate
- Reviewing all function skills before a test
- Function machines as an introduction to function notation
- Comparing two functions to see which grows faster
- Show that command word: using the remainder to find an unknown coefficient
- Show that command word: laws of indices
- Non-calculator paper: simplify an index expression
- Show that command word: surd simplification
- Show that command word: logarithm laws
- Cambridge-style question: change of base formula
- Worked Example: Simplifying an expression carrying both a negative and a fractional index
- Week 2
- Quiz: The gradient of the line through two points
- Structured question: use Pascal's triangle to expand a binomial
- Structured question: binomial expansion with a negative term
- WAEC exam question: full binomial expansion to a stated power
- Solve x^2 - 6x - 16 = 0 with full exam working
- Solve x^2 + 13x + 40 = 0 with full exam working
- Solve 2x^2 - 5x - 12 = 0 with full exam working
- Solve 5x^2 + 13x - 6 = 0 with full exam working
- Solve x^2 - 2x - 35 = 0 with full exam working
- Solve 4x^2 - 12x + 5 = 0 with full exam working
- Solve x^2 + 7x - 18 = 0 with full exam working
- Solve 3x^2 + x - 14 = 0 with full exam working
- Structured question: three-part algebra question mixing methods
- Structured question: fully factorise x^3 - 2x^2 - 5x + 6
- Structured question: fully factorise x^3 + 4x^2 - 11x - 30
- Structured question: unknown coefficient from a factor condition
- WAEC exam question: two unknowns from remainder and factor conditions
- Structured question: three linked function sub-questions
- WAEC exam question: solving f(x) = c for an unknown x
- Solve 2^(3x-1) = 16 with full working
- Solve 5^(x+2) = 125^(x-1) with full working
- Solve log2(x+1) + log2(x-1) = 3 with full working
- Structured question: line through (1,4) and (5,-2)
- Structured question: circle centre (3,-2) radius 5
- Structured question: perpendicular bisector of (2,1) and (-4,7)
- Solve 2sin(x) + 1 = 0 for 0 to 360 degrees
- Solve 2cos^2(x) - cos(x) - 1 = 0 for 0 to 360 degrees
- Prove that (1-cos^2x)/sinx = sinx
- Prove that tanx + 1/tanx = 1/(sinx cosx)
- Spot my mistake: Treating the rth term of a binomial expansion as the one carrying the rth power
- Show that command word: Pascal's triangle row sums to a power of 2
- Express x^2 + 10x - 3 in completed-square form
- Express x^2 - 14x + 20 in completed-square form
- Express x^2 + 5x + 1 in completed-square form
- Express 2x^2 - 8x + 3 in completed-square form
- Express x^2 - 9x + 4 in completed-square form
- Choose your own method command word: solving a quadratic
- Find the remainder command word: cubic divided by a linear factor
- Show that command word: a function is its own inverse
- Simplify (8^(2/3) times 4^(-1/2)) fully
- Rationalise 7/(sqrt5 - 2) fully
- Show that (sqrt3+1)^2 = 4+2sqrt3
- Evaluate log5(200) using change of base
- Find the centre and radius from x^2+y^2-6x+8y-11=0
- Show that (1,2), (3,6), and (5,10) are collinear
- AP with 3rd term 11 and 8th term 31: find a and d
- GP with 3rd term 18 and 6th term 486: find a and r
- Find the sum of the first 15 terms of an AP
- Show that a geometric series converges and find its sum to infinity
- Differentiate y = 4x^3 - 6x^2 + 2x - 9 and evaluate the gradient at x=1
- Find the tangent to y = x^2 - 4x + 7 at x = 3
- Find and classify the stationary points of y = x^3 - 6x^2 + 9x + 1
- Find the range of x for which y = x^3 - 3x is increasing
- Give me an exam formula sheet for quadratic factorisation
- Give me an exam formula sheet for solving quadratics by factorisation
- Learn: Finding the coefficient of a stated power in an expansion
- Week 3
- Quiz: The remainder theorem, and the remainder on dividing by x minus a
- WAEC Prep: Dividing a polynomial by a linear divisor by long division
- Spot my mistake: forming a quadratic equation from its roots
- Spot my mistake: dividing a polynomial by long division
- Spot my mistake: solving an exponential equation
- Spot my mistake: checking each line of a trigonometric identity proof
- Spot my mistake: finding a stationary point
- Spot my mistake: geometric progression sum to infinity
- Give me an exam formula sheet for completing the square
- Give me an exam formula sheet for the quadratic formula
- Give me an exam formula sheet for sum and product of roots
- Give me an exam formula sheet for quadratic inequalities
- Give me an exam formula sheet for linear-quadratic simultaneous equations
- Give me an exam formula sheet for line and curve tangency conditions
- Give me an exam formula sheet for polynomial operations
- Give me an exam formula sheet for the remainder theorem
- Give me an exam formula sheet for the factor theorem
- Give me an exam formula sheet for cubic equations
- Give me an exam formula sheet for functions
- Give me an exam formula sheet for indices
- Give me an exam formula sheet for surds
- Give me an exam formula sheet for logarithms
- Give me an exam formula sheet for coordinate geometry
- Give me an exam formula sheet for trigonometric identities and equations
- Give me an exam formula sheet for sequences and series
- Give me an exam formula sheet for the binomial theorem
- Give me an exam formula sheet for propositional logic
- Give me an exam formula sheet for permutations and combinations
- Give me an exam formula sheet for differentiation basics
- Give me a past-paper style question bank for quadratic factorisation
- Give me a past-paper style question bank for solving quadratics by factorisation
- Give me a past-paper style question bank for completing the square
- Give me a past-paper style question bank for the quadratic formula
- Give me a past-paper style question bank for sum and product of roots
- Give me a past-paper style question bank for quadratic inequalities
- Give me a past-paper style question bank for linear-quadratic simultaneous equations
- Give me a past-paper style question bank for line and curve tangency conditions
- Give me a past-paper style question bank for polynomial operations
- Give me a past-paper style question bank for the remainder theorem
- Give me a past-paper style question bank for the factor theorem
- Give me a past-paper style question bank for cubic equations
- Give me a past-paper style question bank for functions
- Give me a past-paper style question bank for indices
- Give me a past-paper style question bank for surds
- Give me a past-paper style question bank for logarithms
- Give me a past-paper style question bank for coordinate geometry
- Give me a past-paper style question bank for trigonometric identities and equations
- Give me a past-paper style question bank for sequences and series
- Give me a past-paper style question bank for the binomial theorem
- Give me a past-paper style question bank for propositional logic
- Give me a past-paper style question bank for permutations and combinations
- Give me a past-paper style question bank for differentiation basics
- Worked example: solving simultaneous equations with matrices
- Week 4
- Give me an Additional Maths command word matching quiz
- Diagnose my weak topics using a short quiz
- Simulate a WAEC Further Maths Section A objective test
- Quiz me on choosing the correct method for a quadratic
- Quiz me on recognising which command word fits a blank exam question
- Worked example: solve the simultaneous system y = 2x - 1 and y = x^2 - 4
- Worked example: solve 3^(2x) - 4(3^x) + 3 = 0
- Worked example: prove an argument invalid using a truth table
- Structured word problem: fencing a rectangular garden
- Structured word problem: WAEC exam hall seating combinations
- Structured word problem: market trader's geometric savings
- Structured word problem: population growth as a geometric sequence
- Structured question: quadratic with roots that differ by a given amount
- Structured question: find k so a cubic has a repeated root
- Structured question: composite function domain restriction
- Structured question: surd equation with a solution to reject
- Structured question: logarithmic equation reducing to a quadratic
- Structured question: circle and line with an unknown intercept
- Structured question: trig equation with a coefficient inside the angle
- Structured question: AP and GP sharing the same first three terms
- Structured question: binomial expansion coefficient equation
- Structured question: stationary point on a curve with an unknown constant
- Structured question: quadratic word problem about a thrown ball
- Structured question: two quadratics sharing a common root
- Structured question: polynomial with a given remainder pattern
- Structured question: cubic equation from three given roots
- Structured question: stating the domain of the inverse of a fractional function
- Structured question: simplifying a surd raised to a fractional index
- Structured question: logarithmic inequality
- Structured question: circle with a diameter given by two endpoints
- Structured question: solving a trig equation involving a double angle
- Structured question: sum of a series expressed using sigma notation
- Structured question: binomial expansion used for a probability estimate
- Structured question: curve sketching using stationary points
- Structured question: logic puzzle solved with a truth table
- Structured question: combination problem with 'at least' condition
- Structured question: quadratic formula question with a negative discriminant
- WAEC exam question: forming a quadratic equation from a word problem
- WAEC exam question: polynomial division real-world framing
- WAEC exam question: sequences applied to loan repayment
- WAEC Prep: Adding three matrices of the same order
- WAEC Prep: The commutative law for matrix addition
- WAEC Prep: Adding two 2 by 2 matrices entry by entry
- Spot My Mistake: The equation of the tangent to a circle at a point on it
- Simulate a WAEC Further Maths Section B theory question
- Check whether my exam layout would satisfy a Cambridge examiner
