The ESAT Shortcut Library

You see: Coefficients that mirror each other, or a question asking only for a sum or difference.

You do: Add for $x+y$, subtract for $x-y$. Halve a perimeter before you start.

Trap: Solving fully for both variables when the question never asked for them separately.

Practise: Set 1 Maths Q26 · Set 3 Maths Q23 · Set 4 Maths Q1 · Set 6 Maths Q16

Worked solutions: Paper 1 Maths Q9 · Paper 2 Adv Maths Q4 · Paper 2 Maths Q26 · Paper 3 Adv Maths Q18 · Paper 4 Maths Q27

You see: A statement joined by $\equiv$, or a question asking for constants that make two expressions agree for every value of $x$.

You do: An identity is true for every value, so compare the coefficients of each power on both sides and solve the resulting system. An equation is true only for particular values, so solve it.

Trap: Substituting one convenient value and concluding an identity holds; that only shows the two sides agree at that single point.

Practise: Set 4 Maths Q23 · Set 4 Maths Q27 · Set 5 Maths Q1

Worked solutions: None yet

You see: An answer demanded in a unit different from the one the data arrives in.

You do: Convert only the unit that is wrong, and carry the units through the multiplication so they cancel.

Trap: Using a linear factor for an area or volume, where it should be squared or cubed.

Practise: Set 1 Maths Q27 · Set 4 Maths Q16 · Set 4 Maths Q26

Worked solutions: Paper 4 Chemistry Q20

You see: One square subtracted from another — numbers, brackets, radii or trigonometric terms.

You do: Factorise to $(a-b)(a+b)$. The gap is usually small and the sum is usually round.

Trap: Expanding both squares in full, then subtracting six-figure numbers under time pressure.

Practise: Set 1 Maths Q8 · Set 2 Maths Q3 · Set 2 Maths Q26 · Set 4 Maths Q4

Worked solutions: None yet

You see: A quadratic over a quadratic, or a fraction whose denominator is itself a sum of fractions.

You do: Factorise everything first; cancel only whole brackets. Combine a compound denominator before inverting.

Trap: Cancelling individual terms, or using $\dfrac{1}{a+b} = \dfrac1a+\dfrac1b$.

Practise: Set 1 Maths Q6 · Set 1 Maths Q21 · Set 3 Maths Q6 · Set 3 Maths Q24 · Set 5 Maths Q18 · Set 5 Maths Q24 · Set 6 Maths Q13

Worked solutions: Paper 2 Maths Q15 · Paper 2 Maths Q21 · Paper 3 Adv Maths Q20 · Paper 3 Maths Q8 · Paper 3 Maths Q20 · Paper 3 Maths Q23

You see: Fractional, negative or stacked indices; standard form; a root of a decimal.

You do: Root first, then power, then reciprocal — three independent decisions, applied in that order.

Trap: Treating a negative index as a negative number, or halving decimal places instead of the index.

Practise: Set 1 Maths Q25 · Set 2 Maths Q14 · Set 3 Maths Q22 · Set 3 Maths Q26 · Set 4 Maths Q7 · Set 5 Maths Q25 · Set 8 Adv Maths Q24 · Set 10 Adv Maths Q12 · Set 10 Adv Maths Q24 · Set 12 Adv Maths Q15

Worked solutions: Paper 2 Maths Q4 · Paper 2 Maths Q7 · Paper 2 Maths Q16 · Paper 3 Adv Maths Q14 · Paper 3 Maths Q21

You see: HCF, LCM, recurring decimals, bounds, or anything about the prime make-up of a number.

You do: Prime factorise. HCF takes the lowest powers, LCM the highest, and $\text{HCF}\times\text{LCM} = ab$.

Trap: Swapping the HCF and LCM rules, or quoting a bound as the largest value strictly below it.

Practise: Set 3 Maths Q13 · Set 3 Maths Q16 · Set 3 Maths Q18 · Set 4 Maths Q18 · Set 5 Maths Q3 · Set 5 Maths Q12 · Set 6 Maths Q4

Worked solutions: Paper 2 Adv Maths Q20 · Paper 4 Adv Maths Q9

You see: Roots that do not look alike, or a surd sitting in a denominator.

You do: Pull out the largest square factor so every term shares one root, then collect like algebra.

Trap: Adding under the root sign, or rationalising each fraction separately when they are conjugates.

Practise: Set 1 Maths Q1 · Set 2 Maths Q10 · Set 3 Maths Q1 · Set 12 Adv Maths Q19

Worked solutions: Paper 1 Maths Q1

You see: A formula to be solved for a letter, or a worded situation to be turned into an equation.

You do: Undo the operations in reverse order, treating the target letter as the destination.

Trap: Moving a term across the equals sign without inverting the operation that binds it.

Practise: Set 1 Maths Q15 · Set 2 Maths Q8 · Set 4 Maths Q10 · Set 4 Maths Q24 · Set 5 Maths Q22 · Set 10 Adv Maths Q20 · Set 14 Physics Q10

Worked solutions: Paper 1 Maths Q3 · Paper 1 Maths Q18 · Paper 2 Maths Q6 · Paper 3 Maths Q4

You see: A quadratic whose maximum, minimum, vertex or line of symmetry is wanted.

You do: Factor out the leading coefficient, complete the square, and read $q$ off $a(x-p)^2+q$.

Trap: Forgetting to multiply the correction term back by the leading coefficient.

Practise: Set 1 Maths Q2 · Set 2 Maths Q5 · Set 8 Adv Maths Q3

Worked solutions: Paper 2 Adv Maths Q1 · Paper 4 Adv Maths Q19

You see: Tangency, 'exactly one solution', 'two distinct points', or an unknown constant in a quadratic.

You do: $b^2-4ac$: positive means two, zero means tangent, negative means none.

Trap: Answering the tangency case when the question asked for two distinct intersections.

Practise: Set 8 Adv Maths Q11 · Set 10 Adv Maths Q6 · Set 11 Adv Maths Q3

Worked solutions: Paper 1 Adv Maths Q5

You see: A polynomial divided by a linear factor, or a cubic with one or two roots handed to you.

You do: The remainder on dividing by $(x-a)$ is $f(a)$. With two roots known, Vieta gives the third.

Trap: Substituting $x=-a$ for the divisor $(x-a)$, or doing the long division anyway.

Practise: Set 9 Adv Maths Q5 · Set 10 Adv Maths Q4 · Set 10 Adv Maths Q11

Worked solutions: Paper 1 Maths Q11 · Paper 2 Adv Maths Q7

You see: Something squared and the same thing to the first power — $9^x$ with $3^x$, $(\log x)^2$ with $\log x$.

You do: Substitute a single letter for the repeated block, solve the ordinary quadratic, then convert back.

Trap: Stopping at the substituted variable instead of returning to the one the question asked about.

Practise: Set 5 Maths Q27 · Set 8 Adv Maths Q10 · Set 8 Adv Maths Q27 · Set 9 Adv Maths Q10 · Set 11 Adv Maths Q7 · Set 12 Adv Maths Q22

Worked solutions: Paper 1 Adv Maths Q7 · Paper 1 Maths Q8 · Paper 3 Adv Maths Q13 · Paper 3 Maths Q7 · Paper 4 Maths Q4

You see: Roots named but not wanted — the question asks for something symmetric in them.

You do: $\alpha+\beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$, read straight off the coefficients.

Trap: Solving the quadratic first — which fails outright as soon as the roots are irrational.

Practise: Set 3 Maths Q5 · Set 6 Maths Q12 · Set 6 Maths Q20 · Set 8 Adv Maths Q7 · Set 10 Adv Maths Q17

Worked solutions: Paper 1 Maths Q25

You see: An expression unchanged when two variables swap places.

