Maths Methods Semester 2 COMPLETE STUDY GUIDE (Units 1&2 – A-Grade Sheet)

Read this once. It is everything that appears in every Sem 2 WA paper (WAEP 2018, CCGS 2019, RSHS 2021 analysed). Red = traps.

Exam Structure (same every year)

• Section One (Calculator-free): 8 questions, 52 marks, 35%, 50 min. Short answer; no calculator of any kind. Exact values, simple proofs, sketching, symbol-pushing.
• Section Two (Calculator-assumed): 13 questions (Q9–Q21), 98 marks, 65%, 100 min (after 10 min reading). Same types but with nasty numbers — set everything up, then let the calculator finish. Formula sheet retained from Section One.
• Marks are mostly 1–4 each; Section Two has the long apply-to-context questions (models, word problems, optimisation, probability tables). Calculator is only worth ~5% of marks — set-up and theory still earn almost everything; never skip showing the equation you solved.

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Golden Equations (write these on your formula sheet – Unit 1&2 Sem 2)

Sequences: AP: $u_n = a + (n-1)d$, $S_n = \tfrac{n}{2}\big[2a + (n-1)d\big] = \tfrac{n}{2}(a + u_n)$. GP: $u_n = ar^{n-1}$, $S_n = \dfrac{a(1-r^n)}{1-r}$, infinite sum $S_\infty = \dfrac{a}{1-r}$ only if $|r| < 1$. Exponential/log: $a^m a^n = a^{m+n}$, $(ab)^m = a^m b^m$, $(a^m)^n = a^{mn}$, $a^{-n} = \dfrac{1}{a^n}$, $a^{m/n} = \sqrt[n]{a^m}$. $\log_a(xy) = \log_a x + \log_a y$; $\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y$; $\log_a(x^b) = b\log_a x$; $\log_a a = 1$; $\log_a 1 = 0$; change of base $\log_a x = \dfrac{\log_b x}{\log_b a}$. Exponential eqn: $a\,b^x = c \Rightarrow x = \dfrac{\ln(c/a)}{\ln b}$. Circle measure: $\theta = \dfrac{s}{r}$ (radians); sector $A = \tfrac12 r^2\theta$, arc $s = r\theta$; segment area = sector − triangle $\big(\tfrac12 r^2\theta - \tfrac12 r^2\sin\theta\big)$ — the formula sheet gives sector only, you subtract the triangle yourself for segments. Rates (difference quotient): $\dfrac{f(x+h)-f(x)}{h}$ = average rate over $[x, x+h]$; instantaneous rate $= \lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h} = f'(x)$. Derivatives: $\dfrac{d}{dx}x^n = nx^{n-1}$; $\dfrac{d}{dx}\big(a\,f(x)\big) = a\,f'(x)$; sum rule; $\dfrac{d}{dx}(ax+b)^n = an(ax+b)^{n-1}$ (chain rule result). $\dfrac{d}{dx}k = 0$. First principles: $f'(x) = \lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$. Optimisation recipe: “draw the diagram → express the quantity to maximise/minimise in one variable → differentiate → set $f' = 0$ → solve → test (sign of $f'$ either side or endpoints) → answer with units.” Trig model: $a\sin(bt)+c$ or $a\cos(bt)+c$: amplitude $a$, period $\dfrac{2\pi}{b}$ (radians), centre/mean line $c$, max $a+c$, min $c-a$. Phase: goes through max at $t=0$ → cosine; through mean then up → sine.

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1. Sequences (every paper)

Arithmetic (AP)

• $u_n = a + (n-1)d$; $S_n = \tfrac{n}{2}\big[2a + (n-1)d\big]$. Given two terms, solve simultaneously for $a$ and $d$.
• Find d from any two terms: difference ÷ gap in positions (e.g. $u_7 - u_3 = 4d$).