- Build me a two-week countdown revision schedule
- Give me exam-technique advice for running out of time
- Compare WAEC and Cambridge phrasing for the same topic
- Explain back: quadratic inequality solution notation
- Explain back: why we check the domain in logarithmic equations
- Non-calculator paper: exact value question on sum and product of roots
- Non-calculator paper: coordinate geometry with fractional gradients
- Give me a mixed revision pack blending three topics
- Give me a mixed revision pack blending logic and combinatorics
- Help me interpret my own WAEC Further Maths past mistakes
- Worked example: fully factorise x^3 - 7x + 6
- Worked example: find where the tangent to y = x^3 crosses the x-axis
- Worked example: expand (2-x)^5 using Pascal's triangle
- Worked example: seat 4 boys and 3 girls alternately in a row
Revisit from three weeks ago - Week 5
- Quiz: Subtracting one complex number from another
- Quiz: The general term of a binomial expansion
- WAEC Prep: Adding three complex numbers
- WAEC Prep: The geometric meaning of adding two complex numbers
- WAEC Prep: Adding two complex numbers
- Practice: Equating real and imaginary parts to find two unknowns
- WAEC Prep: Multiplying a complex number by a real number
- WAEC Prep: Dividing by a purely imaginary number
- WAEC Prep: Multiplying two complex numbers
- WAEC Prep: How many terms a geometric progression has
- WAEC Prep: A geometric progression in which one term is given as a product
- WAEC Prep: The nth term of a geometric progression
- WAEC Prep: Why the sum to infinity fails when the ratio reaches one
- WAEC Prep: The first term found from the sum to infinity and the ratio
- WAEC Prep: The condition for a geometric series to converge
- Practice: The sum to infinity of a convergent geometric series
- WAEC Prep: The coefficient of the term in x to the power r
- WAEC Prep: Expanding a bracket raised to the power six
- WAEC Prep: Expanding a bracket by the binomial theorem
- Practice: The middle term when the power is odd
- WAEC Prep: Clearing the denominators before comparing coefficients
- WAEC Prep: Substituting a convenient value to find one constant
- WAEC Prep: Splitting a fraction with two distinct linear factors
- WAEC Prep: The remainder when a polynomial is divided by a quadratic
- WAEC Prep: A polynomial leaving the same remainder for two different divisors
- Practice: Finding a remainder without carrying out the division
- WAEC Prep: The mean of a frequency distribution
- WAEC Prep: The effect of one extra value on the mean
- WAEC Prep: The mean of a set of ungrouped data
- Spot my mistake: Assuming AB and BA give the same matrix
- Spot My Mistake: Dividing one complex number by another using the conjugate
- Spot my mistake: Assuming a quadratic with a negative discriminant has no roots at all
- Spot my mistake: Differentiating a product by multiplying the derivatives of its two factors
- Spot My Mistake: Finding the common ratio from two given terms
- Spot My Mistake: Partial fractions with an irreducible quadratic factor
- Explain Back: The sum of the first n terms of a geometric progression
- Explain Back: Writing a recurring decimal as a geometric series
- Explain Back: Partial fractions with a repeated linear factor
- Worked Example: What a determinant of nought means for a 2 by 2 matrix
- Worked Example: Finding a particular term of a binomial expansion
- Worked Example: The powers of i, and the cycle they follow
- Learn: The real and the imaginary parts of a complex number
- Worked Example: The quotient rule for differentiation
- Worked Example: Differentiating a bracket raised to a power
Revisit from three weeks ago - Week 6
- Quiz: The complement of a set within a stated universal set
- Quiz: Testing a binary operation for commutativity
- Quiz: Multiplying two surds and simplifying the result
- Practice: The difference of two sets, A minus B
- WAEC Prep: The number of elements in the union of two sets
- Show All Working: A three-set Venn diagram drawn from a subject survey
- Practice: Testing a binary operation for associativity
- WAEC Prep: Completing a Cayley table for an operation on a small set
- Show All Working: Solving an equation written in terms of a binary operation
- Practice: Rationalising a denominator of the form root a
- WAEC Prep: Comparing the sizes of two surds without a calculator
- Show All Working: Expressing a number in the form a root b
- Spot my mistake: Reading a strong correlation between two variables as proof that one causes the other
- Spot My Mistake: De Morgan's law that the complement of a union is the intersection of the complements
- Common Mistakes: Showing that intersection distributes over union
- Spot My Mistake: Finding the inverse of an element under a binary operation
- Common Mistakes: A binary operation defined by arithmetic modulo n
- Spot My Mistake: Squaring a bracket that contains a surd
- Explain Back: The symmetric difference of two sets
- Final Revision: The power set, and how many subsets a set of n elements has
- Explain Back: Finding the identity element of a binary operation
- Final Revision: Deciding whether a set under an operation forms a group
- Explain Back: Rationalising a denominator of the form a plus root b using the conjugate
- Final Revision: Proving that root 2 is irrational
- Teach me the Poisson distribution from scratch
- Learn: The union of two sets written in set-builder notation
- Worked Example: The intersection of two overlapping sets
- Learn: Reading a binary operation defined by a formula such as a*b = a + b - ab
- Worked Example: Testing whether a binary operation is closed on a given set
- Learn: Simplifying a surd by extracting the largest perfect square
- Worked Example: Collecting like surds in a single sum
Revisit from three weeks ago - Week 7
- Quiz: Solving an index equation by writing both sides to the same base
- Quiz: Forming a quadratic equation whose roots are given
- Quiz: Finding alpha cubed plus beta cubed
- Practice: Converting between index form and logarithm form
- WAEC Prep: The change of base formula for logarithms
- Show All Working: Solving a logarithmic equation and rejecting the invalid root
- Practice: The condition for a quadratic equation to have equal roots
- WAEC Prep: Forming an equation whose roots are each two more than the original
- Show All Working: Solving a quadratic by completing the square
- Practice: Finding alpha minus beta from the sum and the product
- WAEC Prep: Forming an equation whose roots are alpha over beta and beta over alpha
- Show All Working: Finding a coefficient given a stated relation between the roots
- Spot My Mistake: The power law of logarithms
- Common Mistakes: Solving an exponential equation using logarithms
- Spot My Mistake: Forming a new equation whose roots are the reciprocals of the original
- Common Mistakes: Proving that a quadratic expression is positive for every real x
- Spot My Mistake: Forming an equation whose roots are alpha squared and beta squared
- Common Mistakes: Symmetric functions when the three roots of a cubic are involved
- Explain Back: The laws of logarithms for a product and for a quotient
- Final Revision: Sketching the graph of a logarithmic function
- Explain Back: The range of values of k for which the roots are real
- Final Revision: Solving a disguised quadratic by substitution
- Explain Back: Finding alpha squared beta plus alpha beta squared
- Final Revision: Checking a symmetric-function answer by solving the equation outright
- Learn: The multiplication and division laws of indices used together
- Learn: The sum and the product of the roots read from the coefficients
- Worked Example: The discriminant, and what each of its three cases means
- Learn: Finding alpha squared plus beta squared from the sum and product
- Worked Example: Finding one over alpha plus one over beta
Revisit from three weeks ago - Week 8
- Quiz: The factor theorem, and testing whether x minus a is a factor
- Quiz: The product of the roots of a cubic
- Quiz: Solving a quadratic inequality using a sign table
- Practice: Factorising a cubic completely once one factor is known
- WAEC Prep: Sketching a cubic from its factorised form
- Show All Working: Solving a cubic equation by factorisation
- Practice: Finding the third root of a cubic when two are known
- WAEC Prep: Finding a coefficient of a cubic given one of its roots
- Show All Working: Three roots in arithmetic progression in a cubic
- Practice: Solving an inequality with the unknown in the denominator
- WAEC Prep: Writing a solution set in interval notation
- Show All Working: Solving a pair of simultaneous linear inequalities
- Spot My Mistake: The factor theorem for a divisor of the form ax minus b
- Common Mistakes: Adding, subtracting and multiplying two polynomials
- Spot My Mistake: The relations between roots and coefficients for a quartic
- Common Mistakes: Three roots in geometric progression in a cubic
- Spot My Mistake: Showing the solution of an inequality on a number line
- Common Mistakes: The range of values of x for which an expression stays positive
- Explain Back: Finding an unknown coefficient given that a stated factor divides the polynomial
- Final Revision: The zeros of a polynomial and where its graph cuts the x-axis
- Explain Back: Forming a cubic equation whose three roots are given
- Final Revision: Checking that the roots found satisfy the original equation
- Explain Back: Solving an inequality that involves a modulus
- Final Revision: The region of the plane satisfying a set of linear inequalities
- Learn: The degree of a polynomial and its leading coefficient
- Worked Example: Evaluating a polynomial by substitution
- Learn: The sum of the roots of a cubic read from its coefficients
- Worked Example: The sum of the products of the roots of a cubic taken two at a time
- Learn: Solving a linear inequality, and reversing the sign on dividing by a negative
