The TMUA Method Library

You see: Measurements given to a stated accuracy, with a maximum or minimum of some combination of them wanted.

You do: For a quotient, the largest value takes the upper bound of the numerator with the lower bound of the denominator. For a difference, the largest takes the upper bound of the first with the lower bound of the second.

Trap: Using the upper bound of everything, which maximises a sum and a product but not a difference or a quotient.

Practise: No question drills this yet

You see: A count of arrangements, routes, selections or codes, with a restriction attached to one position.

You do: Deal with the restricted position first, multiply the choices stage by stage, and divide by the number of orderings if order does not matter.

Trap: Multiplying the stages when the restriction changes what is left. Fix the constrained slot before counting the free ones.

Practise: Mock 1 Paper 1 Q16 · Mock 1 Paper 2 Q13

You see: Numbers spanning several orders of magnitude, or options that differ by a factor of ten rather than by a digit.

You do: Round every quantity to one significant figure and to a power of ten, settle the exponent first, and only then decide the leading digits.

Trap: Carrying every digit through a long multiplication, then losing the answer to a single misplaced power of ten at the end.

Practise: No question drills this yet

You see: One quantity described as proportional to a power of another, often with one pair of values given.

You do: Write $y = kx^{n}$ or $y = k/x^{n}$, substitute the known pair to find $k$, and only then answer the question asked.

Trap: Scaling linearly when the relationship is not linear. Doubling the radius multiplies an area by four, not by two.

Practise: No question drills this yet

You see: A chain of increases and decreases, or an original value to be recovered from a final one.

You do: Turn each change into a multiplier and multiply them. To reverse a change, divide by its multiplier rather than applying the opposite percentage.

Trap: Adding the percentages. A rise of 20 percent followed by a fall of 20 percent is a multiplier of 0.96, not 1.

Practise: No question drills this yet

You see: A recurring decimal to be written as a fraction, or a fraction whose decimal expansion is asked about.

You do: Multiply by the power of ten matching the period, subtract the original, and solve. A fraction terminates exactly when its denominator in lowest terms has no prime factor other than 2 and 5.

Trap: Multiplying by the wrong power of ten, so the recurring blocks do not line up and the subtraction leaves a tail.

Practise: No question drills this yet

You see: A highest common factor, a lowest common multiple, a count of divisors, or a question about when a product becomes a perfect square.

You do: Write every number as a product of prime powers. HCF takes the smallest exponent of each prime, LCM the largest, and a perfect square is exactly a number whose every exponent is even.

Trap: Listing factors by hand. It works for 36 and fails for 3240, which is why the question chose 3240.

Practise: No question drills this yet

You see: A quantity split in a given ratio, or two ratios that share one term and have to be combined.

You do: Count the parts, find the value of one part, then rebuild whatever the question asked for. To combine ratios, scale each so the shared term matches.

Trap: Reading a part-to-part ratio as a fraction of the whole, so 2 : 3 becomes two thirds instead of two fifths.

Practise: No question drills this yet

You see: An expression stated to be true for all values of the variable, with unknown coefficients to be found.

You do: Compare coefficients of each power, or substitute values chosen to kill terms. Both are valid; substitution is faster when a factor vanishes at a nice number.

Trap: Treating an identity as an equation and solving for the variable, which answers a question nobody asked.

Practise: No question drills this yet

You see: A formula to be rearranged where the new subject appears more than once, often on both sides of a fraction.

You do: Clear denominators, bring every term containing the target to one side, factorise it out, then divide once.

Trap: Dividing before collecting, which leaves the target on both sides and forces a second rearrangement.

Practise: No question drills this yet

You see: A quotient of polynomials, or a sum of algebraic fractions with different denominators.

You do: Factorise every numerator and denominator completely first. Common factors then cancel on sight and the common denominator is usually already visible.

Trap: Cancelling a term rather than a factor. In (x + 2)/(x + 4) the twos and fours are not going anywhere.

Practise: No question drills this yet

You see: A fractional or negative exponent on a rational base, such as $\left(\frac{27}{8}\right)^{-2/3}$.