You do: Rewrite in terms of the sum and product: $x^2+y^2 = (x+y)^2-2xy$, $x^3+y^3 = (x+y)^3-3xy(x+y)$.

Trap: Solving for the individual variables when only their sum and product are given.

Practise: Set 1 Maths Q20 · Set 3 Maths Q27 · Set 4 Maths Q20

Worked solutions: None yet

You see: Successive percentage changes, compound growth, or a percentage of a percentage.

You do: Convert each change to a multiplier and multiply. In a formula, raise each multiplier to its own power.

Trap: Adding the percentages, or applying a reverse percentage to the wrong base.

Practise: Set 1 Maths Q12 · Set 1 Maths Q22 · Set 1 Maths Q24 · Set 2 Maths Q1 · Set 2 Maths Q19 · Set 3 Maths Q10 · Set 4 Maths Q13 · Set 6 Maths Q17

Worked solutions: Paper 4 Maths Q5 · Paper 4 Maths Q6

You see: Equal distances covered at two different speeds.

You do: $\bar v = \dfrac{2uv}{u+v}$, always below the arithmetic mean.

Trap: Averaging the two speeds — more time is spent at the slower one.

Practise: Set 1 Maths Q18 · Set 3 Maths Q19 · Set 3 Maths Q20

Worked solutions: Paper 1 Maths Q27 · Paper 4 Maths Q2

You see: Similar shapes, a model, a map, or 'proportional to' in any form.

You do: Lengths scale by $k$, areas by $k^2$, volumes by $k^3$. For $y \propto x^n$, scaling $x$ by $f$ scales $y$ by $f^n$.

Trap: Comparing part to whole when the question asked part to remainder, or forgetting to raise the factor.

Practise: Set 1 Maths Q13 · Set 1 Maths Q23 · Set 2 Maths Q17 · Set 3 Maths Q21 · Set 4 Maths Q3 · Set 4 Maths Q19 · Set 5 Maths Q21

Worked solutions: Paper 2 Adv Maths Q11 · Paper 2 Adv Maths Q26 · Paper 2 Maths Q13 · Paper 3 Maths Q9 · Paper 3 Maths Q26 · Paper 4 Maths Q24

You see: Two agents working together, or objects approaching each other.

You do: Add rates, never times: $\dfrac{1}{T} = \sum\dfrac{1}{t_i}$, minus for anything working against you.

Trap: Averaging or adding the times — which often yields a result slower than one agent alone.

Practise: Set 2 Maths Q6 · Set 2 Maths Q23 · Set 2 Maths Q25 · Set 3 Maths Q25 · Set 4 Maths Q21 · Set 4 Maths Q22

Worked solutions: Paper 1 Maths Q12 · Paper 3 Physics Q16 · Paper 4 Chemistry Q3 · Paper 4 Maths Q15 · Paper 4 Physics Q17 · Paper 4 Physics Q20

You see: Two groups of different sizes, or a mean that changes when a value is added.

You do: Convert every mean to a total, combine the totals, divide by the combined count.

Trap: Averaging the averages, which is only right when the groups are the same size.

Practise: Set 1 Maths Q17 · Set 2 Maths Q24 · Set 4 Maths Q2 · Set 5 Maths Q9 · Set 6 Maths Q14 · Set 6 Maths Q27 · Set 12 Adv Maths Q13 · Set 12 Adv Maths Q14

Worked solutions: None yet

You see: 'At least one', 'not', 'excluding', or a restriction that creates overlapping cases.

You do: Count everything, then subtract the single unwanted case: $1-P(\text{none})$.

Trap: Enumerating the cases directly and double-counting the overlaps between them.

Practise: Set 1 Maths Q4 · Set 1 Maths Q10 · Set 2 Maths Q2 · Set 7 Maths Q4 · Set 7 Maths Q6 · Set 7 Maths Q23

Worked solutions: None yet

You see: Two overlapping groups, with a 'both' or a 'neither' in the question.

You do: $|A\cup B| = |A|+|B|-|A\cap B|$, then check the four regions sum to the whole.

Trap: Adding the groups without removing the overlap, or subtracting it twice.

Practise: Set 2 Maths Q7 · Set 3 Maths Q7 · Set 3 Maths Q17

Worked solutions: None yet

You see: Two trials, with or without replacement, and a compound outcome.

You do: Multiply along a branch and add across branches; drop both numerator and denominator without replacement.

Trap: Treating a without-replacement draw as independent, leaving the denominator unchanged.

Practise: Set 1 Maths Q3 · Set 1 Maths Q14 · Set 3 Maths Q2 · Set 4 Maths Q5 · Set 6 Maths Q1 · Set 7 Maths Q2 · Set 7 Maths Q9 · Set 7 Maths Q12 · Set 7 Maths Q19 · Set 7 Maths Q22

Worked solutions: Paper 1 Maths Q14 · Paper 2 Maths Q11 · Paper 3 Maths Q3 · Paper 3 Maths Q5 · Paper 3 Maths Q24 · Paper 4 Adv Maths Q22 · Paper 4 Biology Q1 · Paper 4 Maths Q7 · Paper 4 Maths Q8 · Paper 4 Maths Q9 · Paper 4 Maths Q16 · Paper 4 Maths Q19

You see: A team, a code, an arrangement, or a hand of objects.

You do: Order matters gives a permutation; order-free divides by $r!$; repeats divide by each repeat's factorial.

Trap: Using a permutation for a selection, so every group is counted once per ordering.

Practise: Set 2 Maths Q11 · Set 2 Maths Q20 · Set 2 Maths Q22 · Set 7 Maths Q3 · Set 7 Maths Q5 · Set 7 Maths Q10 · Set 7 Maths Q11 · Set 7 Maths Q18 · Set 7 Maths Q24 · Set 7 Maths Q26

Worked solutions: Paper 1 Adv Maths Q15 · Paper 1 Adv Maths Q19 · Paper 3 Adv Maths Q7 · Paper 4 Adv Maths Q2 · Paper 4 Adv Maths Q13 · Paper 4 Maths Q17

You see: A stated group size with percentages or probabilities attached to two successive stages.

You do: Convert the percentages into people at once and fill the tree in whole numbers; a conditional probability is then one branch count over one column total.

Trap: Quoting the accuracy of a test as the probability that a positive result is correct, which reverses the conditional and ignores the false positives from the larger group.

Practise: Set 7 Maths Q13 · Set 7 Maths Q14 · Set 7 Maths Q15 · Set 7 Maths Q21

Worked solutions: None yet

You see: A circle given in expanded form, a sector, or two ends of a diameter.

You do: Complete the square for centre and radius. For $S=0$, the tangent length from a point is $\sqrt{S(\text{point})}$.

Trap: Using the diameter as the radius, or forgetting the equation carries $r^2$ rather than $r$.

Practise: Set 2 Maths Q12 · Set 3 Maths Q12 · Set 4 Maths Q12 · Set 6 Maths Q19

Worked solutions: Paper 2 Maths Q1 · Paper 2 Maths Q27 · Paper 4 Maths Q18 · Paper 4 Maths Q22

You see: A diagram or description carrying no lengths worth computing with, where the question turns on a named property.

You do: Name the fact first: the angle at the centre is twice the angle at the circumference on the same arc, a rhombus has perpendicular bisecting diagonals of unequal length, and $\text{SSS}$, $\text{SAS}$, $\text{ASA}$ and $\text{RHS}$ prove congruence while $\text{SSA}$ does not.