- Worked Example: Solving a quadratic inequality by sketching the parabola
Revisit from three weeks ago - Week 9
- Quiz: The compound angle formula for sine of A plus B
- Quiz: Solving tan x equals a given value, and the period of the tangent
- Quiz: The tangent curve and its asymptotes
- Practice: The compound angle formula for cosine of A minus B
- WAEC Prep: The three forms of the double angle formula for cosine
- Show All Working: The half angle formulae and the t-substitution
- Practice: Solving an equation of the form sin of 2x equals a given value
- WAEC Prep: Solving an equation by first applying a double angle formula
- Show All Working: Solving a cos x plus b sin x equals c by the R method
- Practice: The effect of a in y equals a sin x
- WAEC Prep: A phase shift in y equals sin of x minus d
- Show All Working: Reading the solutions of a trigonometric equation off a graph
- Spot My Mistake: The double angle formula for sine
- Common Mistakes: Expressing a cos x plus b sin x in the form R cos of x minus alpha
- Spot My Mistake: Solving an equation by first applying a Pythagorean identity
- Common Mistakes: Giving the general solution of a trigonometric equation
- Spot My Mistake: The effect of c in y equals sin x plus c
- Common Mistakes: Sketching two trigonometric curves on the same axes to find where they meet
- Explain Back: The compound angle formula for tan of A plus B
- Final Revision: Proving an identity by working on one side only
- Explain Back: Solving a quadratic in sin x
- Final Revision: Losing solutions by dividing through by cos x
- Explain Back: The effect of b in y equals sin bx on the period
- Final Revision: The graphs of y equals sec x, cosec x and cot x
- Learn: The Pythagorean identity, sin squared plus cos squared equals one
- Worked Example: The identities for one plus tan squared and one plus cot squared
- Learn: Solving sin x equals a given value between 0 and 360 degrees
- Worked Example: Solving cos x equals a negative value, and finding both solutions
- Learn: The shape, period and range of the sine curve
- Worked Example: The shape, period and range of the cosine curve
Revisit from three weeks ago
Sequences, Series & Trigonometry
↑ Jump to week- Week 10
- Quiz: Solving a matrix equation of the form A plus X equals B
- Quiz: Finding the centre and the radius from the general equation
- WAEC Prep: Checking a matrix subtraction by adding back
- WAEC Prep: Subtracting a matrix from itself
- WAEC Prep: Subtracting one 2 by 2 matrix from another
- WAEC Prep: The product of a 2 by 2 matrix with the identity
- WAEC Prep: Raising a 2 by 2 matrix to the third power
- WAEC Prep: Multiplying two 2 by 2 matrices, row into column
- Practice: A 2 by 2 matrix multiplied by a column vector
- WAEC Prep: Multiplying a 3 by 3 matrix by the identity matrix
- WAEC Prep: Checking a single entry of a 3 by 3 product
- WAEC Prep: Multiplying two 3 by 3 matrices, row into column
- Practice: The equation of a line through a point with a given gradient
- WAEC Prep: Proving two given lines parallel from their gradients
- Show All Working: Proving two given lines perpendicular from their gradients
- Practice: The equation of a circle on a given diameter
- WAEC Prep: The length of the tangent from an external point
- Show All Working: The intersection of a circle and a straight line
- Spot My Mistake: Why matrix subtraction is not commutative
- Spot My Mistake: Squaring a 2 by 2 matrix
- Spot My Mistake: Multiplying a 3 by 3 matrix by a 3 by 1 column
- Spot My Mistake: The general form ax plus by plus c equals nought
- Common Mistakes: The angle between two lines found from their gradients
- Common Mistakes: The condition for a line to be a tangent to a circle
- Explain Back: Subtracting two 3 by 3 matrices
- Explain Back: Showing that AB and BA are usually different for 2 by 2 matrices
- Explain Back: Keeping track of the nine entries when multiplying 3 by 3 matrices
- Explain Back: The equation of the line through two given points
- Final Revision: The perpendicular distance from a point to a line
- Explain Back: Deciding whether a point lies inside, on or outside a circle
- Final Revision: The equation of the circle through three given points
- Worked Example: Adding two 3 by 3 matrices
- Learn: Why two matrices can be added only when their orders match
- Learn: The distance between two points
- Worked Example: The midpoint formula applied to a given segment
- Learn: The equation of a circle with its centre at the origin
- Worked Example: The equation of a circle with a given centre and radius
Revisit from three weeks ago
Coordinate Geometry & Conics
↑ Jump to week- Week 11
- Quiz: Multiplying a matrix by minus one
- Quiz: Solving AX equals B and XA equals B by different routes
- Quiz: The zero matrix, and how it differs from nought in arithmetic
- Quiz: The equation of the tangent to a parabola
- WAEC Prep: Multiplying a matrix by a fraction
- WAEC Prep: The effect of a scalar on the determinant
- WAEC Prep: Multiplying every entry of a matrix by a scalar
- WAEC Prep: Pre-multiplying a column vector by a matrix
- WAEC Prep: Post-multiplying a row vector by a matrix
- WAEC Prep: The difference between pre-multiplying and post-multiplying by a matrix
- WAEC Prep: The transpose of a column vector
- WAEC Prep: The transpose of a transpose
- WAEC Prep: Writing the transpose of a matrix by turning rows into columns
- Practice: The transpose of a 2 by 3 matrix
- WAEC Prep: A skew-symmetric matrix
- WAEC Prep: The identity matrix of order three
- WAEC Prep: The identity matrix, and what it does in a product
- Practice: A diagonal matrix
- Practice: The ellipse in standard form, and its two axes
- WAEC Prep: The hyperbola in standard form, and its asymptotes
- Show All Working: The rectangular hyperbola xy equals c squared
- Spot My Mistake: Orders that make a product possible one way round but not the other
- Spot My Mistake: A symmetric matrix, equal to its own transpose
- Spot My Mistake: The foci of an ellipse
- Common Mistakes: Identifying a conic from a general second-degree equation
- Explain Back: Why the order of the letters matters in a matrix product
- Explain Back: The rule that the transpose of a product reverses the order
- Explain Back: The eccentricity of an ellipse
- Final Revision: The focus-directrix definition shared by all the conics
- Worked Example: The associative law for the product of three matrices
- Worked Example: Taking a common factor out of a matrix
- Learn: The distributive law for a scalar over a sum of matrices
- Learn: The parabola y squared equals 4ax, with its focus and directrix
- Worked Example: The parametric form of a point on a parabola
Revisit from three weeks ago
Complex Numbers
↑ Jump to week- Week 12
- Quiz: Finding an unknown entry from a stated matrix equation
- Quiz: Expanding a 3 by 3 determinant along a row or column containing noughts
- WAEC Prep: The order of a matrix formed from a table of data
- WAEC Prep: Stating the order of a product before working it out
- WAEC Prep: Reading the order of a matrix as rows by columns
- Practice: The condition on the orders for a product to exist
- WAEC Prep: Evaluating A squared minus 2A
- WAEC Prep: A matrix expression with brackets to expand
- WAEC Prep: Evaluating 2A plus 3B for two given matrices
- Practice: Evaluating AB minus BA
- WAEC Prep: The determinant of a matrix with a row of noughts
- WAEC Prep: The determinant of the identity matrix
- WAEC Prep: The determinant of a 2 by 2 matrix as ad minus bc
- WAEC Prep: The determinant of a 3 by 3 matrix with two equal rows
- WAEC Prep: Expanding a 3 by 3 determinant down a column
- WAEC Prep: Expanding a 3 by 3 determinant along the first row
- Practice: The sign pattern of the cofactors in a 3 by 3 determinant
- WAEC Prep: When Cramer's rule is quicker than elimination
- WAEC Prep: Cramer's rule applied to a pair with fractional coefficients
- WAEC Prep: Setting out the coefficient determinant for two simultaneous equations
- Spot My Mistake: Forming the determinant Dx by replacing the first column
- Explain Back: The order of the answer when two matrices are multiplied
- Explain Back: Solving a matrix equation containing both a sum and a product
- Explain Back: The determinant of a product as the product of the determinants
- Worked Example: A singular matrix, and why it has no inverse
- Learn: Deciding whether a stated matrix calculation is possible at all
- Learn: The effect on a determinant of swapping two rows
- Worked Example: Evaluating a 3 by 3 determinant by the rule of Sarrus
Revisit from three weeks ago - Week 13
- Quiz: Arranging the equations in standard form before applying Cramer's rule
- Quiz: Checking an inverse by multiplying to get the identity
- WAEC Prep: The determinant Dz in a system of three unknowns
- WAEC Prep: What Cramer's rule gives when the system is inconsistent
- WAEC Prep: Setting out Cramer's rule for three simultaneous equations
- WAEC Prep: The cofactor of a corner entry
- WAEC Prep: Why the cofactor signs alternate
- WAEC Prep: The minor of an entry of a 3 by 3 matrix
- Practice: Turning a minor into a cofactor with the sign rule
- WAEC Prep: The adjoint of the identity matrix
- WAEC Prep: The order in which the cofactors are transposed
- WAEC Prep: The adjoint as the transpose of the matrix of cofactors
- WAEC Prep: The inverse of a 2 by 2 matrix with a fractional determinant
- WAEC Prep: The inverse of an inverse
- WAEC Prep: The inverse of a 2 by 2 matrix as one over the determinant times the adjoint
- Practice: Why a 2 by 2 matrix with determinant nought cannot be inverted
- WAEC Prep: How long the inverse of a 3 by 3 matrix takes, and how to shorten it
- WAEC Prep: Checking the determinant before attempting a 3 by 3 inverse
- WAEC Prep: The inverse of a 3 by 3 matrix from its adjoint and its determinant
- Spot My Mistake: Checking a Cramer's rule solution by substituting back