You do: Invert the base to clear the minus sign, then take the root before the power. The root of a perfect power is a small integer and nothing large is ever computed.

Trap: Taking the power first, so a two-digit base becomes a five-digit number that then has to be cube-rooted by hand.

Practise: Mock 1 Paper 1 Q1

You see: A surd in a denominator, especially a sum or difference of two surds.

You do: Multiply top and bottom by the conjugate of the denominator. The difference of two squares clears the root in one step.

Trap: Multiplying by the denominator itself rather than its conjugate, which leaves a cross term and a surd still downstairs.

Practise: No question drills this yet

You see: Two unknowns given only through symmetric combinations such as $x + y$ and $xy$, with another symmetric expression wanted.

You do: Express the target in terms of the sum and the product: $x^{2} + y^{2} = (x+y)^{2} - 2xy$ and $x^{3} + y^{3} = (x+y)^{3} - 3xy(x+y)$.

Trap: Solving the pair for x and y, which usually means a quadratic with ugly roots that cancel again at the end.

Practise: Mock 1 Paper 1 Q20

You see: An equation or inequality containing an absolute value, often with a linear expression inside.

You do: Either split into the two cases and check each solution back in the original, or square both sides when both are known to be non-negative.

Trap: Squaring without checking, which manufactures solutions that satisfy the squared equation and not the original.

Practise: Mock 1 Paper 1 Q15

You see: A minimum, a maximum, a range, or a proof that an expression is always positive.

You do: Write $a(x + p)^{2} + q$. The turning point is at $x = -p$, the extreme value is $q$, and the sign of $q$ settles positivity outright.

Trap: Differentiating to find the turning point of a quadratic. It works and it is slower, and it tells you nothing about the range.

Practise: No question drills this yet

You see: A quadratic inequality, or an inequality with the variable in a denominator.

You do: Bring everything to one side, factorise, and read the sign from the sketch. For a denominator, multiply by its square, which is positive and cannot flip the inequality.

Trap: Multiplying by the denominator itself. Its sign is unknown, so the inequality may need reversing and usually does for half the values.

Practise: No question drills this yet

You see: A straight line and a curve, with intersections, a tangency condition, or a count of solutions wanted.

You do: Substitute the linear equation into the quadratic one and read the discriminant of the result: two roots meet twice, one root is a tangent, none means they miss.

Trap: Solving the pair graphically or by elimination in the wrong direction, which turns one substitution into a system.

Practise: No question drills this yet

You see: A polynomial with an unknown coefficient, together with a stated factor or a stated remainder.

You do: The remainder on division by $(x - a)$ is $\mathrm{f}(a)$, so a stated factor gives $\mathrm{f}(a) = 0$ and a stated remainder gives $\mathrm{f}(a) = r$. One substitution replaces a long division.

Trap: Performing the algebraic division in full, which is four lines of work to reach a number one substitution already gave.

Practise: Mock 1 Paper 1 Q18

You see: A condition on the number of real roots, with a parameter in the coefficients.

You do: Solve the discriminant inequality, then ask separately whether every value it allows still gives a quadratic. A vanishing leading coefficient is a second condition, not a detail.

Trap: Stopping at $b^{2} - 4ac \gt 0$. The value that makes the equation linear usually sits inside the interval the discriminant allows.

Practise: Mock 1 Paper 1 Q2

You see: A sum to infinity, or a question asking for which values of a parameter a series converges.

You do: Use $S_\infty = \frac{a}{1-r}$ only after checking $|r| \lt 1$, and treat that inequality as part of the answer when a parameter is involved.

Trap: Quoting a sum to infinity for a series that diverges, or losing the second solution because only one of two candidate ratios was tested for convergence.

Practise: Mock 1 Paper 1 Q4

You see: A geometric progression given through two non-adjacent terms, or through a sum.

You do: Divide one term equation by the other so that $a$ cancels and a power of $r$ remains. Take the root, then recover $a$.

Trap: Forgetting that an even power of r has two roots, so a legitimate negative ratio is quietly dropped.