Trap: Reaching for trigonometry or coordinates when a one-line property settles it, and accepting two sides and a non-included angle as proof of congruence.

Practise: Set 5 Maths Q2 · Set 5 Maths Q5 · Set 5 Maths Q8 · Set 5 Maths Q11 · Set 5 Maths Q14 · Set 5 Maths Q17 · Set 5 Maths Q20 · Set 5 Maths Q23 · Set 5 Maths Q26 · Set 6 Maths Q2

Worked solutions: None yet

You see: Parallel, perpendicular, a normal, or a perpendicular bisector.

You do: Parallel keeps $m$; perpendicular takes $-\dfrac1m$. A bisector also needs the midpoint.

Trap: Negating a gradient instead of inverting it as well.

Practise: Set 1 Maths Q9 · Set 8 Adv Maths Q17 · Set 10 Adv Maths Q13 · Set 10 Adv Maths Q23 · Set 12 Adv Maths Q6

Worked solutions: Paper 2 Physics Q2 · Paper 2 Physics Q11 · Paper 3 Physics Q15 · Paper 4 Adv Maths Q15 · Paper 4 Physics Q8

You see: A solid or polygon whose defining length is implied rather than given.

You do: Recover the defining length first. Interior angles sum to $(n-2)180^\circ$; exterior always to $360^\circ$.

Trap: Working forwards from a formula when the question supplies the output and wants the input.

Practise: Set 1 Maths Q16 · Set 3 Maths Q3 · Set 3 Maths Q15 · Set 4 Maths Q6 · Set 4 Maths Q15 · Set 6 Maths Q11 · Set 6 Maths Q15

Worked solutions: Paper 1 Maths Q7 · Paper 2 Adv Maths Q13 · Paper 2 Maths Q12 · Paper 3 Maths Q14 · Paper 4 Chemistry Q10 · Paper 4 Chemistry Q19 · Paper 4 Maths Q23 · Paper 4 Maths Q25

You see: $x^2+y^2$, a space diagonal, a tangent length, or a chord.

You do: Read it as a squared distance. Check for $(3,4,5)$, $(5,12,13)$, $(6,8,10)$ before computing.

Trap: Giving the distance when the question wanted its square, or the half-chord instead of the chord.

Practise: Set 1 Maths Q19 · Set 2 Maths Q16 · Set 4 Maths Q9 · Set 6 Maths Q22 · Set 8 Adv Maths Q4

Worked solutions: Paper 2 Adv Maths Q15 · Paper 4 Adv Maths Q16 · Paper 4 Maths Q1

You see: 'How many solutions', or a multiple angle inside the function.

You do: Widen the range by the multiple, solve for the whole angle, then divide. Reject roots outside $[-1,1]$.

Trap: Solving first and widening afterwards, which loses solutions that can never be recovered.

Practise: Set 9 Adv Maths Q11 · Set 11 Adv Maths Q2 · Set 11 Adv Maths Q20 · Set 12 Adv Maths Q17

Worked solutions: Paper 1 Adv Maths Q8 · Paper 1 Adv Maths Q27 · Paper 2 Adv Maths Q5 · Paper 2 Adv Maths Q25 · Paper 4 Physics Q1

You see: An angle given as a multiple of $\pi$, or a sector question in a Mathematics 2 context.

You do: In radians the fractions of $360^{\circ}$ disappear: arc $s = r\theta$, sector $A = \tfrac12 r^{2}\theta$, and a segment is the sector minus the triangle $\tfrac12 r^{2}\sin\theta$.

Trap: Using the degree formulas with a radian angle, or forgetting that the segment needs the triangle subtracted.

Practise: Set 11 Adv Maths Q5 · Set 11 Adv Maths Q12 · Set 11 Adv Maths Q18

Worked solutions: Paper 4 Maths Q13

You see: A non-right-angled triangle with a mix of sides and angles.

You do: Cosine rule finds sides, sine rule finds angles, $\tfrac12 ab\sin C$ finds areas.

Trap: Dropping one of the two halves in $\tfrac12 ab \sin C$ — the formula's and the sine's.

Practise: Set 9 Adv Maths Q15 · Set 11 Adv Maths Q9 · Set 11 Adv Maths Q15 · Set 12 Adv Maths Q4

Worked solutions: None yet

You see: A mix of functions, a squared trigonometric term, or a sum of two waves.

You do: Pick the identity matching what is present: $1+\cos2\theta \to 2\cos^2\theta$; $a\sin\theta+b\cos\theta \to R = \sqrt{a^2+b^2}$.

Trap: Choosing the form of a double-angle identity that leaves the expression no simpler.

Practise: Set 8 Adv Maths Q6 · Set 8 Adv Maths Q13 · Set 8 Adv Maths Q19 · Set 9 Adv Maths Q4 · Set 10 Adv Maths Q7

Worked solutions: Paper 1 Adv Maths Q13 · Paper 1 Adv Maths Q26 · Paper 3 Adv Maths Q10

You see: An angle outside the first quadrant with an exact answer demanded.

You do: Reference angle first, quadrant sign second — two separate decisions.

Trap: Getting the magnitude right and the sign wrong, particularly for tangent in the third quadrant.

Practise: Set 1 Maths Q7 · Set 3 Maths Q8 · Set 6 Maths Q3 · Set 10 Adv Maths Q1 · Set 12 Adv Maths Q12

Worked solutions: None yet

You see: A constant ratio between terms, or a sum that converges.

You do: $u_n = ar^{n-1}$, $S_n = \dfrac{a(r^n-1)}{r-1}$, $S_\infty = \dfrac{a}{1-r}$ when $|r| \lt 1$.

Trap: Confusing the $r^{n-1}$ of the nth term with the $r^n$ of the sum.

Practise: Set 2 Maths Q4 · Set 2 Maths Q13 · Set 8 Adv Maths Q9 · Set 9 Adv Maths Q13 · Set 9 Adv Maths Q21 · Set 10 Adv Maths Q21 · Set 10 Adv Maths Q27 · Set 11 Adv Maths Q11

Worked solutions: Paper 1 Adv Maths Q2 · Paper 2 Adv Maths Q2 · Paper 2 Adv Maths Q19 · Paper 3 Adv Maths Q23 · Paper 4 Adv Maths Q14 · Paper 4 Adv Maths Q25 · Paper 4 Adv Maths Q27

You see: A linear nth term, a run of consecutive numbers, or a sum of multiples.

You do: $S_n = \dfrac{n}{2}(a+l)$. Factor out any common multiple first; $u_n = S_n-S_{n-1}$.

Trap: Using $n$ steps where there are $n-1$ between $n$ terms.

Practise: Set 1 Maths Q11 · Set 2 Maths Q9 · Set 2 Maths Q21 · Set 3 Maths Q9 · Set 8 Adv Maths Q21 · Set 9 Adv Maths Q17 · Set 10 Adv Maths Q10 · Set 10 Adv Maths Q16

Worked solutions: Paper 2 Adv Maths Q23

You see: A denominator that is a product of consecutive terms, or a sum of logs of ratios.

You do: Split into partial fractions; the interior cancels and only the boundary terms survive.

Trap: An off-by-one in the surviving final term.

Practise: Set 8 Adv Maths Q15 · Set 11 Adv Maths Q13 · Set 11 Adv Maths Q17

Worked solutions: Paper 3 Adv Maths Q3 · Paper 4 Adv Maths Q17

You see: First differences that change by a constant amount.

You do: Second difference is $2a$; subtract $an^2$ and what remains is linear.

Trap: Accepting an nth term that only matches the first term — the distractors are built to survive that.