- Spot My Mistake: Checking cofactors by expanding the determinant along a second row
- Spot My Mistake: Finding the adjoint of a 3 by 3 matrix
- Spot my mistake: Negating the leading diagonal and swapping the other two entries when inverting a 2 by 2 matrix
- Explain Back: What a coefficient determinant of nought says about the system
- Explain Back: Writing out the full matrix of cofactors
- Explain Back: Finding the adjoint of a 2 by 2 matrix quickly
- Worked Example: Forming the determinant Dy by replacing the second column
- Learn: Solving two simultaneous equations by Cramer's rule
- Learn: Swapping and negating the entries when inverting a 2 by 2 matrix
Revisit from three weeks ago - Week 14
- Quiz: Checking a 3 by 3 inverse by multiplication
- Quiz: Solving three simultaneous equations with the inverse matrix
- Quiz: Reading the wording of a question to decide which distribution applies
- Quiz: Writing a two-dimensional vector as a column
- Quiz: The direction cosines of a vector in three dimensions
- WAEC Prep: Simultaneous equations with fractional coefficients solved by matrix
- WAEC Prep: Reading the solution off the product of the inverse and the constants
- WAEC Prep: Writing two simultaneous equations as a single matrix equation
- Practice: What a singular coefficient matrix says about the equations
- WAEC Prep: A three-unknown system arising from a word problem
- WAEC Prep: When the matrix method is worth the work for three unknowns
- WAEC Prep: Writing three simultaneous equations as a single matrix equation
- Practice: Comparing the inverse method with elimination for three equations
- WAEC Prep: A particle, and what the word assumes
- WAEC Prep: A rigid body, and what the word assumes
- WAEC Prep: Mass, and how it differs from weight
- WAEC Prep: The zero vector, and what it means
- WAEC Prep: Two vectors that are equal but drawn in different places
- WAEC Prep: Writing a two-dimensional vector in i and j form
- Practice: The components of a vector given its magnitude and direction
- WAEC Prep: The position of a point in space written as a vector
- WAEC Prep: A three-dimensional vector parallel to another
- WAEC Prep: Writing a three-dimensional vector in i, j and k form
- WAEC Prep: The magnitude of a vector after multiplication by a scalar
- WAEC Prep: A vector of magnitude one in three dimensions
- WAEC Prep: The magnitude of a two-dimensional vector
- Spot My Mistake: Why the inverse must be applied on the correct side
- Spot My Mistake: Drawing a vector as a directed line segment
- Spot my mistake: Assuming multiplying a vector by a negative scalar leaves its magnitude negative
- Explain Back: The inverse of a diagonal matrix
- Explain Back: Solving two simultaneous equations by multiplying by the inverse
- Learn: The order of the steps in inverting a 3 by 3 matrix
- Worked Example: Checking the solution of three equations by substitution
- Worked Example: The continuity correction when approximating a discrete distribution
- Learn: The coordinates of a point in three dimensions
- Worked Example: The distance between two points in three dimensions
Revisit from three weeks ago - Week 15
- Quiz: The direction of a vector as an angle with the x-axis
- Practice: The magnitude of the sum of two vectors
- WAEC Prep: The unit vector in the direction of a displacement
- WAEC Prep: Using a unit vector to write a force of given size
- WAEC Prep: A unit vector, and what makes it a unit
- Practice: A vector of stated magnitude in a given direction
- WAEC Prep: The position vector of a point dividing a line in a ratio
- WAEC Prep: The distance between two points from their position vectors
- WAEC Prep: The position vector of a point relative to the origin
- WAEC Prep: The parallelogram law of vector addition
- WAEC Prep: Adding three vectors in component form
- WAEC Prep: Adding two vectors given in component form
- WAEC Prep: The dot product of a vector with itself
- WAEC Prep: The dot product used to find the work done by a force
- WAEC Prep: The dot product worked out from components
- Practice: The dot product as the magnitudes times the cosine of the angle
- WAEC Prep: The angle between two three-dimensional vectors
- WAEC Prep: Showing a triangle is right-angled using the dot product
- WAEC Prep: Finding the angle between two vectors from their dot product
- Spot my mistake: Treating any vector with small components as a unit vector
- Spot My Mistake: The vector from A to B as the difference of two position vectors
- Spot my mistake: Treating the dot product of two vectors as another vector
- Explain Back: The dot product of two perpendicular vectors
- Learn: The magnitude of a three-dimensional vector
- Worked Example: Finding the unit vector in the direction of a given vector
- Learn: The unit vectors i, j and k
- Worked Example: The position vector of the midpoint of AB
- Worked Example: The triangle law of vector addition
- Learn: Subtracting one vector from another
- Worked Example: Testing two vectors for perpendicularity
Revisit from three weeks ago - Week 16
- Quiz: The vector equation of the line through two given points
- Quiz: The product of a complex number and its conjugate
- Quiz: The argument of a complex number
- Quiz: Writing a complex number in polar form
- WAEC Prep: The parameter in the vector equation of a line
- WAEC Prep: Converting the vector equation of a line into Cartesian form
- WAEC Prep: The vector equation of a line through a point in a given direction
- Practice: Deciding whether a point lies on a given line
- WAEC Prep: Three forces in equilibrium written as vectors
- WAEC Prep: The resultant force on a particle given in i and j form
- WAEC Prep: Writing a force as a vector in i and j form
- WAEC Prep: Why the square root of a negative number needs i
- WAEC Prep: Writing the square root of minus nine in terms of i
- WAEC Prep: The imaginary unit i, and why i squared is minus one
- WAEC Prep: The conjugate of a purely imaginary number
- WAEC Prep: The conjugate of a product of two complex numbers
- WAEC Prep: The conjugate of a complex number, and how it is written
- Practice: The sum of a complex number and its conjugate
- WAEC Prep: The modulus of a product of two complex numbers
- WAEC Prep: The argument measured in radians
- WAEC Prep: The modulus and the argument taken together
- Practice: The argument of a complex number in the second or third quadrant
- WAEC Prep: The conjugate shown on an Argand diagram
- WAEC Prep: The locus of a complex number equidistant from two points
- WAEC Prep: Plotting a complex number as a point on the Argand plane
- Practice: The addition of complex numbers shown on an Argand diagram
- WAEC Prep: The polar form written as r cos theta plus i r sin theta
- WAEC Prep: Multiplying two complex numbers in polar form
- WAEC Prep: Converting a complex number from polar form into the form a plus bi using tables
- Practice: Converting a complex number from polar form back to a plus bi
- WAEC Prep: De Moivre's theorem used with a negative index
- WAEC Prep: Expanding cos of 3 theta using de Moivre's theorem
- WAEC Prep: The statement of de Moivre's theorem
- WAEC Prep: The square roots of a complex number
- WAEC Prep: The arguments of the n roots, spaced equally round the circle
- WAEC Prep: The n nth roots of a complex number
- Spot My Mistake: The equilibrium condition written in vector form
- Spot My Mistake: Deriving a multiple angle formula from de Moivre's theorem
- Explain Back: The resultant of forces given as vectors
- Explain Back: The locus of a complex number at a fixed distance from a point
- Explain Back: Raising a complex number to a power by de Moivre's theorem
- Learn: The angle between a vector and one of the axes
- Worked Example: Squaring a complex number
Revisit from three weeks ago - Week 17
- Quiz: The roots of a complex number shown on an Argand diagram
- WAEC Prep: The sum and product of a pair of complex roots
- WAEC Prep: A quadratic whose roots are 2 plus i and 2 minus i
- WAEC Prep: Solving a quadratic whose discriminant is negative
- Practice: Why complex roots of a real quadratic come in conjugate pairs
- WAEC Prep: Differentiating a constant from first principles
- WAEC Prep: Why the limit is taken as h tends to nought
- WAEC Prep: The definition of the derivative as a limit
- Practice: Differentiating x squared from first principles
- WAEC Prep: Differentiating a polynomial with a fractional index
- WAEC Prep: Differentiating a sum of a polynomial and a trigonometric term
- WAEC Prep: Differentiating a polynomial term by term
- Practice: Differentiating sin x and cos x
- WAEC Prep: The product rule applied to three factors
- WAEC Prep: Simplifying the answer after using the quotient rule
- WAEC Prep: The product rule for differentiation
- WAEC Prep: The chain rule applied twice in one problem
- WAEC Prep: Differentiating the square root of a function
- WAEC Prep: The chain rule for a function of a function
- WAEC Prep: A stationary point at which the second derivative is nought
- WAEC Prep: The greatest and least values of a function in a closed interval
- WAEC Prep: Finding the stationary points of a curve
- WAEC Prep: The indefinite integral, and its notation
- WAEC Prep: Integrating a derivative to recover the original function
- WAEC Prep: Why integration undoes differentiation
- WAEC Prep: Integrating a bracket that must be expanded first
- WAEC Prep: Integrating a polynomial divided by x
- WAEC Prep: Integrating a power of x
- Spot My Mistake: Classifying a stationary point by the second derivative
- Spot My Mistake: Finding the constant of integration from a given point
- Explain Back: Forming a quadratic equation with given complex roots
- Explain Back: Differentiating one over x from first principles
- Explain Back: Differentiating tan x
- Explain Back: The constant of integration, and why it appears
- Learn: The cube roots of unity
- Learn: Choosing between the product rule and the quotient rule