Practise: No question drills this yet

You see: A sequence defined by $x_{n+1} = \mathrm{f}(x_n)$ with a distant term or a long sum wanted.

You do: Compute the first several terms. These sequences are almost always periodic or eventually constant, and the period then answers any term number by taking a remainder.

Trap: Trying to find a closed form. The question chose a recurrence precisely because the closed form is unpleasant and the period is short.

Practise: No question drills this yet

You see: A single named coefficient wanted from a binomial power, often with a coefficient inside the bracket.

You do: Write the general term $\binom{n}{r} a^{n-r} b^{r}$, solve for the $r$ that gives the power you want, and evaluate that one term.

Trap: Expanding everything to reach the term wanted, and losing the powers of the internal coefficient on the way.

Practise: Mock 1 Paper 1 Q5

You see: A sum in sigma notation whose summand is a polynomial in the index, or a sum starting somewhere other than one.

You do: Split the sum term by term, use $\sum_{1}^{n} r = \frac{n(n+1)}{2}$, and handle a shifted start by subtracting the missing head from a full sum.

Trap: Off-by-one on the number of terms. A sum from 5 to 20 has sixteen terms, not fifteen.

Practise: No question drills this yet

You see: An arithmetic progression with two facts given about terms or sums, and the first term or common difference wanted.

You do: Write each fact with $a + (n-1)d$ or $\frac{n}{2}(2a + (n-1)d)$ and subtract the two equations. The first term cancels and $d$ falls out in one line.

Trap: Listing terms until the pattern appears. It works for the fifth term and not for the fiftieth, which is why the question asked for the fiftieth.

Practise: Mock 1 Paper 1 Q3

You see: A circle given in the expanded form $x^{2} + y^{2} + cx + dy + e = 0$.

You do: Complete the square in x and in y separately. The centre is read off the brackets and the radius squared is what is left on the right.

Trap: Reading the centre straight off the coefficients without halving and negating them, or forgetting that the constant moves across.

Practise: No question drills this yet

You see: A tangent, a chord, or an angle standing on an arc.

You do: The tangent is perpendicular to the radius at the point of contact, and the perpendicular from the centre bisects a chord. Both turn a circle question into a right-angled triangle.

Trap: Reaching for the equation of the tangent by calculus. The perpendicular radius gives the gradient without differentiating anything.

Practise: Mock 1 Paper 1 Q6

You see: Two points, a gradient condition, or a perpendicular bisector.

You do: Find the gradient, use $y - y_1 = m(x - x_1)$, and for a perpendicular take $-1/m$. A perpendicular bisector also needs the midpoint.

Trap: Negating the gradient without taking the reciprocal, which gives a reflection rather than a perpendicular.

Practise: No question drills this yet

You see: A distance, a midpoint, or the area of a polygon given by its vertices.

You do: Use the difference of coordinates as the two shorter sides. For an area, split the shape into triangles with a horizontal or vertical base.

Trap: Assuming a quadrilateral given by four points is a rectangle. Check the gradients before using length times width.

Practise: No question drills this yet

You see: A non-right-angled triangle with three pieces of information.

You do: Two sides and the angle between them, or three sides, calls for the cosine rule. Anything else calls for the sine rule. Watch for the ambiguous case when the given angle is not between the given sides.

Trap: Taking the acute solution of the sine rule without checking whether the obtuse one also fits the triangle.

Practise: No question drills this yet

You see: An arc length, a sector area, or a segment, with an angle that could be given in radians.

You do: In radians, arc is $r\theta$ and sector is $\frac{1}{2}r^{2}\theta$. A segment is the sector minus the triangle $\frac{1}{2}r^{2}\sin\theta$.

Trap: Using the degree formulas with a radian angle, which is out by a factor of about 57 and rarely looks wrong among the options.

Practise: No question drills this yet

You see: A trigonometric equation in a compound argument such as $2x + \frac{\pi}{3}$, with a solution interval attached.

You do: Transform the interval to match the argument first, find every solution inside the transformed interval, then invert to get back to x.

Trap: Finding the principal value and then adding periods in the original variable, which loses solutions whenever the argument has a coefficient.