Practise: Set 3 Maths Q4 · Set 9 Adv Maths Q2 · Set 10 Adv Maths Q3

Worked solutions: None yet

You see: A bracket raised to a power with one particular term wanted.

You do: $T_k = \binom{n}{k}a^{n-k}b^k$. Set the index of $x$ to what you want and solve for $k$.

Trap: Leaving a minus sign outside the bracket, so the sign is not raised to the power.

Practise: Set 6 Maths Q8 · Set 8 Adv Maths Q2 · Set 9 Adv Maths Q9 · Set 9 Adv Maths Q19 · Set 11 Adv Maths Q4

Worked solutions: Paper 2 Adv Maths Q17 · Paper 2 Maths Q24 · Paper 3 Adv Maths Q21 · Paper 4 Adv Maths Q8 · Paper 4 Adv Maths Q11 · Paper 4 Adv Maths Q18 · Paper 4 Adv Maths Q26 · Paper 4 Maths Q26

You see: A rule of the form $u_{n+1} = f(u_n)$ with a starting value, rather than a formula in $n$.

You do: Generate the terms one at a time and stop as soon as you reach the one asked for. A term-to-term rule is followed, not solved.

Trap: Treating $u_1$ as $u_0$ and generating one term too many or too few. Write the index beside every term.

Practise: Set 5 Maths Q16 · Set 5 Maths Q19 · Set 12 Adv Maths Q8

Worked solutions: Paper 4 Adv Maths Q6

You see: Maximum, minimum, optimisation, or a turning point.

You do: Differentiate, solve $f'(x)=0$, classify by $f''$ or by the shape, then filter against the domain.

Trap: Reporting the x-coordinate when the y-value was asked, or keeping a physically impossible root.

Practise: Set 8 Adv Maths Q1 · Set 8 Adv Maths Q16 · Set 9 Adv Maths Q26 · Set 10 Adv Maths Q8 · Set 10 Adv Maths Q26 · Set 11 Adv Maths Q14 · Set 12 Adv Maths Q3

Worked solutions: Paper 3 Adv Maths Q22 · Paper 3 Adv Maths Q24 · Paper 4 Adv Maths Q10

You see: 'Estimate', a strip count, or an integrand with no elementary antiderivative.

You do: $\dfrac{h}{2}\left[\text{ends}+2(\text{middles})\right]$, with $h$ the strip width.

Trap: Doubling the end ordinates, or using the whole interval as the strip width.

Practise: Set 9 Adv Maths Q27 · Set 10 Adv Maths Q14 · Set 10 Adv Maths Q19

Worked solutions: Paper 3 Maths Q18

You see: Integrals quoted as values rather than functions, with a new combination asked for.

You do: Use the structure rather than the integrand: contiguous ranges add, $\int_a^b = -\int_b^a$, constants come out, and differentiating $\int_a^x f(t)\,\mathrm{d}t$ returns $f(x)$.

Trap: Trying to recover the function from the given values. The question is about the properties of the integral, not about integrating anything.

Practise: Set 12 Adv Maths Q11 · Set 12 Adv Maths Q16 · Set 12 Adv Maths Q20

Worked solutions: None yet

You see: A function inside another, or two rates linked by a formula.

You do: Derivative of the outside with the inside unchanged, times the derivative of the inside.

Trap: Dropping the inner derivative entirely — the single most common calculus slip.

Practise: Set 8 Adv Maths Q23 · Set 9 Adv Maths Q12 · Set 9 Adv Maths Q18 · Set 9 Adv Maths Q24 · Set 11 Adv Maths Q19 · Set 11 Adv Maths Q26

Worked solutions: Paper 1 Adv Maths Q20 · Paper 3 Adv Maths Q26

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

You do: Point from the curve, gradient from the derivative; the normal is the negative reciprocal.

Trap: Answering with the tangent gradient when the normal was wanted.

Practise: Set 6 Maths Q7 · Set 9 Adv Maths Q20 · Set 10 Adv Maths Q22 · Set 10 Adv Maths Q25

Worked solutions: Paper 1 Adv Maths Q14 · Paper 4 Maths Q10

You see: A root or a reciprocal in something to be differentiated or integrated.

You do: Rewrite as $x^n$ with a fractional or negative index and apply the ordinary rule.

Trap: Differentiating $\dfrac1x$ to $\dfrac{1}{x^2}$ and losing the minus sign.

Practise: Set 11 Adv Maths Q1 · Set 11 Adv Maths Q23 · Set 11 Adv Maths Q24 · Set 12 Adv Maths Q23

Worked solutions: None yet

You see: Two functions multiplied or divided.

You do: $u'v+uv'$; or $\dfrac{u'v-uv'}{v^2}$ — numerator derivative first. Simplify before substituting.

Trap: Reversing the quotient rule numerator, which flips the sign of the answer.

Practise: Set 8 Adv Maths Q20 · Set 8 Adv Maths Q25 · Set 9 Adv Maths Q22 · Set 11 Adv Maths Q25

Worked solutions: None yet

You see: Limits symmetric about zero.

You do: Odd terms integrate to zero; even terms give twice the half-interval. Cross the odd terms out first.

Trap: Assuming the whole integrand vanishes when a constant or even term survives.

Practise: Set 8 Adv Maths Q8 · Set 9 Adv Maths Q1 · Set 11 Adv Maths Q27

Worked solutions: None yet

You see: A derivative plus a point the curve passes through.

You do: Integrate, substitute the point to find $c$, and only then evaluate anywhere else.

Trap: Assuming $c=0$, so the curve misses the given point entirely.

Practise: Set 9 Adv Maths Q8 · Set 9 Adv Maths Q25 · Set 11 Adv Maths Q22

Worked solutions: None yet

You see: A region enclosed by two curves, or by a curve and a line.

You do: Find the intersections, integrate (upper − lower) once, and use symmetry where it exists.

Trap: Integrating across a sign change without splitting, giving net area instead of total.

Practise: Set 5 Maths Q4 · Set 8 Adv Maths Q14 · Set 9 Adv Maths Q16 · Set 9 Adv Maths Q23 · Set 11 Adv Maths Q8

Worked solutions: Paper 1 Adv Maths Q16 · Paper 4 Physics Q19

You see: A sum or difference of logs, a log in an index, or a chain of bases.

You do: Combine to one log, convert to exponential form, and check the domain. $\log_a b\times\log_b c = \log_a c$.

Trap: Keeping a phantom root that satisfies the quadratic but not the original logarithm.

Practise: Set 8 Adv Maths Q5 · Set 8 Adv Maths Q12 · Set 8 Adv Maths Q26 · Set 9 Adv Maths Q3 · Set 10 Adv Maths Q5 · Set 11 Adv Maths Q21

Worked solutions: Paper 2 Adv Maths Q8 · Paper 2 Adv Maths Q10 · Paper 3 Adv Maths Q25 · Paper 4 Adv Maths Q7 · Paper 4 Adv Maths Q12

You see: A described sequence of reflections, translations or stretches.

You do: Track the vertex or a single known point. Inside the bracket acts on x and does the opposite.

Trap: Applying an inside change as though it were an outside one, so a compression reads as a stretch.

Practise: Set 10 Adv Maths Q2 · Set 11 Adv Maths Q10 · Set 11 Adv Maths Q16 · Set 12 Adv Maths Q9

Worked solutions: Paper 1 Adv Maths Q10 · Paper 3 Maths Q15 · Paper 4 Adv Maths Q5 · Paper 4 Maths Q11

You see: An inequality with the variable on both sides, a modulus, or a quadratic.