- Learn: Differentiating sin of a multiple of x
- Worked Example: Classifying a stationary point by the sign of the gradient either side
Revisit from three weeks ago - Week 18
- Quiz me on unit vectors
- Quiz me on modulus and argument of complex numbers
- Show all working: vector geometry question
- WAEC past-question practice: vectors #1
- WAEC past-question practice: vectors #2
- WAEC past-question practice: vectors #3
- WAEC past-question practice: vectors #4
- Complex number arithmetic practice #1
- Complex number arithmetic practice #2
- Complex number arithmetic practice #3
- Complex number arithmetic practice #4
- Complex number arithmetic practice #5
- Solve a quadratic equation with complex roots #1
- Solve a quadratic equation with complex roots #2
- Solve a quadratic equation with complex roots #3
- Show all working: complex number equation question
- WAEC past-question practice: complex numbers #1
- WAEC past-question practice: complex numbers #2
- WAEC past-question practice: complex numbers #3
- WAEC past-question practice: complex numbers #4
- Differentiation practice question #1
- Differentiation practice question #2
- Differentiation practice question #3
- Differentiation practice question #4
- Differentiation practice question #5
- Integration practice question #1
- Integration practice question #2
- Integration practice question #3
- Integration practice question #4
- Spot my mistake finding a vector magnitude
- Explain parallel and perpendicular vectors
- Vectors and simple geometry proof
- Modulus, argument and the Argand diagram #1
- Modulus, argument and the Argand diagram #2
- Modulus, argument and the Argand diagram #3
- Modulus, argument and the Argand diagram #4
- Explain why a complex number times its conjugate is real
- Explain equal complex numbers
- Find and classify stationary points #1
- Find and classify stationary points #2
- Find and classify stationary points #3
- Teach me 3D vectors from scratch
- Teach me complex numbers from scratch
Revisit from three weeks ago
Matrices
↑ Jump to week- Week 19
- Quiz: Why the power rule fails for the integral of one over x
- Quiz: Using the initial conditions to fix the constants in a kinematics problem
- WAEC Prep: A definite integral evaluated with the limits reversed
- WAEC Prep: The area under a straight line checked by a trapezium
- WAEC Prep: Evaluating a definite integral between two limits
- WAEC Prep: The distance travelled in the first five seconds
- WAEC Prep: The starting velocity recovered as a constant of integration
- WAEC Prep: Finding velocity by integrating acceleration
- WAEC Prep: The rate at which the height of a liquid rises in a cone
- WAEC Prep: A small percentage change estimated by calculus
- WAEC Prep: The derivative read as a rate of change
- Spot my mistake: Applying the power rule to the integral of one over x
- Spot my mistake: Reporting an area computed below the x-axis as a negative distance travelled
- Spot my mistake: Leaving out the constant of integration when velocity is found from acceleration
- Learn: Integrating a polynomial term by term
- Worked Example: The area between a curve and the x-axis
- Learn: An area lying below the x-axis, and its sign
- Learn: Finding displacement by integrating velocity
Revisit from three weeks ago - Week 20
- Quiz: The zero matrix as the identity for matrix addition
- Identify symmetric matrices practice
- Show all working: adding three matrices in one question
- WAEC past-question practice: matrices #1
- WAEC past-question practice: matrices #2
- WAEC past-question practice: matrices #3
- WAEC past-question practice: matrices #4
- WAEC past-question practice: matrices #5
- WAEC past-question practice: matrices #6
- WAEC past-question practice: matrices #7
- Spot my mistake in a matrix transpose
- Spot my mistake in a determinant expansion
- Scalar multiplication of a matrix #1
- Scalar multiplication of a matrix #2
- Scalar multiplication of a matrix #3
- Scalar multiplication of a matrix #4
- Scalar multiplication of a matrix #5
- Explain pre-multiplication and post-multiplication
- Why is matrix multiplication not commutative
- Find the transpose of a 3x3 matrix
- Explain the identity matrix
- Explain the zero matrix in matrix algebra
- Order of matrices before multiplying
- Multi-part matrix question: add then multiply
- Evaluate a 2x2 determinant #1
- Evaluate a 2x2 determinant #2
- Evaluate a 2x2 determinant #3
- Evaluate a 2x2 determinant #4
- Evaluate a 2x2 determinant #5
- Evaluate a 2x2 determinant #6
- Evaluate a 3x3 determinant by expansion #1
- Evaluate a 3x3 determinant by expansion #2
- Evaluate a 3x3 determinant by expansion #3
- Evaluate a 3x3 determinant by expansion #4
- Evaluate a 3x3 determinant by expansion #5
- Explain the rule of signs for determinants
- Multiply matrices step by step #1
- Multiply matrices step by step #2
- Multiply matrices step by step #3
- Multiply matrices step by step #4
- Multiply matrices step by step #5
- Multiply matrices step by step #6
Revisit from three weeks ago - Week 21
- Quiz me on 2x2 and 3x3 determinants together
- Quiz me on units of force
- Show all working: evaluate a 3x3 determinant
- Solve simultaneous equations with Cramer's rule (order 2) #1
- Solve simultaneous equations with Cramer's rule (order 2) #2
- Solve simultaneous equations with Cramer's rule (order 2) #3
- Solve simultaneous equations with Cramer's rule (order 2) #4
- Solve a 3x3 simultaneous equation system with Cramer's rule #1
- Solve a 3x3 simultaneous equation system with Cramer's rule #2
- Solve a 3x3 simultaneous equation system with Cramer's rule #3
- WAEC past-question practice: determinants #1
- WAEC past-question practice: determinants #2
- WAEC past-question practice: determinants #3
- WAEC past-question practice: determinants #4
- WAEC past-question practice: determinants #5
- Show all working: the determinant of a three by three matrix
- Solve simultaneous equations using an inverse matrix #1
- Solve simultaneous equations using an inverse matrix #2
- Solve simultaneous equations using an inverse matrix #3
- Solve a 3x3 system using the inverse matrix method #1
- Solve a 3x3 system using the inverse matrix method #2
- WAEC past-question practice: inverse matrices #1
- WAEC past-question practice: inverse matrices #2
- WAEC past-question practice: inverse matrices #3
- WAEC past-question practice: inverse matrices #4
- Spot my mistake in finding a matrix inverse
- Compare expanding a determinant along different rows
- Find the inverse of a 2x2 matrix #1
- Find the inverse of a 2x2 matrix #2
- Find the inverse of a 2x2 matrix #3
- Find the inverse of a 2x2 matrix #4
- Find the inverse of a 2x2 matrix #5
- Find the inverse of a 3x3 matrix using cofactors #1
- Find the inverse of a 3x3 matrix using cofactors #2
- Find the inverse of a 3x3 matrix using cofactors #3
- Explain the adjoint of a matrix
- Explain why a matrix with zero determinant has no inverse
- Verify A times A-inverse equals the identity
- Define mass, force and weight with correct units
- Define particle and rigid body in mechanics
- Worked Example: The identity that A times its adjoint gives the determinant times the identity
Revisit from three weeks ago - Week 22
- Quiz me on equilibrium conditions
- Solve a three coplanar forces in equilibrium problem #1
- Solve a three coplanar forces in equilibrium problem #2
- Solve a three coplanar forces in equilibrium problem #3
- Solve a friction and forces of equilibrium problem #1
- Solve a friction and forces of equilibrium problem #2
- Solve a friction and forces of equilibrium problem #3
- Show all working: moments and equilibrium question
- WAEC past-question practice: equilibrium #1
- WAEC past-question practice: equilibrium #2
- WAEC past-question practice: equilibrium #3
- WAEC past-question practice: equilibrium #4
- Scalar (dot) and vector (cross) product practice #1
- Scalar (dot) and vector (cross) product practice #2
- Scalar (dot) and vector (cross) product practice #3
- Scalar (dot) and vector (cross) product practice #4
- Spot my mistake taking moments about a pivot
- Vector addition, subtraction and magnitude #1
- Vector addition, subtraction and magnitude #2
- Vector addition, subtraction and magnitude #3
- Vector addition, subtraction and magnitude #4
- Vector addition, subtraction and magnitude #5
- Position vectors and displacement between points #1
- Position vectors and displacement between points #2
- Position vectors and displacement between points #3
- Position vectors and displacement between points #4
- Explain the difference between scalar and vector product
- Teach me equilibrium of forces from scratch
Revisit from three weeks ago - Week 23
- Quiz me on rates of change problems
- Quiz me: matrices, determinants and inverses mixed
- Quiz me: vectors and complex numbers mixed
- Quiz me: forces and equilibrium mixed
- Quiz me: calculus differentiation rules
- Show all working: differentiation and stationary points question
- WAEC past-question practice: calculus #1
- WAEC past-question practice: calculus #2
- WAEC past-question practice: calculus #3
- WAEC past-question practice: calculus #4
- Kinematics: equations of motion practice #1
- Kinematics: equations of motion practice #2
- Kinematics: equations of motion practice #3
- Kinematics: equations of motion practice #4
- Kinematics: equations of motion practice #5
- Show all working: mixed matrices and determinants exam question
- Show all working: vectors and forces combined question
- Show all working: complex numbers and quadratics combined
- Show all working: calculus and stationary points multi-part
- Spot my mistake using the product rule
- Spot my mistake in a Cramer's rule solution
- Spot my mistake finding a resultant force
- Spot my mistake in a Poisson probability calculation
- Spot my mistake integrating a polynomial
- Spot my mistake finding a vector cross product
- Spot my mistake solving a moments problem