Geometric (GP)

• $u_n = ar^{n-1}$; $S_n = \dfrac{a(1-r^n)}{1-r}$. Given two terms and an index, divide one equation by the other to eliminate $a$, solve for $r$ first.
• Infinite sum: $S_\infty = \dfrac{a}{1-r}$ valid only if $|r| < 1$. If $|r| \geq 1$ the sum does not exist — say “diverges / no finite sum”.
• RED TRAP (CCGS19 Q13): quadratic in $r$ gives two values (e.g. $r = 1.4$ or $-0.4$). The finite-$S_\infty$ condition $|r| < 1$ selects $-0.4$; you lose marks by reporting the other root. Always check which $r$ actually fits the situation.
• “Grows by $k\%$ per unit” → $r = 1 + \dfrac{k}{100}$; “decays by $k\%$” → $r = 1 - \dfrac{k}{100}$.

Word problems (WAEP18 Q9×, RSHS21 Q13 aeroplane)

1. Identify AP or GP and write $a$ and $d$/$r$ from the first two pieces of data.
2. Show the explicit model: $u_n = \dots$ or $S_n = \dots$.
3. Answer the specific index / cumulative question — read whether it asks single term ($u_n$) or cumulative ($S_n$).
4. Check the unit (metres, mL, people) and a sensible range.

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2. Functions & Relations (Section One heavy)

• Relation vs function: a function has exactly one $y$ for each $x$ — every vertical line cuts the graph at most once.
• Domain/range: state the natural domain ($x \neq 0$ for $\tfrac{1}{x}$, $x \geq k$ under a square root, all reals for linear/quadratic) and the matching range. Say them from the graph, not from memory for circles/parabolas.
• Transformations: $y = a\,f\big(b(x-h)\big) + k$ — $a$ vertical stretch/reflection, $b$ horizontal stretch/$\tfrac{1}{b}$, $h$ shift right, $k$ shift up. Reflections: $-f(x)$ across x-axis, $f(-x)$ across y-axis.
• Inverse notation: $f^{-1}$ swaps x and y and reflects in $y = x$; not $\dfrac{1}{f(x)}$.
• Distance/midpoint: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, $M = \left(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\right)$. Gradients of perpendicular lines multiply to $-1$.
• Circle: centre $(h,k)$, $(x-h)^2 + (y-k)^2 = r^2$. Tangent at a point is perpendicular to the radius to that point.

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3. Trigonometry & Circle Measure (every paper)

Radians + exact values

• $\pi = 180°$; convert by $\times \dfrac{\pi}{180}$ or $\times \dfrac{180}{\pi}$. RED TRAP: leave answers in radians when the question says so — radian mode on the calculator.

Exact-value table (learn it – Section One guarantees it):

$\theta$	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$
$\sin\theta$	$0$	$\tfrac12$	$\tfrac{\sqrt2}{2}$	$\tfrac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac{\sqrt2}{2}$	$\tfrac12$	$0$
$\tan\theta$	$0$	$\tfrac{1}{\sqrt3}$	$1$	$\sqrt3$	—

• Unit circle: $\sin = y$, $\cos = x$. Quadrants (CAST): 1st all $+$, 2nd $\sin$ $+$, 3rd $\tan$ $+$, 4th $\cos$ $+$. Sign of a trig value decides which quadrant solutions live in.
• Reference-angle method for equations: solve $\sin(x) = k$ in $[0, 2\pi)$ by finding the acute reference angle then placing it in the quadrants where the function is positive/negative as required.
• Identities: $\sin^2\theta + \cos^2\theta = 1$; $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$; period of sin/cos $= 2\pi$, tan $= \pi$.

Sectors & segments

• Area of sector $= \frac12 r^2\theta$, arc $s = r\theta$ — $\theta$ in radians.
• Segment area $= \frac12 r^2\theta - \frac12 r^2\sin\theta$ (sector minus isosceles triangle). The triangle is always $\tfrac12 r^2\sin\theta$ regardless of how big $\theta$ is.
• RED TRAP: $r$ must be the radius of the sector, not a side given elsewhere; and angle mode must be radians when using $\frac12 r^2\theta$.

