Free resource

TMUA Survival Kit

The formulas you must know cold, the speed techniques that win a no-calculator paper, and the ten traps that catch people out most. Skim it now, print it for the fridge, and reach for it the week of your sitting.

Formulas to know cold

Algebra & indices

Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Discriminant $b^{2} - 4 a c$ : $> 0$ two roots, $= 0$ one repeated root, $< 0$ none
Difference of squares: $a^{2} - b^{2} = (a - b) (a + b)$
Indices: $a^{m} a^{n} = a^{m + n}$ , $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{- n} = \frac{1}{a ^{n}}$ , $a^{1/ n} = n a$

Logarithms

$lo g_{a} x = y ⟺ a^{y} = x$
$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ and $lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$
$lo g_{a} (x^{k}) = k lo g_{a} x$ and $a^{l o g_{a} x} = x$
Change of base: $lo g_{a} x = \frac{lo g _{b} x}{lo g _{b} a}$

Sequences & series

Arithmetic: $u_{n} = a + (n - 1) d$ , $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$
Geometric: $u_{n} = a r^{n - 1}$ , $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$
Sum to infinity: $S_{\infty} = \frac{a}{1 - r}$ , only when $∣ r ∣ < 1$

Coordinate geometry

Gradient $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ ; perpendicular gradients multiply to $- 1$
Distance $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ ; midpoint $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Line: $y - y_{1} = m (x - x_{1})$
Circle, centre $(a, b)$ radius $r$ : $(x - a)^{2} + (y - b)^{2} = r^{2}$

Trigonometry

$sin^{2} θ + cos^{2} θ = 1$ and $tan θ = \frac{sin θ}{cos θ}$
Exact: $sin 3 0^{\circ} = \frac{1}{2}$ , $cos 3 0^{\circ} = \frac{3}{2}$ , $tan 4 5^{\circ} = 1$ , $sin 6 0^{\circ} = \frac{3}{2}$
Double angle: $sin 2 θ = 2 sin θ cos θ$ , $cos 2 θ = 2 cos^{2} θ - 1$

Calculus & counting

Differentiate: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ ; stationary where $\frac{d y}{d x} = 0$
Integrate: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c$ (for $n \neq = - 1$ )
Choose: $(r n) = \frac{n !}{r ! ( n - r )!}$
Probability: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ ; independent $\Rightarrow P (A \cap B) = P (A) P (B)$

Exam-day speed techniques

1
Read the options first
They reveal the form of the answer and often let you work backwards instead of solving forwards.
2
Test small values
For 'which is always true' or unknown coefficients, substitute $x = 0, 1, - 1$ to eliminate wrong options fast.
3
Eliminate before you compute
Rule out options with the wrong sign or wrong magnitude first; you may be left with one.
4
Estimate and bound
You rarely need the exact value to choose. Bounding the answer is often enough and much faster.
5
Keep everything exact
Work in surds and fractions, never decimals. The answers are exact, and rounding loses you the match.
6
Answer the question asked
If it asks 'how many solutions', use the discriminant. Do not waste time finding the actual roots.
7
Mind the clock
About 75 minutes for 20 questions is under 4 minutes each. Flag a hard one, move on, come back.

The 10 most common traps

Trap 1. Counting a repeated root twice

Fix. $x^{2} - 6 x + 9 = (x - 3)^{2}$ has one solution, not two. When the discriminant is $0$ , there is a single repeated root.

Trap 2. Keeping a rejected root

Fix. Logs and square roots need a valid domain. After solving, discard any root that makes an argument negative or undefined.

Trap 3. Dropping the $\pm$

Fix. $x^{2} = 4 \Rightarrow x = \pm 2$ , not just $2$ . Tangency conditions often give $k = \pm 2$ as well.

Trap 4. Strict vs inclusive inequality

Fix. $<$ excludes the boundary, $\leq$ includes it. Whether the endpoint counts changes the answer.

Trap 5. Off-by-one in a sequence

Fix. The $n$ th term is $a + (n - 1) d$ , not $a + n d$ . The 10th term uses $n = 10$ , so $+ 9 d$ .

Trap 6. Misusing sum to infinity

Fix. $S_{\infty} = \frac{a}{1 - r}$ only converges when $∣ r ∣ < 1$ . Otherwise the series has no finite sum.

Trap 7. AND vs OR in probability

Fix. 'A and B' multiplies (if independent); 'A or B' adds and then subtracts the overlap $P (A \cap B)$ .

Trap 8. Losing the constant of integration

Fix. An indefinite integral needs $+ c$ . It matters whenever you are asked to find the actual curve or function.

Trap 9. Log law slips

Fix. $lo g (x + y) \neq = lo g x + lo g y$ . The product law is $lo g (x y) = lo g x + lo g y$ ; there is no law for a sum inside.

Trap 10. Quantifier and logic traps (Paper 2)

Fix. The negation of 'for all $x$ , $P$ ' is 'there exists an $x$ with not $P$ '. And 'necessary' is not the same as 'sufficient'.

Now put it into practice

Knowing the traps is half the battle. The other half is spotting them under time pressure, with a full worked solution every time.

Start practising free