- Explain the difference between differentiation and integration
- Explain finding area under a curve using integration
- Projectile motion problem #1
- Projectile motion problem #2
- Projectile motion problem #3
- Projectile motion problem #4
- Teach me the chain rule with an example
- Worked example: 3x3 matrix multiplication
- Worked example: resultant of two forces
- Worked example: binomial distribution problem
- Worked example: differentiating using the quotient rule
- Worked example: finding the matrix of cofactors and the adjugate
- Worked example: projectile range and time of flight
- Worked example: finding the argument in the correct quadrant
Revisit from three weeks ago
Statistics: Probability Distributions
↑ Jump to week- Week 24
- Quiz me on logic symbols and their meanings
- Quiz me on nPr versus nCr notation
- Structured question: equation of a line through two points
- Structured question: perpendicular bisector of a line segment
- Structured question: equation of a circle from centre and a point
- Structured question: intersection of a line and a circle
- Structured question: prove a trigonometric identity
- Structured question: solve a trig equation for all solutions in range
- Solve for command word: quadratic trigonometric equation
- Structured question: find the nth term and a specific term of an AP
- Structured question: geometric progression with two given terms
- WAEC exam question: mixed AP and GP problem
- Structured question: the constant term in a binomial expansion
- Structured question: use binomial expansion to approximate a value
- Structured question: find the equation of a tangent to a curve
- Structured question: find the equation of a normal to a curve
- Structured question: find and classify all stationary points
- WAEC exam question: rate of change problem
- Give me a full Additional Maths style Paper 2 structured question
- Timed 20-minute Further Maths exam practice
- WAEC exam question: the converse, the inverse and the contrapositive
- Structured question: converse, inverse, and contrapositive together
- Structured question: identify a tautology using a truth table
- WAEC exam question: predicate logic quantifiers
- Structured question: two compound statements compared row by row
- WAEC exam question: testing an argument with a truth table
- Structured question: arrangements with a restriction
- Structured question: choosing a committee with a condition
- Structured question: permutations with repeated letters
- WAEC exam question: circular permutation with a condition
- Structured question: permutation versus combination in one paper
- Structured question: probability using combinations, exam style
- WAEC exam question: arranging books on a shelf with a rule
- Spot my mistake in a Pascal's row truth table
- Spot my mistake in a combination calculation
- Show that command word: three points are collinear
- Find the equation command word: circle through the origin
- Show that command word: line is tangent to a circle
- Show that command word: trigonometric identity
- Non-calculator paper: exact trigonometric values
- Show that command word: sum of an arithmetic progression
- Find the sum to infinity command word
- Show that command word: binomial coefficient identity
- Find the coefficient command word: binomial expansion
- Show that command word: stationary point classification
- Find the range of values command word: increasing function
- Give me a full Additional Maths style Paper 1 question set
- Show that command word: biconditional equivalence
- Determine command word: validity of an argument
- Explain back: converse and contrapositive
- Show all your working command word: permutation problem
- Find the number of ways command word: combinations
- Explain back: difference between permutations and combinations
- Worked Example: Finding the first term and the common difference from two given terms
Revisit from three weeks ago - Week 25
- Quiz: Why mass stays the same wherever the body is taken
- Quiz: The situations a Poisson distribution models
- WAEC Prep: The difference between a kilogram and a kilogram weight
- WAEC Prep: The weight of a body on the moon compared with on earth
- WAEC Prep: The newton, and how it is defined
- WAEC Prep: A probability distribution shown as a bar chart
- WAEC Prep: The variance of a discrete random variable
- WAEC Prep: A random variable, and the values it may take
- WAEC Prep: Expanding p plus q, all to the power n, as a probability model
- WAEC Prep: The probabilities of nought, one and two successes read off an expansion
- WAEC Prep: The binomial expansion of a plus b all to the power n
- Practice: Using a binomial expansion to approximate a number
- WAEC Prep: The probability of at most r successes
- WAEC Prep: The most likely number of successes in n trials
- WAEC Prep: The four conditions a binomial situation must satisfy
- Practice: The probability of at least one success
- WAEC Prep: The probability of at least one event in a Poisson distribution
- WAEC Prep: A Poisson distribution taken over a different length of interval
- WAEC Prep: The Poisson formula, and what lambda stands for
- Practice: The probability of exactly r events in a Poisson distribution
- Spot my mistake: Treating the kilogram and the kilogram weight as the same unit
- Spot My Mistake: When Lami's theorem may be used, and when it may not
- Spot My Mistake: Why the probabilities in a distribution must sum to one
- Spot My Mistake: The probability of exactly r successes in n trials
- Explain Back: A probability distribution table for a discrete random variable
- Explain Back: The binomial probability formula
- Worked Example: The units of mass, and the SI unit
- Learn: Inertia, and what mass actually measures
- Worked Example: The weight of a body as mass times g
- Worked Example: Drawing the polygon of forces to scale
- Worked Example: Matching each force to the sine of the angle opposite it
- Learn: Finding two unknown forces by Lami's theorem
- Learn: The expectation of a discrete random variable
Revisit from three weeks ago - Week 26
- Quiz: The standard deviation of a Poisson distribution
- Quiz: The area under the normal curve, and what it represents
- Quiz: The probability in the upper tail of a normal distribution
- WAEC Prep: Comparing the mean with the variance of a binomial distribution
- WAEC Prep: The expected number of successes in a practical problem
- WAEC Prep: The mean of a binomial distribution as np
- Practice: The variance of a binomial distribution as npq
- WAEC Prep: The variance of a Poisson distribution over two intervals
- WAEC Prep: Using the mean of a Poisson distribution to find a probability
- WAEC Prep: Why the mean and the variance of a Poisson distribution are equal
- WAEC Prep: The points of inflexion of the normal curve
- WAEC Prep: Why the total area under the normal curve is one
- WAEC Prep: The shape and the symmetry of the normal curve
- WAEC Prep: Comparing two values from different normal distributions by z-score
- WAEC Prep: The z-score of a value that lies below the mean
- WAEC Prep: The standard normal distribution, with mean nought and variance one
- WAEC Prep: The probability that a normal variable exceeds the mean by more than one standard deviation
- WAEC Prep: Finding the mean or the standard deviation from a given probability
- WAEC Prep: The probability that a normal variable lies below a given value
- Practice: Using the normal tables when the z-score is negative
- WAEC Prep: When a Poisson distribution approximates a binomial one
- WAEC Prep: A question whose wording hides which distribution is meant
- WAEC Prep: Choosing between a binomial and a Poisson model
- Spot my mistake: Treating the variance of a binomial distribution as its standard deviation
- Spot my mistake: Assuming the normal curve meets the horizontal axis at its points of inflexion
- Spot My Mistake: The proportion of a normal distribution within one, two and three standard deviations
- Spot My Mistake: Reading a probability from the standard normal table
- Spot My Mistake: When the normal distribution may approximate the binomial
- Explain Back: Finding n and p from a given mean and variance
- Explain Back: Testing whether data are Poisson by comparing mean and variance
- Explain Back: Converting a value to a z-score
- Worked Example: The probability of no events at all in a Poisson distribution
- Learn: The standard deviation of a binomial distribution
- Learn: Finding lambda from a stated mean
- Learn: The mean, median and mode of a normal distribution
- Worked Example: Finding a value from a given probability by working backwards
- Learn: The probability that a normal variable lies between two values
Revisit from three weeks ago - Week 27
- Quiz: The modal class of a grouped frequency distribution
- WAEC Prep: Positive, negative and zero correlation
- WAEC Prep: Using a regression line to estimate a value
- WAEC Prep: The scatter diagram, and what it shows about correlation
- Practice: The rank correlation coefficient
- Spot my mistake: Assuming moments may only be taken about the pivot and never about another point
- Explain Back: The line of best fit, and the regression line
- Add two matrices, 2x2 #1
- Subtract two matrices, 2x2 #2
- Add two matrices, 2x2 #3
- Subtract two matrices, 2x2 #4
- Add two matrices, 3x3 #5
- Subtract two matrices, 3x3 #6
- Add two matrices, 2x2 #7
- Subtract two matrices, 3x3 #8
- Learn: The median of a set of ordered data
Revisit from three weeks ago - Week 28
- Quiz me on the symmetry of the normal distribution curve
- WAEC past-question practice: coplanar forces #1
- WAEC past-question practice: coplanar forces #2
- WAEC past-question practice: coplanar forces #3
- WAEC past-question practice: coplanar forces #4
- Solve a binomial probability problem #1
- Solve a binomial probability problem #2
- Solve a binomial probability problem #3
- Solve a binomial probability problem #4