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4. Exponential Growth & Decay (WAEP9/11, CCGS15)

• General form $y = A\,b^{kt}$ or $y = A\,a^t$. Write the model first, then substitute the point(s).
• Typical: $T = 840(0.94)^t$ — after which time does $T$ halve/reach a target? → set expression = target, take logs, solve.
• % change per unit: “gets 6% smaller each period” → base $0.94$ (i.e. $1 - 0.06$); “increases by $r\%$” → base $1 + \tfrac{r}{100}$. RED TRAP: the base is NOT the percentage — “decays by 6%” means multiply by 0.94, not 0.06.
• Half-life/finding t: take $\ln$ of both sides; require at least 2 correct decimal places and give units.
• Reading rate from a word problem (CCGS15): identify the starting value ($A$, when $t=0$), the final value, and how the quantity changes per unit time → write $y = A\,b^{kt}$ then solve for $k$ with one known point.

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5. Probability: Two-Way Tables (every paper — biggest Section Two block)

Structure

A two-way table classifies by two attributes (e.g. Android vs iPhone × battery life). Fill the row totals, column totals and grand total FIRST — most cells are found by subtraction.

Conditioning (the whole game)

• $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ — denominator is the row/column total you were given the condition on. If asked “…given that it has a long battery” then denominator = battery column total, not the grand total.
• $P(A \text{ and } B)$ at the intersection cell (top-left style). $P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$.
• Complement: $P(A') = 1 - P(A)$; $P(A \cap B') = P(A) - P(A \cap B)$.

Independence (definition marks)

• Independent: $P(A \cap B) = P(A) \cdot P(B)$. Equivalent check: $P(A \mid B) = P(A)$.
• Mutually exclusive: $P(A \cap B) = 0$ (can’t both happen) — different from independent; independent events can overlap.
• RED TRAP: “independent” and “mutually exclusive” are not synonyms. Write both definitions if a question asks one.
• Given $P(A)$, $P(A \cap B')$ (or other region) as unknowns → build a full table/region diagram with a letter for the missing cell, solve.

Geometric distribution (CCGS19 Q20)

• Repeated independent trials, probability of success $p$ each trial.
• $P(\text{first success on trial } n) = q^{\,n-1} \cdot p$ where $q = 1 - p$. RED TRAP: the formula has $q^{\,n-1}p$, not $q^{n}p$ — count the failures BEFORE, not after.
• “At most $n$ attempts” → sum geometric terms or $1 - P(\text{no success in } n)$.

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6. Combinatorics (WAEP16, RSHS18)

• $\displaystyle \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$ — unordered selections; $^nP_r$ — ordered arrangements. RED TRAP: wording — “arrangements/orders” = $P$ (order matters), “groups/choose/committee” = $C$.
• Probability numerator = favourable arrangements (choose the wanted group) × any remaining picks; denominator = total arrangements.
• “At least one” → $1 - P(\text{none})$. Separate “at least 2 of a kind” → count directly or complement up to it.
• Small numbers: list carefully instead of formulas for marks in method anyway.

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7. Circle/Hyperbola/Linear Intersections + Discriminant (RSHS19, CCGS12)

• Intersect two curves: solve simultaneously (substitute $y$). The number of solutions is decided by the discriminant $\Delta = b^2 - 4ac$:
  • $\Delta > 0$ → two distinct solutions (secant/intersects twice), $\Delta = 0$ → tangent (touches once), $\Delta < 0$ → no real intersection.
• Tangency condition = $\Delta = 0$. Example: line $y = mx + 3$ tangent to hyperbola $y = \dfrac{a}{x+b}$ → substitute, multiply through, set discriminant $= 0$, solve for $m$ and the contact point.
• Hyperbola $y = \dfrac{a}{x+b} + c$: vertical asymptote $x = -b$, horizontal asymptote $y = c$.
• Sign of $\Delta$ decides “touches / cuts / misses” — answer literally using one of those three words.

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8. Difference Quotient & Rate of Change (CCGS14, RSHS15)

• Average rate of change of $f$ on $[a,b]$ $= \dfrac{f(b) - f(a)}{b - a}$ — slope of the secant.
• Instantaneous rate at $x$ $= \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h} = f'(x)$.
• Typical calc-assumed task: table with $h = 0.01$ available → compute $\dfrac{f(x+h) - f(x)}{h}$ directly = IROC approximation. $h = 0.001$ gives an even better approximation — smaller $h$ = closer to the true IROC.
• Units of a rate always “quantity per time” (e.g. m/s, people/year).
• RED TRAP: on the calculator, brackets matter: $((f(b) - f(a))/(b - a))$, and for instantaneous use the smaller $h$ you’re given.