You do: Collect so the coefficient stays positive. $|A| \lt k$ gives one interval; $|A| \gt k$ gives two branches.

Trap: Dividing by a negative without reversing, or giving the inside of a parabola for the outside.

Practise: Set 2 Maths Q27 · Set 4 Maths Q25 · Set 5 Maths Q6 · Set 5 Maths Q7 · Set 9 Adv Maths Q7

Worked solutions: Paper 2 Adv Maths Q6 · Paper 4 Maths Q3 · Paper 4 Maths Q20

You see: A graph described in words rather than drawn, or a list of equations to be matched to a shape.

You do: Read the family off the equation: $\dfrac{k}{x}$ has both axes as asymptotes, $a^{x}$ has one horizontal asymptote and never reaches it, a cubic turns at most twice. Then fix the detail with one substituted point.

Trap: Confusing an intercept with an asymptote, or reading distance-time as speed-time — on a distance-time graph a horizontal section means stopped, not constant speed.

Practise: Set 5 Maths Q10 · Set 5 Maths Q13 · Set 12 Adv Maths Q7 · Set 12 Adv Maths Q10 · Set 12 Adv Maths Q25

Worked solutions: None yet

You see: Two powers with different bases that share a prime, or a doubling or half-life model.

You do: Rewrite both sides to the smallest common base and equate the indices; count the halvings.

Trap: Reaching for logarithms when the target is already a clean power of the growth factor.

Practise: Set 1 Maths Q5 · Set 6 Maths Q9 · Set 8 Adv Maths Q18 · Set 9 Adv Maths Q14 · Set 10 Adv Maths Q18 · Set 11 Adv Maths Q6

Worked solutions: Paper 1 Adv Maths Q12 · Paper 1 Maths Q13 · Paper 3 Adv Maths Q12 · Paper 4 Adv Maths Q4

You see: $f^{-1}$, or a composite such as $fg(x)$.

You do: Swap and solve for the inverse. $fg$ means $g$ acts first — read right to left.

Trap: Computing $gf$ when $fg$ was asked, which gives an entirely different answer.

Practise: Set 5 Maths Q15 · Set 8 Adv Maths Q22 · Set 10 Adv Maths Q9 · Set 10 Adv Maths Q15 · Set 12 Adv Maths Q2

Worked solutions: Paper 2 Maths Q19 · Paper 4 Biology Q26

You see: A population split into groups with a sample to be drawn from it.

You do: Stratified sampling keeps each group's share: $\dfrac{\text{sample}}{\text{population}}\times\text{group size}$.

Trap: Splitting the sample equally between groups, which over-represents the small ones.

Practise: Set 7 Maths Q25

Worked solutions: None yet

You see: A data set with an obvious extreme value, or a question asking which average to use.

You do: Mean uses every value, median ignores extremes, mode is the only one for categories. Pick by what the outlier does.

Trap: Quoting the mean of a skewed set, where a single extreme value drags it away from the bulk of the data.

Practise: Set 4 Maths Q14 · Set 7 Maths Q8 · Set 12 Adv Maths Q27

Worked solutions: None yet

You see: Every value transformed the same way — $y = ax+b$, a change of units, or a subtracted constant.

You do: The mean follows the whole transformation; the spread ignores the shift: $\bar y = a\bar x+b$ but $\sigma_y = |a|\sigma_x$.

Trap: Adding the constant to the standard deviation. Shifting every value moves the data without spreading it.

Practise: Set 6 Maths Q24 · Set 7 Maths Q27

Worked solutions: None yet

You see: A discrete random variable with its probabilities listed.

You do: $E(X) = \sum xP(X=x)$, and $E(aX+b) = aE(X)+b$. For variance, $\text{Var}(X) = E(X^{2})-\left[E(X)\right]^{2}$.

Trap: Computing $E(X^{2})$ by squaring $E(X)$ — the two differ by exactly the variance.

Practise: Set 12 Adv Maths Q5 · Set 12 Adv Maths Q18 · Set 12 Adv Maths Q21 · Set 12 Adv Maths Q24

Worked solutions: None yet

You see: Data in class intervals, or a histogram with unequal class widths.

You do: Midpoint as the value, frequency as the weight: $\bar x \approx \dfrac{\sum fx}{\sum f}$. On a histogram, frequency is density times class width.

Trap: Calling the result exact. Grouping loses the individual values, so a grouped mean is always an estimate.

Practise: Set 2 Maths Q18 · Set 6 Maths Q18 · Set 7 Maths Q1 · Set 7 Maths Q16 · Set 9 Adv Maths Q6

Worked solutions: None yet

You see: An ordered list, or a cumulative frequency total, with a median or quartile wanted.

You do: Order first, then locate by position: the median sits at $\dfrac{n+1}{2}$, and for even $n$ it is the mean of the two middle values.

Trap: Reading the middle of the unordered list, or halving n and taking that value rather than that position.

Practise: Set 6 Maths Q5 · Set 6 Maths Q6 · Set 6 Maths Q10 · Set 6 Maths Q21 · Set 6 Maths Q26 · Set 7 Maths Q20

Worked solutions: None yet

You see: Quartiles, interquartile range, or a value that looks far from the rest.

You do: $\text{IQR} = Q_3-Q_1$, and a value is an outlier if it lies more than $1.5\times\text{IQR}$ beyond a quartile.

Trap: Using the range rather than the IQR, so a single extreme value decides the whole measure of spread.

Practise: Set 3 Maths Q14 · Set 6 Maths Q23 · Set 6 Maths Q25

Worked solutions: None yet

You see: A pie chart, two-way table, bar chart, pictogram or time series to be read rather than calculated.

You do: Ask what the chart encodes: a pie sector encodes a proportion of 360 degrees, a two-way table encodes four regions that must total, a time series encodes a difference per step.

Trap: Reading an angle, a bar height or a row total as though it were the frequency the question asked for.

Practise: Set 2 Maths Q15 · Set 3 Maths Q11 · Set 4 Maths Q8

Worked solutions: Paper 4 Physics Q14

You see: A scatter graph, a described trend, or a claim that one variable causes another.

You do: State direction and strength from the trend, and stop there. Correlation constrains explanation; it does not supply one.

Trap: Turning a strong correlation into a causal claim, or extrapolating far outside the range of the data.

Practise: Set 4 Maths Q11 · Set 4 Maths Q17 · Set 7 Maths Q17

Worked solutions: None yet

You see: $\sum x$ and $\sum x^{2}$ given, rather than the values themselves.

You do: $\sigma^{2} = \dfrac{\sum x^{2}}{n}-\bar x^{2}$ — the mean of the squares minus the square of the mean.

Trap: Reversing the two terms, which produces a negative variance and should be caught instantly.

Practise: Set 7 Maths Q7 · Set 12 Adv Maths Q1 · Set 12 Adv Maths Q26

Worked solutions: None yet

You see: Turns on a transformer, a wire in a field, or a magnet being moved near a coil.

You do: For a transformer, $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$, and if it is ideal the power is unchanged, so stepping voltage down steps current up. For the motor effect, $F = BIL$.

Trap: Assuming a step-down transformer reduces the current as well. It does the opposite, which is the whole point of the national grid.

Practise: Set 13 Physics Q17 · Set 13 Physics Q19 · Set 13 Physics Q21 · Set 13 Physics Q23 · Set 14 Physics Q5

Worked solutions: None yet

You see: A resistor network, a reading from an ammeter or voltmeter, or a power or cost figure.

You do: Collapse the network to one resistance first: series add, parallel combine as $\dfrac{1}{R} = \sum\dfrac{1}{R_i}$. Then $V = IR$ once, and pick the power formula that matches the two quantities you already have.