- Solve a binomial probability problem #5
- Solve a Poisson distribution problem #1
- Solve a Poisson distribution problem #2
- Solve a Poisson distribution problem #3
- Solve a Poisson distribution problem #4
- Show all working: combined binomial and normal question
- WAEC past-question practice: probability distributions #1
- WAEC past-question practice: probability distributions #2
- WAEC past-question practice: probability distributions #3
- WAEC past-question practice: probability distributions #4
- WAEC past-question practice: probability distributions #5
- Spot my mistake in a binomial probability question
- Spot my mistake in standardising a normal variable
- Explain the properties of the normal distribution curve
- Explain standard deviation and the normal curve
- Use the normal distribution function to find a probability #1
- Use the normal distribution function to find a probability #2
- Use the normal distribution function to find a probability #3
- Use the normal distribution function to find a probability #4
- Explain the meaning of a probability distribution
- Explain when to use binomial versus Poisson
- Find the moment of a force about a point #1
- Find the moment of a force about a point #2
- Find the moment of a force about a point #3
- Find the moment of a force about a point #4
- Explain the conditions for equilibrium of a rigid body
- Explain the polygon of forces
- Explain friction and the coefficient of friction
- Teach me composition of coplanar forces
Revisit from three weeks ago - Week 29
- Quiz: The number of arrangements of r objects chosen from n
- Quiz: Choosing a committee of r people from n
- Quiz: The semi-interquartile range
- Practice: Arrangements when some of the objects are identical
- WAEC Prep: Circular permutations of n objects round a table
- Show All Working: Arrangements of the letters of a given word
- Practice: Choosing a committee that must contain one particular person
- WAEC Prep: Pascal's triangle and the combination numbers in it
- Show All Working: Counting the handshakes, or the lines joining n points
- Practice: The mean deviation from the mean
- WAEC Prep: The standard deviation of grouped data using an assumed mean
- Show All Working: The effect on the standard deviation of adding a constant to every value
- Spot My Mistake: Arrangements in which two particular objects must be kept apart
- Common Mistakes: Arrangements restricted at the first or the last place
- Spot My Mistake: The identity that n choose r equals n choose n minus r
- Common Mistakes: Selections in which at least one of a kind must appear
- Spot My Mistake: The standard deviation of an ungrouped set of data
- Common Mistakes: The coefficient of variation, for comparing two sets
- Explain Back: Arrangements in which two particular objects must stay together
- Final Revision: Counting the number plates or codes that can be formed under a rule
- Explain Back: Choosing a committee with a stated number from each of two groups
- Final Revision: Combinations used with the addition and multiplication principles
- Explain Back: The variance of an ungrouped set of data
- Final Revision: Reading the quartiles off a cumulative frequency curve
- Learn: The meaning of n factorial, and how to evaluate it
- Worked Example: The number of arrangements of n different objects in a row
- Learn: The meaning of n choose r, and how to evaluate it
- Worked Example: The difference between a permutation and a combination
- Learn: The range, and why it is a weak measure of spread
- Worked Example: The interquartile range of a set of ordered data
Revisit from three weeks ago
Mechanics: Statics
↑ Jump to week- Week 30
- Quiz: Recovering the original force from its two components
- Quiz: The magnitude of the resultant of several coplanar forces
- Practice: The value of g used in Nigerian examination papers
- WAEC Prep: The resultant of two equal forces at sixty degrees
- WAEC Prep: Resolving before compounding several forces
- WAEC Prep: The resultant of two forces acting at the same point
- Practice: The parallelogram law of forces
- WAEC Prep: The angle the resultant makes with the larger of two forces
- WAEC Prep: Two forces acting in opposite directions
- WAEC Prep: The magnitude of the resultant of two forces inclined at an angle
- WAEC Prep: The component of a force at thirty degrees to the horizontal
- WAEC Prep: Why one component vanishes when the direction is chosen well
- WAEC Prep: Resolving a force into horizontal and vertical components
- Practice: Choosing the most convenient pair of directions to resolve along
- WAEC Prep: A table of components used to find a resultant
- WAEC Prep: The resultant of four concurrent forces
- WAEC Prep: Summing the components of several concurrent forces
- Practice: The direction of the resultant of several coplanar forces
- WAEC Prep: The forces on a body resting on a rough horizontal plane
- WAEC Prep: The forces on a body suspended by a single string
- WAEC Prep: Drawing a clear force diagram for a body at rest
- WAEC Prep: The triangle of forces used together with the cosine rule
- WAEC Prep: Labelling the triangle of forces from the space diagram
- WAEC Prep: The triangle of forces for three forces in equilibrium
- Spot My Mistake: Compounding three forces acting at one point
- Spot My Mistake: The greatest and the least resultant of two given forces
- Spot my mistake: Adding the magnitudes of concurrent forces to get the magnitude of their resultant
- Spot My Mistake: Drawing the triangle of forces to scale to find an unknown force
- Explain Back: The resultant of two forces acting in the same straight line
- Explain Back: Showing that a system of concurrent forces is in equilibrium
- Explain Back: The forces acting on a body hanging from two strings
- Learn: Converting a weight given in kilogram-force into newtons
- Worked Example: The direction of the resultant of two forces
- Learn: The resultant of two perpendicular forces
- Learn: The component of a force along a stated direction
- Worked Example: Marking the weight, the normal reaction and the friction on a diagram
- Learn: A space diagram and a force diagram drawn side by side
Revisit from three weeks ago - Week 31
- Quiz: The coefficient of friction between two surfaces
- WAEC Prep: A rigid body held in equilibrium by four forces
- WAEC Prep: The three unknowns a rigid body problem usually carries
- WAEC Prep: The two conditions for a rigid body to be in equilibrium
- Practice: Why the forces must sum to nought in two perpendicular directions
- WAEC Prep: The polygon of forces for five concurrent forces
- WAEC Prep: The resultant read off an unclosed polygon
- WAEC Prep: The polygon of forces for several forces in equilibrium
- Practice: Finding an unknown force from a closed polygon
- WAEC Prep: Why friction is called a self-adjusting force
- WAEC Prep: A body just about to slide on a horizontal surface
- WAEC Prep: Friction, and the direction in which it acts
- Practice: Limiting friction, and the greatest value it can reach
- WAEC Prep: The moment of a force about the foot of a pole
- WAEC Prep: The moment of a weight about a hinge
- WAEC Prep: Calculating a moment as force times perpendicular distance
- WAEC Prep: The resultant moment of a couple and a force together
- WAEC Prep: The moment of two equal and opposite parallel forces
- WAEC Prep: The total moment of two forces about the same point
- Spot My Mistake: Why friction does not depend on the area of contact
- Spot My Mistake: The moment of a force whose line of action is not perpendicular
- Explain Back: The order in which the forces are drawn round the triangle
- Explain Back: Why the total moment about any point must be nought
- Explain Back: The units of a moment, and why they are newton metres
- Worked Example: Using the sine rule on the triangle of forces
- Learn: Choosing the point about which to take moments
- Learn: What an unclosed polygon of forces tells you
- Worked Example: The moment of a force about a point on its own line of action
Revisit from three weeks ago - Week 32
- Quiz: The point about which the total moment of a set of forces vanishes
- Quiz: A rod resting against a wall, and the concurrence of its three forces
- Quiz: A rate of change problem about a vessel filling with water
- Quiz: Choosing which equation of motion to use
- Quiz: The time of flight of a projectile
- Quiz: Impulse as the change in momentum
- WAEC Prep: The triangle of forces and concurrence taken together
- WAEC Prep: Using concurrence to locate an unknown line of action
- WAEC Prep: Why three non-parallel forces in equilibrium must be concurrent
- WAEC Prep: A particle in equilibrium on a smooth inclined plane
- WAEC Prep: A knot held in equilibrium by three strings
- WAEC Prep: The conditions for a particle under three forces to be in equilibrium
- Practice: Resolving three forces in two perpendicular directions
- WAEC Prep: Lami's theorem derived from the triangle of forces
- WAEC Prep: A lamp hanging from two wires, solved by Lami's theorem
- WAEC Prep: The statement of Lami's theorem
- WAEC Prep: A body decelerating to rest
- WAEC Prep: The distance covered in the nth second
- WAEC Prep: The equations of motion for constant acceleration
- Practice: The sign convention for displacement, velocity and acceleration
- WAEC Prep: The time of flight of a body thrown vertically upwards
- WAEC Prep: A body dropped from a rising balloon
- WAEC Prep: A body dropped from rest under gravity
- Practice: A body thrown vertically upwards
- WAEC Prep: The greatest height reached by a projectile
- WAEC Prep: The angle of projection that gives the greatest range
- WAEC Prep: The horizontal and vertical components of a projectile's velocity
- Practice: The range of a projectile on level ground
- WAEC Prep: The impulse of a force acting for a short time
- WAEC Prep: Two bodies that coalesce on impact
- WAEC Prep: The momentum of a moving body
