Free printable resource

TMUA Survival Kit

There's no formula booklet and no calculator, so you have to recall all of this. Every formula below is on the official syllabus. Print it and keep it by your desk for the run-up to your sitting.

Formulas to recall

Algebra & indices

Quadratic roots: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Discriminant $b^{2} - 4 a c$ : $> 0$ two roots, $= 0$ one repeated, $< 0$ none
Complete the square: $x^{2} + b x + c = (x + \frac{b}{2})^{2} + c - \frac{b ^{2}}{4}$
Difference of squares: $a^{2} - b^{2} = (a - b) (a + b)$
Indices: $a^{m} a^{n} = a^{m + n}, (a^{m})^{n} = a^{mn}, a^{- n} = \frac{1}{a ^{n}}, a^{1/ n} = n a$
Rationalise: $\frac{1}{a} = \frac{a}{a}$
Factor theorem: $(x - a)$ is a factor of $f (x) ⟺ f (a) = 0$
Remainder theorem: dividing $f (x)$ by $(x - a)$ leaves remainder $f (a)$

Logarithms & exponentials

$lo g_{a} x = y ⟺ a^{y} = x$
$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y, lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$
$lo g_{a} (x^{k}) = k lo g_{a} x, a^{l o g_{a} x} = x$
$lo g_{a} a = 1, lo g_{a} 1 = 0$
(Change of base is NOT tested — don't waste time on it.)

Sequences & series

Arithmetic: $u_{n} = a + (n - 1) d, S_{n} = \frac{n}{2} (2 a + (n - 1) d)$
Sum of first $n$ naturals: $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$
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 $∣ r ∣ < 1$
Binomial ( $n$ a positive integer): $(1 + x)^{n} = \sum_{r = 0}^{n} (r n) x^{r}$
$(r n) = \frac{n !}{r ! ( n - r )!}$

Coordinate geometry

Gradient $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ ; parallel $\Rightarrow$ equal, perpendicular $\Rightarrow$ product $- 1$
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

$sin^{2} θ + cos^{2} θ = 1, tan θ = \frac{sin θ}{cos θ}$
Sine rule: $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ (watch the ambiguous case)
Cosine rule: $a^{2} = b^{2} + c^{2} - 2 b c cos A$
Area of a triangle: $\frac{1}{2} ab sin C$

Radians & circles

$π rad = 18 0^{\circ}$
Arc length: $s = r θ$
Sector area: $\frac{1}{2} r^{2} θ$
(Angles in radians throughout.)

Differentiation & integration

$\frac{d}{d x} (x^{n}) = n x^{n - 1}$ (differentiate term by term)
Stationary points where $\frac{d y}{d x} = 0$ ; max/min from the sign of $\frac{d ^{2} y}{d x ^{2}}$
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c (n \neq = - 1)$
Area under a curve: $\int_{a}^{b} y d x$
Trapezium rule estimates area (over/underestimate depends on curvature)

Probability & counting

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Independent $\Rightarrow P (A \cap B) = P (A) P (B)$
Conditional: $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$
Arrangements use $n!$ ; selections use $(r n)$

Graphs & transformations

$y = f (x) + a$ shifts up $a$ ; $y = f (x + a)$ shifts left $a$
$y = a f (x)$ stretches vertically by $a$ ; $y = f (a x)$ stretches horizontally by $\frac{1}{a}$
$y = - f (x)$ reflects in the $x$ -axis; $y = f (- x)$ in the $y$ -axis
$a (x + b)^{2} + c$ has vertex $(- b, c)$

Logic & proof (Paper 2)

Contrapositive of 'if $P$ then $Q$ ' is 'if not $Q$ then not $P$ ' — always equivalent
Converse 'if $Q$ then $P$ ' is NOT equivalent to the original
$P$ necessary for $Q$ : $Q \Rightarrow P$ . $P$ sufficient for $Q$ : $P \Rightarrow Q$
Negate 'for all $x$ , $P$ ' $\to$ 'there exists $x$ with not $P$ ' (and vice versa)
Proof: direct, by cases, by contradiction, by counterexample (induction is NOT on the TMUA)

Exact values

	$0^{\circ}$	$3 0^{\circ}$	$4 5^{\circ}$	$6 0^{\circ}$	$9 0^{\circ}$
$sin$	$0$	$\frac{1}{2}$	$\frac{2}{2}$	$\frac{3}{2}$	$1$
$cos$	$1$	$\frac{3}{2}$	$\frac{2}{2}$	$\frac{1}{2}$	$0$
$tan$	$0$	$\frac{3}{3}$	$1$	$3$	$—$

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 kill wrong options fast.
3. Eliminate before you compute. Rule out options with the wrong sign or wrong magnitude first; you may be left with just 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. There is no formula booklet and no calculator, so recall matters.
6. Answer the question asked. If it asks 'how many solutions', use the discriminant. Don't waste time finding the actual roots.
7. Mind the clock. 75 minutes for 20 questions is under 4 minutes each. Flag a hard one, move on, come back.

The 10 most common traps

1. Counting a repeated root twice

Fix: $x^{2} - 6 x + 9 = (x - 3)^{2}$ has one solution, not two. Discriminant $0$ means a single repeated root.

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.

3. Dropping the $\pm$

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

4. Strict vs inclusive inequality

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

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 $+ 9 d$ .

6. Misusing sum to infinity

Fix: $S_{\infty} = \frac{a}{1 - r}$ only converges when $∣ r ∣ < 1$ . Otherwise there is no finite sum.

7. AND vs OR in probability

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

8. Losing the constant of integration

Fix: An indefinite integral needs $+ c$ . It matters whenever you're asked for the actual curve or function.

9. Log law slip

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's no law for a sum inside.

10. Quantifiers & logic (Paper 2)

Fix: The negation of 'for all $x$ , $P$ ' is 'there exists $x$ 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