Practise: No question drills this yet

You see: An exact value at 30, 45, 60 or 90 degrees, or a surd among the options of a trigonometric question.

You do: Recall the half equilateral triangle and the isosceles right-angled triangle, and read every exact value off them rather than memorising a table.

Trap: Mixing up sine and cosine at 30 and 60 degrees, which is the one slip these questions are built to catch.

Practise: No question drills this yet

You see: An equation mixing $\tan$ with $\sin$ or $\cos$, or a mix of squared and linear trigonometric terms.

You do: Replace $\tan$ by $\sin / \cos$ and use $\sin^{2} + \cos^{2} = 1$ to reach a single function, which usually leaves a quadratic.

Trap: Dividing through by $\cos\theta$, which silently discards every solution where the cosine is zero.

Practise: Mock 1 Paper 1 Q7 · Mock 1 Paper 1 Q19

You see: An equation with two or more logarithms of the same base, or a logarithm of a product or a power.

You do: Use the sum, difference and power laws to reach a single logarithm on each side, then equate the arguments. Check every solution back in the original.

Trap: Keeping a root that makes an argument negative or zero. $\log$ of a non-positive number does not exist, so that root was never a solution.

Practise: Mock 1 Paper 1 Q8

You see: Compound interest, a doubling time, or a quantity multiplied by a fixed factor each period.

You do: Write $A = A_0 k^{t}$ with $k$ the per-period multiplier, then take logarithms of both sides only once the model is written down.

Trap: Treating repeated percentage growth as linear, which underestimates badly over many periods and matches one of the options exactly.

Practise: No question drills this yet

You see: An equation such as $25^{x} - 3 \times 5^{x} + 2 = 0$, or two powers that can be written with one base.

You do: Rewrite both sides with a common base and equate exponents, or substitute $u = a^{x}$ when the powers are $a^{x}$ and $a^{2x}$ and solve the quadratic.

Trap: Rejecting nothing. A substitution can produce a negative value of $u$, and $a^{x}$ is never negative, so that root is discarded rather than logged.

Practise: Mock 1 Paper 1 Q9

You see: An area between a curve and an axis where the curve crosses that axis inside the interval.

You do: Find the crossing points, integrate each piece separately, and add the absolute values. Add curves together only where the region is genuinely between them.

Trap: Integrating straight across a sign change, so the part below the axis cancels part of the part above and the area comes out too small.

Practise: Mock 1 Paper 1 Q12

You see: An approximation to an area by trapezia, with a question about whether the estimate is too big or too small.

You do: Apply the rule, then sketch. A curve that bends upwards over the interval sits below its chords, so the trapezium rule overestimates.

Trap: Deciding over or under from whether the function is increasing. It is the second derivative that settles it, not the first.

Practise: No question drills this yet

You see: A maximum or minimum of one quantity subject to a fixed perimeter, area, volume or budget.

You do: Use the constraint to write the target as a function of one variable, differentiate, and check that the stationary point is the right kind.

Trap: Differentiating with two variables still present, which cannot work, or forgetting to convert back to the quantity actually asked for.

Practise: No question drills this yet

You see: A tangent or a normal at a named point on a curve.

You do: Differentiate, substitute the x coordinate to get the gradient, and use $y - y_1 = m(x - x_1)$. A normal takes $-1/m$.

Trap: Substituting the point into the derivative and then treating the result as a y value rather than as a gradient.

Practise: Mock 1 Paper 1 Q11

You see: An indefinite integral of a polynomial or of a rational power, sometimes after simplification.

You do: Split into terms of the form $ax^{n}$ and integrate each. Remember the constant of integration when the question asks for a function rather than a number.

Trap: Applying the rule to $x^{-1}$, which is the one exponent it does not cover.

Practise: Mock 1 Paper 1 Q13

You see: A maximum, a minimum, or a range of x on which a function increases or decreases.

You do: Solve $\mathrm{f}'(x) = 0$ for the locations, and use the sign of $\mathrm{f}''$ or the sign of $\mathrm{f}'$ either side to classify. Increasing means $\mathrm{f}'(x) \gt 0$ throughout.