Trap: Adding parallel resistances, or using the supply voltage across a component that only receives part of it.

Practise: Set 13 Physics Q2 · Set 13 Physics Q6 · Set 13 Physics Q9 · Set 13 Physics Q12 · Set 13 Physics Q15 · Set 14 Physics Q15

Worked solutions: Paper 1 Physics Q6

You see: Three of the five quantities among initial speed, final speed, acceleration, time and distance.

You do: List $s$, $u$, $v$, $a$, $t$, mark the one you were not given and do not want, and choose the equation that omits it. Most often that is $v^{2} = u^{2}+2as$, which skips time entirely.

Trap: Reaching for the time-based equations and solving a quadratic when the question never mentioned time.

Practise: Set 13 Physics Q1 · Set 13 Physics Q5 · Set 13 Physics Q8 · Set 14 Physics Q8

Worked solutions: None yet

You see: A height, a speed and a height together, or a time attached to an energy transfer.

You do: Follow the energy: $\tfrac12 mv^{2}$ converts to $mgh$ and back. Power is energy over time, and efficiency is useful output over total input.

Trap: Cancelling the mass without noticing, then using the answer for a question where the mass does matter. In $\tfrac12 mv^{2} = mgh$ it genuinely cancels; in a power calculation it does not.

Practise: Set 13 Physics Q25 · Set 13 Physics Q26 · Set 13 Physics Q27 · Set 14 Physics Q18

Worked solutions: Paper 4 Physics Q24

You see: Several forces on one body, a spring extension, or a weight quoted on another planet.

You do: Find the resultant force first, then $a = \dfrac{F}{m}$ with the total mass. Weight is $mg$ and changes with location; mass does not.

Trap: Using the driving force rather than the resultant, or treating a Newton's third law pair as two forces on the same body — they always act on different bodies.

Practise: Set 13 Physics Q11 · Set 13 Physics Q14 · Set 13 Physics Q16 · Set 13 Physics Q18 · Set 13 Physics Q20 · Set 14 Physics Q13

Worked solutions: Paper 2 Physics Q16 · Paper 3 Physics Q1 · Paper 3 Physics Q21 · Paper 4 Physics Q23

You see: A collision, an explosion, a recoil, or a force acting for a stated time.

You do: Total momentum before equals total momentum after, taking one direction as positive throughout. Impulse $Ft$ is the change in momentum.

Trap: Assuming kinetic energy is conserved as well. It is conserved only in an elastic collision, and never when the bodies stick together.

Practise: Set 13 Physics Q22 · Set 13 Physics Q24 · Set 14 Physics Q3 · Set 14 Physics Q22

Worked solutions: Paper 1 Physics Q8 · Paper 2 Physics Q7 · Paper 2 Physics Q25 · Paper 3 Physics Q3 · Paper 3 Physics Q22 · Paper 4 Physics Q21

You see: A gas with one quantity held fixed, a density, or a force spread over an area.

You do: For a gas, hold the constant quantity fixed and use the ratio: $p_1V_1 = p_2V_2$ at constant temperature. Pressure is $\dfrac{F}{A}$, and in a fluid at depth it is $\rho gh$.

Trap: Mixing units — grams with metres, or centimetres cubed with kilograms per cubic metre. Convert before substituting, not after.

Practise: Set 13 Physics Q4 · Set 14 Physics Q4 · Set 14 Physics Q9 · Set 14 Physics Q14 · Set 14 Physics Q19 · Set 14 Physics Q23

Worked solutions: Paper 3 Maths Q12 · Paper 3 Physics Q9

You see: A temperature rise, or a change of state at constant temperature.

You do: A temperature change costs $E = mc\Delta\theta$; a change of state costs $E = mL$ with no temperature change at all. If both happen, add the stages.

Trap: Using $mc\Delta\theta$ across a melting or boiling point, where the temperature is not changing and $\Delta\theta$ is meaningless.

Practise: Set 13 Physics Q3 · Set 13 Physics Q7 · Set 13 Physics Q10 · Set 13 Physics Q13 · Set 18 Chemistry Q25

Worked solutions: Paper 1 Physics Q25 · Paper 4 Chemistry Q9 · Paper 4 Physics Q18

You see: An angle quoted at a surface, or light passing between two materials.

You do: Measure every angle from the normal. Entering a denser medium bends light towards the normal; leaving it bends away, and beyond the critical angle it does not leave at all.

Trap: Taking the angle from the surface rather than the normal, which turns an angle into its complement and silently changes the answer.

Practise: Set 14 Physics Q16 · Set 14 Physics Q20 · Set 14 Physics Q24

Worked solutions: Paper 2 Physics Q26

You see: A frequency, a wavelength or a wave speed, with the third quantity wanted.

You do: $v = f\lambda$, every time. The medium fixes the speed, so when a wave crosses a boundary the frequency stays and the wavelength changes.

Trap: Forgetting the return trip in an echo question. The sound covers twice the distance to the reflector.

Practise: Set 14 Physics Q1 · Set 14 Physics Q6 · Set 14 Physics Q11 · Set 14 Physics Q25 · Set 14 Physics Q26 · Set 14 Physics Q27

Worked solutions: Paper 4 Physics Q22

You see: A nuclide written with a mass number and a proton number, or an activity that is falling.

You do: Balance the top and bottom numbers across the equation: alpha removes $4$ and $2$, beta-minus leaves the mass number alone and raises the proton number by one. For decay, count halvings rather than solving anything.

Trap: Treating half-life as though the activity fell to zero after two halvings. Each half-life halves whatever is left, so it never quite reaches zero.

Practise: Set 14 Physics Q2 · Set 14 Physics Q7 · Set 14 Physics Q12 · Set 14 Physics Q17 · Set 14 Physics Q21

Worked solutions: None yet

You see: A percentage of one base, a count of amino acids, or a described change to a nucleotide sequence.

You do: A pairs with T and C with G, so $\%A = \%T$, $\%C = \%G$, and all four sum to $100\%$. Three bases code for one amino acid, so multiply or divide by three.

Trap: Assuming every mutation changes the protein. The genetic code is degenerate, so many base changes give the same amino acid and no effect at all.

Practise: Set 15 Biology Q4 · Set 15 Biology Q9 · Set 15 Biology Q14 · Set 15 Biology Q19 · Set 16 Biology Q6 · Set 16 Biology Q10 · Set 16 Biology Q12

Worked solutions: Paper 4 Biology Q11

You see: Two parent genotypes, or a stated ratio among the offspring.

You do: Draw the two-by-two grid. $\text{Bb}\times\text{Bb}$ gives $3:1$ by phenotype and $1:2:1$ by genotype; crossing with a homozygous recessive gives $1:1$ and reveals an unknown genotype.

Trap: Quoting a genotype ratio when the question asked for phenotypes, or treating the four boxes as four offspring rather than four equally likely outcomes.

Practise: Set 15 Biology Q16 · Set 15 Biology Q21 · Set 15 Biology Q24 · Set 15 Biology Q26 · Set 16 Biology Q15 · Set 16 Biology Q19

Worked solutions: Paper 4 Biology Q9 · Paper 4 Biology Q12

You see: A named organelle, a comparison between cell types, or an image size quoted against a real size.

You do: Go from function to structure: the job the question describes names the organelle. For magnification, convert both lengths to the same unit before dividing.

Trap: Listing what plant cells have without checking what animal cells also have. Mitochondria, ribosomes and a nucleus are in both.