- Practice: The conservation of momentum in a collision
- WAEC Prep: A body pulled along a rough horizontal plane
- WAEC Prep: The normal reaction in a lift accelerating upwards
- WAEC Prep: Newton's first law, and what it says about equilibrium
- Spot My Mistake: Using concurrence to find the direction of an unknown third force
- Spot My Mistake: Finding two unknown tensions from three forces in equilibrium
- Spot my mistake: Using an equation of constant acceleration on a body whose acceleration is changing
- Explain Back: Locating the point of concurrence on a diagram
- Explain Back: A particle hanging from two strings set at different angles
- Explain Back: The greatest height reached by a body thrown upwards
- Worked Example: The total moment of three parallel forces about one end
- Learn: The algebraic sum of moments when the senses differ
- Learn: Connected rates of change through the chain rule
Revisit from three weeks ago - Week 33
- Quiz: Newton's third law, and identifying the pair of forces
- WAEC Prep: The work done against gravity in raising a body
- WAEC Prep: The power of an engine driving a car at constant speed
- WAEC Prep: The work done by a constant force
- WAEC Prep: How many terms of an arithmetic progression lie between two values
- WAEC Prep: An arithmetic progression whose terms are given in terms of n
- WAEC Prep: The nth term of an arithmetic progression
- Show all working: resultant of forces question
- WAEC Further Mathematics: calculus optimisation word problem
- WAEC Further Mathematics: mechanics word problem on a lift or vehicle
- WAEC Further Mathematics: probability distribution word problem
- Timed drill: five quick Further Maths WAEC-style questions
- Spot my mistake: Treating the action and the reaction of Newton's third law as forces acting on the same body
- Spot My Mistake: The sum of the first n terms of an arithmetic progression
- Spot my mistake resolving a force
- Find the resultant of two forces #1
- Find the resultant of two forces #2
- Find the resultant of two forces #3
- Find the resultant of two forces #4
- Find the resultant of two forces #5
- Resolve a force into horizontal and vertical components #1
- Resolve a force into horizontal and vertical components #2
- Resolve a force into horizontal and vertical components #3
- Resolve a force into horizontal and vertical components #4
- Find the resultant of several concurrent forces #1
- Find the resultant of several concurrent forces #2
- Find the resultant of several concurrent forces #3
- Explain the parallelogram of forces
- Explain the triangle of forces method
- WAEC Further Mathematics: vectors and geometry combined question
- WAEC Further Mathematics: complex numbers and Argand diagram question
- WAEC Further Mathematics: matrices in a real-world context
- Final revision: my three hardest Further Maths topics
- Learn: Newton's second law written as F equals ma
- Worked Example: Power as the rate of doing work
- Learn: The kinetic energy of a moving body
- Teach me the basics of force, mass and weight
- Teach me Cramer's rule from first principles
- Teach me how to find the writing the inverse of a two by two matrix with the adjugate
- Teach me resolving forces on an inclined plane
- Teach me the normal distribution standardisation formula
- Teach me integration by recognising the reverse of differentiation
Revisit from three weeks ago
Mechanics: Moments & Rigid Bodies
↑ Jump to week- Week 34
- Quiz me on kinematics equations of motion
- Show all working: the range and the greatest height of a projectile
- WAEC past-question practice: mechanics #1
- WAEC past-question practice: mechanics #2
- WAEC past-question practice: mechanics #3
- WAEC past-question practice: mechanics #4
- Practice under WAEC exam conditions
- Spot my mistake in a projectile motion question
- Explain the difference between speed and velocity in mechanics
- Explain time of flight and range in projectile motion
- Build my SS3 Further Maths WAEC revision timetable
- Rank my weakest Further Maths topics before WAEC
- Simulate a WAEC Further Mathematics theory paper
- Explain WAEC Further Mathematics marking scheme style
- Common WAEC Further Mathematics mistakes to avoid
- Last-minute WAEC Further Mathematics formula check
- WAEC Further Mathematics command words explained
- Time management strategy for the WAEC Further Maths paper
- Mixed-topic WAEC Further Mathematics past-question set
- Explain how to present working for full WAEC marks
- Diagnose my exam readiness for WAEC Further Maths
- Explain-back: why matrix multiplication order matters
- Explain-back: my understanding of Cramer's rule
- Explain-back: what a matrix inverse actually does
- Explain-back: normal distribution properties
- Explain-back: resultant versus resolution of forces
- Explain-back: complex conjugates
- Explain-back: the chain rule in calculus
- Explain-back: equilibrium of a rigid body
- Teach me the suvat equations of motion
Revisit from three weeks ago - Week 35
- Quiz: The principle of moments for a body in equilibrium
- Quiz: Relative velocity when the two motions are at right angles
- Quiz: The tension in the string joining two connected particles
- Quiz: The acceleration of a body sliding down a smooth incline
- Practice: A uniform beam resting on two supports
- WAEC Prep: A couple, and why its moment is the same about every point
- Show All Working: Replacing two parallel forces by a single resultant
- Practice: The triangle of velocities for a boat crossing a river
- WAEC Prep: The closest approach of two moving ships
- Show All Working: The time at which two bodies are nearest one another
- Practice: A particle on a smooth table connected to a hanging particle
- WAEC Prep: A car towing a trailer, and the force in the tow bar
- Show All Working: A lift, and the person standing in it
- Practice: The frictional force on a body on a rough incline
- WAEC Prep: Limiting equilibrium, with a body on the point of sliding
- Show All Working: The angle at which a body just begins to slide
- Spot My Mistake: The point at which a loaded beam will tip about its support
- Common Mistakes: The centre of gravity of a uniform lamina
- Spot My Mistake: An aeroplane in a cross-wind, and its resultant track
- Common Mistakes: The relative displacement of two bodies after a given time
- Spot My Mistake: Two particles on a pulley at the top of an inclined plane
- Common Mistakes: What becomes of the tension when the string goes slack
- Spot My Mistake: The least force needed to move a body up a rough incline
- Common Mistakes: The work done against friction on an incline
- Explain Back: Finding the reaction at each support of a loaded beam
- Final Revision: A ladder leaning against a wall, and the moments acting on it
- Explain Back: The course to steer so that a boat crosses straight over
- Final Revision: Interception, where one body must catch another
- Explain Back: A particle on a rough table connected to a hanging particle
- Final Revision: The distance connected particles travel before one reaches the floor
- Explain Back: The angle of friction, and the coefficient of friction
- Final Revision: The speed of a body when it reaches the foot of an incline
- Learn: The moment of a force about a point, and its units
- Worked Example: The sign convention for clockwise and anticlockwise moments
- Learn: The velocity of one body relative to another
- Worked Example: Relative velocity when two bodies move along the same line
- Learn: Two particles joined by a light inextensible string over a smooth pulley
- Worked Example: The common acceleration of two connected particles
- Learn: Resolving the weight along and perpendicular to an inclined plane
- Worked Example: The normal reaction on a body resting on a smooth incline
Revisit from three weeks ago
Mechanics: Dynamics
↑ Jump to week- Week 36
- Quiz: The centripetal force, and what supplies it in a given situation
- Quiz: The velocity of a body in simple harmonic motion at a given displacement
- Quiz: The extension of a spring under a given load
- Practice: A car rounding a bend on a level road
- WAEC Prep: The conical pendulum
- Show All Working: A body in a vertical circle, and the tension at the top
- Practice: The maximum speed and the maximum acceleration in simple harmonic motion
- WAEC Prep: The simple pendulum as an example of simple harmonic motion
- Show All Working: A mass on a spring performing simple harmonic motion
- Practice: The difference between an elastic string and a spring in compression
- WAEC Prep: Two elastic strings supporting one particle
- Show All Working: The work done in stretching an elastic string
- Spot My Mistake: A body on the end of a string moving in a horizontal circle
- Common Mistakes: The condition for a body to complete a vertical circle
- Spot My Mistake: The time taken to move between two points in simple harmonic motion
- Common Mistakes: The energy of a body in simple harmonic motion
- Spot My Mistake: A particle hanging in equilibrium on an elastic string
- Common Mistakes: A particle oscillating on the end of an elastic string
- Explain Back: A car on a banked track
- Final Revision: The period and the frequency of circular motion
- Explain Back: The displacement written as x equals a sin omega t
- Final Revision: Showing that a given motion is simple harmonic
- Explain Back: The elastic potential energy stored in a stretched string
- Final Revision: The greatest extension reached by a falling particle on an elastic string
- Learn: Angular velocity, and how it relates to linear speed
- Worked Example: The acceleration of a body moving round a circle at constant speed
- Learn: The defining equation of simple harmonic motion
- Worked Example: The amplitude, period and frequency of a simple harmonic motion
- Learn: Hooke's law, and what the modulus of elasticity means
- Worked Example: The tension in a stretched elastic string
Revisit from three weeks ago