Trap: Reporting the x coordinate when the question asked for the value of the function, or the other way round.

Practise: Mock 1 Paper 1 Q10

You see: A quotient, a product or a surd to be differentiated, such as $\frac{(3x+2)^{2}}{2\sqrt{x}}$.

You do: Expand and split the expression into a sum of terms of the form a x to a rational power, then differentiate term by term.

Trap: Reaching for a product or quotient rule that this specification does not need. Simplifying first is shorter and cannot go wrong in the same way.

Practise: No question drills this yet

You see: Several definite integrals of the same function over adjacent or equal ranges, with a combination wanted.

You do: Add integrals over contiguous ranges by joining the limits, add integrands over equal ranges, and reverse a pair of limits by changing the sign.

Trap: Multiplying integrands. The integral of a product is not the product of the integrals, and one option is always built from that.

Practise: No question drills this yet

You see: A question asking how many solutions an equation has, especially one mixing a polynomial with an exponential or a modulus.

You do: Split the equation into two functions and count how often their graphs cross. Shape and end behaviour settle the count without solving anything.

Trap: Trying to solve exactly. These equations are chosen so that no closed form exists, which is the point of asking only for the number of roots.

Practise: No question drills this yet

You see: A transformed function such as $\mathrm{f}(2x + 6)$, or a described sequence of shifts and stretches.

You do: Changes outside the function act on $y$ in the obvious direction. Changes inside act on $x$ in the opposite one, so $\mathrm{f}(x + a)$ shifts left and $\mathrm{f}(ax)$ compresses.

Trap: Reading $\mathrm{f}(2x + 6)$ as a stretch then a shift of 6. Factorise to $\mathrm{f}(2(x+3))$ first: the shift is 3.

Practise: Mock 1 Paper 1 Q14

You see: A function given in factorised form, or a question about its intercepts, asymptotes or sign.

You do: Roots come from the factors, the y intercept from setting x to zero, the far behaviour from the highest power, and a repeated factor touches the axis instead of crossing it.

Trap: Plotting points. Five points can be joined into several different curves, and the options are chosen to include the wrong ones.

Practise: No question drills this yet

You see: A question about whether two graphs meet, touch or miss, usually with a parameter.

You do: Set the two expressions equal, gather into a quadratic in one variable, and read the answer from the sign of the discriminant.

Trap: Treating tangency as a repeated root of the original curve rather than of the difference of the two functions.

Practise: No question drills this yet

You see: A pie chart, a two-way table, a cumulative frequency graph or a histogram to be read rather than calculated.

You do: A pie sector encodes a proportion of 360 degrees, a cumulative graph encodes a running total, and a histogram encodes frequency as area rather than as height.

Trap: Reading a histogram bar height as a frequency. With unequal class widths the height is a density and the frequency is the area.

Practise: No question drills this yet

You see: A combined experiment, a conditional probability, or a question about at least one of something.

You do: Build the possibility space, or a tree, and count. For at least one, take one minus the probability of none, which is a single product.

Trap: Adding probabilities of events that can happen together, or multiplying probabilities that are not independent.

Practise: Mock 1 Paper 1 Q17

You see: A data set with an extreme value, a missing value recoverable from a stated mean, or grouped data.

You do: The mean uses every value, so a missing value can be recovered from the total. The median ignores extremes. Grouped data gives estimates because the individual values are gone.

Trap: Quoting the mean of a skewed set, or forgetting that a mean calculated from midpoints is an estimate rather than a value.

Practise: No question drills this yet

You see: A compound statement to be judged true or false, or a question asking how many of several statements are true.

You do: An implication is false in exactly one case, both parts of an and must hold, and one part of an or is enough. Test the deciding case rather than tabulating everything.

Trap: Calling an implication false because its hypothesis is false. A statement about nothing is true, not false.

Practise: No question drills this yet

You see: A statement joined by and or or, with its negation wanted.

You do: Not (A and B) is (not A) or (not B), and not (A or B) is (not A) and (not B). The connective changes as well as the parts.

Trap: Negating both parts and keeping the connective, which gives a strictly stronger statement than the negation.