Practise: Set 15 Biology Q2 · Set 15 Biology Q7 · Set 15 Biology Q12 · Set 15 Biology Q17 · Set 16 Biology Q4 · Set 16 Biology Q16

Worked solutions: Paper 4 Biology Q18

You see: A chromosome number before and after division, or a question about how alike the offspring are.

You do: Mitosis gives two genetically identical diploid cells; meiosis gives four genetically different haploid ones. Count the cells and the chromosomes together and only one answer fits both.

Trap: Halving the chromosome number for mitosis. Growth and repair need full copies, so mitosis conserves the number exactly.

Practise: Set 15 Biology Q1 · Set 15 Biology Q6 · Set 15 Biology Q11 · Set 16 Biology Q3

Worked solutions: None yet

You see: A trait being increased over generations, or a gene being moved between organisms.

You do: Ask what does the selecting. In selective breeding a person chooses the parents; in natural selection the environment does. Both act only on variation that already exists.

Trap: Saying an organism develops a useful feature because it needs one. Variation arises first, by mutation and recombination, and selection acts on it afterwards.

Practise: Set 15 Biology Q3 · Set 15 Biology Q8 · Set 15 Biology Q13 · Set 15 Biology Q18 · Set 15 Biology Q23 · Set 16 Biology Q5 · Set 16 Biology Q9 · Set 16 Biology Q11

Worked solutions: Paper 4 Biology Q7 · Paper 4 Biology Q23

You see: Two solutions either side of a membrane, or a question about whether a process needs energy.

You do: Water moves down the water potential gradient, towards the more concentrated solution. Diffusion and osmosis are passive; only active transport moves a substance against its gradient, and only that needs energy from respiration.

Trap: Saying water moves from low to high concentration without saying concentration of what. Water moves towards the higher solute concentration, which is the lower water potential.

Practise: Set 15 Biology Q22 · Set 15 Biology Q25 · Set 15 Biology Q27 · Set 16 Biology Q20

Worked solutions: Paper 4 Biology Q8 · Paper 4 Biology Q17 · Paper 4 Biology Q25

You see: A food chain, a quadrat count, or a named process in the carbon cycle.

You do: Energy flows one way and is lost at each transfer; carbon cycles round and is conserved. For a quadrat estimate, scale the mean count per unit area up to the whole area.

Trap: Scaling a quadrat count by the number of quadrats rather than by the ratio of the areas, and forgetting that photosynthesis is the only process removing carbon dioxide from the air.

Practise: Set 16 Biology Q2 · Set 16 Biology Q8 · Set 16 Biology Q14 · Set 16 Biology Q18 · Set 16 Biology Q22 · Set 16 Biology Q24 · Set 16 Biology Q25 · Set 16 Biology Q26

Worked solutions: Paper 4 Biology Q6 · Paper 4 Biology Q13 · Paper 4 Biology Q21

You see: A quantity being held steady, a hormone, or a described response to a stimulus.

You do: Name the change, then the correction that opposes it. Blood glucose up means insulin; water low means more ADH and more reabsorption. The response always pushes back towards the set point.

Trap: Getting the direction backwards, or naming the gland when the question wanted the hormone. Read whether the trigger, the messenger or the effect is being asked for.

Practise: Set 16 Biology Q1 · Set 16 Biology Q7 · Set 16 Biology Q13 · Set 16 Biology Q17 · Set 16 Biology Q21 · Set 16 Biology Q27

Worked solutions: Paper 4 Biology Q2 · Paper 4 Biology Q5 · Paper 4 Biology Q19 · Paper 4 Biology Q20

You see: A rate plotted against temperature or pH, or a named digestive enzyme.

You do: Rate rises with temperature as collisions increase, then falls sharply past the optimum as the active site denatures. pH behaves similarly, but falls away on both sides of the optimum.

Trap: Calling a denatured enzyme killed. It is a protein, not alive, and the change to its active site is usually permanent rather than merely paused.

Practise: Set 15 Biology Q5 · Set 15 Biology Q10 · Set 15 Biology Q15 · Set 15 Biology Q20 · Set 16 Biology Q23

Worked solutions: Paper 4 Biology Q4 · Paper 4 Biology Q10 · Paper 4 Biology Q14

You see: An observation quoted - a colour, a precipitate, a gas - with an identification wanted.

You do: Work from the observation to the species. Each test has one distinctive result, and the flame colours and precipitate colours are the whole content.

Trap: Confusing the limewater test with the hydrogen test. Limewater turning milky is carbon dioxide; the squeaky pop is hydrogen.

Practise: Set 19 Chemistry Q3 · Set 19 Chemistry Q13 · Set 19 Chemistry Q20 · Set 19 Chemistry Q26

Worked solutions: None yet

You see: A Group or Period number, or a comparison of reactivity between two elements in the same column.

You do: Group number gives the outer-shell electrons and so the ion formed; Period number gives the number of occupied shells. Down a metal Group reactivity rises; down the halogens it falls.

Trap: Applying one reactivity trend to both. Metals lose electrons more easily further down, halogens gain them less easily - so the two Groups trend in opposite directions.

Practise: Set 17 Chemistry Q5 · Set 17 Chemistry Q11 · Set 18 Chemistry Q6 · Set 18 Chemistry Q13

Worked solutions: Paper 4 Chemistry Q12

You see: A nuclide symbol, a list of isotope abundances, or an atomic number to build a configuration from.

You do: Protons fix the element, mass number minus atomic number gives the neutrons, and a weighted mean of the isotope masses gives $A_r$. Fill shells $2, 8, 8, 2$ for the first twenty elements.

Trap: Treating isotopes as chemically different. They differ only in neutrons, so their chemistry is identical and only mass-dependent properties change.

Practise: Set 17 Chemistry Q2 · Set 17 Chemistry Q8 · Set 17 Chemistry Q14 · Set 17 Chemistry Q17 · Set 17 Chemistry Q20

Worked solutions: Paper 4 Chemistry Q1 · Paper 4 Chemistry Q13 · Paper 4 Chemistry Q14 · Paper 4 Chemistry Q22 · Paper 4 Chemistry Q26 · Paper 4 Physics Q15

You see: A named atmospheric gas, a pollutant, or a stage of water treatment.

You do: Dry air is roughly $78\%$ nitrogen, $21\%$ oxygen and $1\%$ argon, with carbon dioxide a trace. Match each pollutant to its origin and its effect.

Trap: Assuming abundance means importance. Carbon dioxide is a fraction of a percent of the atmosphere and still drives the greenhouse effect.

Practise: Set 19 Chemistry Q4 · Set 19 Chemistry Q14 · Set 19 Chemistry Q21

Worked solutions: None yet

You see: A reaction producing a salt, or a substance to be classified as acid or base.

You do: An acid donates $\text{H}^{+}$; a base accepts it. Acid plus base gives salt plus water; acid plus carbonate adds carbon dioxide; acid plus metal adds hydrogen.

Trap: Forgetting the third product. Carbonates give off carbon dioxide and metals give off hydrogen, and those extra products are usually what the question is testing.

Practise: Set 18 Chemistry Q4 · Set 18 Chemistry Q11 · Set 18 Chemistry Q18

Worked solutions: Paper 4 Chemistry Q7 · Paper 4 Chemistry Q18

You see: An unbalanced equation, a missing state symbol, or a half-equation with electrons to account for.

You do: Balance the atoms element by element, leaving oxygen and hydrogen until last, then balance the charge with electrons if it is a half-equation. Never change a formula to make it balance.

Trap: Altering a subscript instead of a coefficient. Changing the formula changes the substance, which is a chemistry error rather than an arithmetic one.