Practise: Mock 1 Paper 2 Q2

You see: A statement using 'A only if B', 'A if B', or 'if and only if'.

You do: 'A if B' means B implies A. 'A only if B' means A implies B. 'If and only if' is both directions and needs both checked.

Trap: Reading 'only if' as 'if'. They are converses of each other, so this reverses the whole statement.

Practise: Mock 1 Paper 2 Q6

You see: An implication with four rearrangements of it among the options.

You do: 'If A then B' is equivalent to 'if not B then not A' and to nothing else on the list. Swapping gives the converse, negating both gives the inverse, and those two are equivalent to each other rather than to the original.

Trap: Choosing the converse. English often uses 'if' where it means 'if and only if', so the converse reads as though it must follow.

Practise: Mock 1 Paper 2 Q1 · Mock 1 Paper 2 Q20

You see: A universal statement over an empty set, or an implication whose hypothesis never holds.

You do: A for-all over no elements is true, and a there-exists over no elements is false. An implication with a false hypothesis is true whatever the conclusion says.

Trap: Calling the statement meaningless or undecidable. It has a truth value and the value is true.

Practise: Mock 1 Paper 2 Q14

You see: A question asking whether such an object exists, or asking which value shows that it does.

You do: To establish a there-exists statement, produce one object and check it. To refute one, you have to rule out every object, which is a different and much longer job.

Trap: Arguing that no such object is obvious. Not finding a witness is not the same as showing there is none.

Practise: No question drills this yet

You see: A statement beginning every, all, some, no, or there exists, with its negation wanted.

You do: The negation of 'every x has P' is 'some x fails P', and the negation of 'some x has P' is 'no x has P'. The quantifier flips and the property is negated.

Trap: Negating the property and keeping the quantifier, giving 'every x fails P', which is far stronger than the negation and false in most cases where the negation is true.

Practise: Mock 1 Paper 2 Q4 · Mock 1 Paper 2 Q9

You see: A statement with two quantifiers, such as for every x there is a y, or an order-of-quantifiers question.

You do: Push the negation through one quantifier at a time, flipping each as it passes. Swapping the order of two different quantifiers changes the statement, so check whether y is allowed to depend on x.

Trap: Treating 'for every x there is a y' as interchangeable with 'there is a y for every x'. The second is much stronger and usually false.

Practise: Mock 1 Paper 2 Q16

You see: Several conditions of different strengths on the same object, with one of them to be selected.

You do: Order the conditions by how much they demand. A condition that demands more is sufficient more easily and necessary less easily, and the answer is usually the one that sits just above or just below the target.

Trap: Choosing a condition that is trivially true, such as x squared being at least zero. It is necessary, but so is everything true, and the question wants the one that says something.

Practise: Mock 1 Paper 2 Q12

You see: A claim that two conditions are the same, or an if and only if.

You do: Check that each implies the other. Finding one object with the first and not the second is enough to break the equivalence.

Trap: Verifying one direction on several examples and concluding equivalence. Examples cannot establish an implication, only refute one.

Practise: No question drills this yet

You see: A condition described as necessary, sufficient, or both, for some property.

You do: A implies B says A is sufficient for B and B is necessary for A. Test the two directions separately: one counterexample settles each.

Trap: Reading the implication backwards, so the sufficient condition is called necessary. Ask which one you could have without the other.

Practise: Mock 1 Paper 2 Q3 · Mock 1 Paper 2 Q18

You see: A general claim with candidate counterexamples among the options.

You do: Check both halves: the candidate has to satisfy the hypothesis and fail the conclusion. A value that fails the hypothesis proves nothing at all.

Trap: Picking a value that makes the conclusion false without checking that the hypothesis holds there, which is not a counterexample.

Practise: Mock 1 Paper 2 Q8

You see: A proof that opens with 'suppose not', or a question about what such a proof should assume.

You do: The opening assumption is the negation of the whole statement, negated properly. Derive a contradiction with something already known, then conclude the original.

Trap: Assuming the statement itself, or negating only part of it. A negation that is not a negation cannot produce a valid proof.

Practise: Mock 1 Paper 2 Q7 · Mock 1 Paper 2 Q11

You see: A sequence, a sum or a divisibility claim where the general result is asked for and the first few cases are cheap.

You do: Compute three or four cases, guess the pattern, then test it on a case you did not use. A pattern that survives an unused case is worth committing to.

Trap: Committing to a pattern from two cases. Many wrong formulas agree with the right one at n equal to 1 and 2.

Practise: Mock 1 Paper 2 Q5

You see: A claim about all integers where parity, sign or a remainder changes the argument.

You do: Split into cases that are exhaustive, prove the claim in each, and check that no object falls outside the list. Remainders modulo a small number are the usual split.

Trap: Cases that overlap harmlessly but do not cover everything. Even and odd covers the integers; positive and negative does not, because zero is neither.

Practise: No question drills this yet

You see: A list of statements to be arranged into a valid proof.

You do: Find the step that uses only the hypothesis, then repeatedly take the step whose ingredients are already established. The conclusion is whatever is left.

Trap: Ordering by how the statements look rather than by what each one needs. A step that uses a result proved later is not a proof.

Practise: No question drills this yet

You see: Several statements given as true, with a question about what follows.

You do: Chain the implications forwards, and check each candidate conclusion against every premise. A conclusion follows only if it holds in every situation the premises allow.

Trap: Importing outside knowledge that the premises did not state. The question is what follows from these, not what happens to be true.

Practise: No question drills this yet

You see: A plausible claim about all integers, all primes, or all real numbers, with the truth of the claim in question.

You do: Test the edge cases first: zero, one, two, a negative, a fraction, a repeated root. A claim that survives those is usually true and one that fails does so within seconds.

Trap: Testing three medium-sized positive integers and concluding the claim holds. Those are exactly the cases the claim was built to survive.

Practise: Mock 1 Paper 2 Q17

You see: A count of objects larger than a count of categories, with a conclusion that two must share something.

You do: Name the objects and the boxes explicitly, and count both. If objects exceed boxes, some box holds two; if they exceed twice the boxes, some box holds three.

Trap: Concluding more than the count forces, such as exactly two sharing when three might, or a shared birthday when only a shared month follows.

Practise: Mock 1 Paper 2 Q19

You see: An argument that divides both sides by an expression, or cancels a common factor.

You do: Before cancelling, split into the case where the factor is zero and the case where it is not. From $ab = ac$ you get $a = 0$ or $b = c$, not $b = c$.

Trap: Dividing by something the argument has just set to zero, which is how every proof that 1 equals 2 works.

Practise: Mock 1 Paper 2 Q15

You see: A proof that is correct line by line but reaches a conclusion that is too strong.

You do: Look for a step that needs the variable to be positive, an integer, non-zero, or in a particular range, and see whether the statement guarantees it.

Trap: Checking only the algebra. These questions put the error in the conditions rather than in the manipulation.

Practise: No question drills this yet

You see: A short argument reaching an absurd conclusion, with its steps listed.

You do: Take each step in turn and ask whether it holds for every value satisfying what came before. The failing step is usually a division, a square root or a cancellation.

Trap: Blaming the last step because the conclusion is obviously false. The error is almost always earlier, and everything after it is valid reasoning from a broken premise.

Practise: No question drills this yet

You see: An argument that squares both sides, takes a square root, or deduces an angle from a trigonometric value.

You do: Squaring is not reversible, so check every solution in the original. The square root symbol means the non-negative root, so $\sqrt{x^{2}} = |x|$ and not $x$.

Trap: Concluding $A = B$ from $\sin A = \sin B$, or $x = 3$ from $x^{2} = 9$. Both discard a whole family of solutions.

Practise: No question drills this yet

You see: An argument that uses 'if A then B' together with B, and concludes A.

You do: Only two moves are valid: A gives B, and not B gives not A. Having B tells you nothing about A, and not having A tells you nothing about B.

Trap: Accepting the argument because the conclusion happens to be true. Validity is about the form, not about whether the conclusion is correct this time.

Practise: Mock 1 Paper 2 Q10