Practise: Set 17 Chemistry Q6 · Set 17 Chemistry Q12 · Set 17 Chemistry Q18

Worked solutions: None yet

You see: A mass of reactant with a mass of product wanted, or an actual yield quoted against a calculation.

You do: Mass to moles, apply the equation's ratio, moles back to mass. Percentage yield is $\dfrac{\text{actual}}{\text{theoretical}}\times 100$.

Trap: Using the reactant that is in excess. The limiting reactant is the one that runs out, and it alone fixes how much product forms.

Practise: Set 17 Chemistry Q22 · Set 17 Chemistry Q24 · Set 17 Chemistry Q26 · Set 19 Chemistry Q15

Worked solutions: Paper 4 Chemistry Q5 · Paper 4 Chemistry Q25

You see: A mass in grams with a formula, or a composition by mass to be turned into a formula.

You do: $\text{moles} = \dfrac{\text{mass}}{M_r}$, always the first step. For an empirical formula, divide each mass by its $A_r$ and then by the smallest result.

Trap: Comparing masses directly between substances. Only moles compare, because the balanced equation gives a ratio of particles rather than of grams.

Practise: Set 17 Chemistry Q1 · Set 17 Chemistry Q7 · Set 17 Chemistry Q13 · Set 17 Chemistry Q19 · Set 19 Chemistry Q5 · Set 19 Chemistry Q22

Worked solutions: Paper 4 Chemistry Q4 · Paper 4 Chemistry Q24

You see: A volume of gas, or a concentration in $\text{mol dm}^{-3}$ with a volume in $\text{cm}^{3}$.

You do: One mole of any gas occupies the same molar volume, $24\ \text{dm}^{3}$ at room conditions. For solutions, $\text{moles} = \text{concentration}\times\text{volume in dm}^{3}$.

Trap: Leaving the volume in $\text{cm}^{3}$. Divide by $1000$ before multiplying, or the answer is out by exactly that factor.

Practise: Set 17 Chemistry Q23 · Set 17 Chemistry Q25 · Set 17 Chemistry Q27

Worked solutions: Paper 4 Chemistry Q11

You see: An equation where something gains or loses oxygen or electrons, or an oxidation state to assign.

You do: Oxidation is loss of electrons, reduction is gain. Assign states from the rules and see which atom rises and which falls; the oxidising agent is the species reduced.

Trap: Naming the oxidising agent as the thing oxidised. The oxidising agent does the oxidising and is itself reduced, which is the reverse of what the name suggests.

Practise: Set 17 Chemistry Q3 · Set 17 Chemistry Q9 · Set 17 Chemistry Q15 · Set 17 Chemistry Q21 · Set 19 Chemistry Q10

Worked solutions: Paper 4 Chemistry Q2 · Paper 4 Chemistry Q23

You see: Two metals and a salt solution, or an ore and a proposed extraction method.

You do: A more reactive metal displaces a less reactive one from its compound. Extraction method follows reactivity: carbon reduction below carbon in the series, electrolysis above it.

Trap: Extracting a reactive metal with carbon. Aluminium sits above carbon in the series, so only electrolysis works, which is why it stayed a precious metal until electricity was cheap.

Practise: Set 18 Chemistry Q20 · Set 19 Chemistry Q2 · Set 19 Chemistry Q12 · Set 19 Chemistry Q19 · Set 19 Chemistry Q25

Worked solutions: None yet

You see: Two electrodes, a molten or aqueous electrolyte, and products to predict.

You do: Cations go to the cathode and gain electrons; anions go to the anode and lose them. In solution, hydrogen competes at the cathode and oxygen at the anode unless a halide is present.

Trap: Predicting the metal at the cathode of an aqueous solution when the metal is more reactive than hydrogen. Then hydrogen is discharged instead.

Practise: Set 18 Chemistry Q2 · Set 18 Chemistry Q9 · Set 18 Chemistry Q16 · Set 18 Chemistry Q22 · Set 19 Chemistry Q9

Worked solutions: Paper 4 Chemistry Q8 · Paper 4 Chemistry Q21

You see: A mixture to be separated, with some stated difference between its components.

You do: Identify the property that differs - boiling point, solubility, particle size, magnetism - and choose the technique that exploits it. Miscible liquids need fractional distillation; immiscible ones need a separating funnel.

Trap: Reaching for a chemical process. Separating a mixture is physical; only a compound needs a chemical reaction to break apart.

Practise: Set 18 Chemistry Q3 · Set 18 Chemistry Q10 · Set 18 Chemistry Q17 · Set 18 Chemistry Q23

Worked solutions: Paper 4 Chemistry Q6

You see: A change of state, or the arrangement and motion of particles to be described.

You do: Solid: fixed positions, vibrating. Liquid: touching but mobile. Gas: far apart and fast. Energy in loosens the arrangement; energy out tightens it.

Trap: Saying particles expand when heated. The particles are unchanged; they move faster and occupy more space between them.

Practise: Set 19 Chemistry Q6 · Set 19 Chemistry Q16 · Set 19 Chemistry Q23

Worked solutions: None yet

You see: A melting point, a conductivity, or a solubility to be explained rather than looked up.

You do: Name the structure first: giant ionic, giant covalent, simple molecular or metallic. Every physical property follows from what has to be overcome to separate the particles.

Trap: Saying covalent bonds break when a simple molecular substance melts. Only the weak intermolecular forces break; the covalent bonds inside each molecule survive.

Practise: Set 17 Chemistry Q4 · Set 17 Chemistry Q10 · Set 17 Chemistry Q16 · Set 18 Chemistry Q5 · Set 18 Chemistry Q7 · Set 18 Chemistry Q12 · Set 18 Chemistry Q19 · Set 19 Chemistry Q8

Worked solutions: None yet

You see: A general formula, a homologous series, or a functional group named in the stem.

You do: Count carbons and apply the general formula: alkanes $\text{C}_n\text{H}_{2n+2}$, alkenes $\text{C}_n\text{H}_{2n}$, alcohols $\text{C}_n\text{H}_{2n+1}\text{OH}$. The functional group decides the reaction.

Trap: Confusing addition with substitution. Alkenes add across the double bond; alkanes only substitute, and only with ultraviolet light.

Practise: Set 18 Chemistry Q14 · Set 19 Chemistry Q1 · Set 19 Chemistry Q11 · Set 19 Chemistry Q18 · Set 19 Chemistry Q24 · Set 19 Chemistry Q27

Worked solutions: None yet

You see: A change in concentration, temperature, surface area or catalyst, with an effect on rate to predict.

You do: Ask what happens to the frequency of successful collisions. Concentration, pressure and surface area raise the frequency; temperature raises both frequency and energy; a catalyst lowers the activation energy.

Trap: Saying a catalyst is used up or changes the yield. It is unchanged at the end and shifts nothing but the speed.

Practise: Set 18 Chemistry Q1 · Set 18 Chemistry Q8 · Set 18 Chemistry Q15 · Set 18 Chemistry Q21 · Set 19 Chemistry Q7

Worked solutions: Paper 4 Chemistry Q17

You see: A temperature change in a calorimeter, an energy level diagram, or a table of bond energies.

You do: Bond breaking is endothermic and bond forming is exothermic, so $\Delta H = \text{bonds broken}-\text{bonds formed}$. In calorimetry, $q = mc\Delta T$.

Trap: Getting the sign backwards. A negative enthalpy change means energy is released and the surroundings warm up, which feels like a gain but is a loss from the system.

Practise: Set 18 Chemistry Q24 · Set 18 Chemistry Q26 · Set 18 Chemistry Q27 · Set 19 Chemistry Q17

Worked solutions: None yet