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9. Differentiation (both sections)

First principles (usually Section One)

• $f'(x) = \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h}$. Expand, cancel the $h$ in the denominator, then let $h \to 0$.
• RED TRAP: you must cancel a factor of $h$ — substituting $h=0$ into the pre-cancelled quotient gives $\frac{0}{0}$ and earns nothing.

Rules (Section Two)

• Power rule $nx^{n-1}$, constant multiple, sum/difference. $\dfrac{d}{dx}(\text{constant}) = 0$.
• Chain (linear function): $\dfrac{d}{dx}(ax+b)^n = an(ax+b)^{n-1}$ — don’t forget the extra factor $a$ from the inner derivative (e.g. $(2x+1)^5 \to 10(2x+1)^4$).
• For expressions like $x^2\sqrt{x}$ / $\dfrac{x+2}{x^2}$ → rewrite with negative/fractional powers first, then differentiate term by term.
• Tangent: $m = f'(a)$ at the point; equation $y - f(a) = m(x - a)$ or $y = mx + c$ with the point subbed in.
• Normal: gradient $= -\dfrac{1}{f'(a)}$.
• $f'(x) = 0$ at stationary points → solve, substitute into $f$. Nature by sign test of $f'$ around the point (or second derivative).
  • $f'' > 0$ → local minimum; $f'' < 0$ → local maximum; $f'' = 0$ / change test → check both sides, might be a horizontal point of inflection (stationary point of inflection) where sign of $f'$ does NOT change.

Sketching polynomials

• $y$-intercept ($x=0$), $x$-intercepts (factor), stationary points + nature, end behaviour from the leading term (degree + sign).
• Cubic with factorable $f'$ (e.g. $f'(x) = 4(x-1)^2(x+2)$) → stationary points at both roots; at a double root of $f'$ the sign doesn’t change → stationary point of inflection, not a turning point.

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10. Optimisation (every paper — the full-marks word question)

Give all 5 steps for full marks:

1. Define variable(s); draw the diagram and label.
2. Write the quantity to maximise/minimise in terms of one variable (eliminate the other using the constraint, e.g. cone volume $V = 25\pi x^2 - 2.5\pi x^3$ from $h$ substituted from a similar-triangles ratio).
3. Differentiate; set $= 0$; solve (calculator for solving).
4. Test for maximum vs minimum (sign of derivative either side, or second derivative, or endpoints of the domain). State which.
5. Answer the question with units (cm³, m²); if asked, give the optimal $x$ too and verify it’s in the domain.

• RED TRAP: most optimisation marks are in setup + testing + units, not the algebra. Never stop at “found $x$”. Domain must be stated (e.g. $0 < x < 10$ if width can’t be negative or exceed material).
• Common solids that appear: cylinder in a cone, box from a sheet, rectangle with a given perimeter/area.

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11. Trigonometric Models (WAEP18 Ferris wheel, RSHS20 springs)

Given a height model like $h = 6.5\cos\left(\dfrac{\pi t}{25}\right) + 8$:

• Amplitude $a$ = distance from the centre-line to max/min; period $\dfrac{2\pi}{b}$; centre-line $c$ (mid-height).
• Max $= c + a$, min $= c - a$. First reaches a given height: solve $a\cos(bt) + c = \text{value}$, take the correct branch (rising vs falling branch of cosine).
• “First time at a height while rising” vs while falling → choose the appropriate solution from the unit circle (rising = the solution after the minimum, falling = the one before/after maximum). Diagram always helps.
• Ferris wheel: $h = 6.5\cos\left(\dfrac{\pi t}{25}\right) + 8$ → amplitude 6.5, period 50 s, min height 1.5 m, max 14.5 m.
• Multiple springs $h_A = 16\cos\left(\tfrac{3\pi t}{4}\right)+20$, $h_B = 12\sin\left(\tfrac{3\pi t}{4}\right)+25$, $h_C = 12\cos\left(\tfrac{5\pi t}{4}\right)+20$: compare amplitude (height range), period (frequency), and max for “which spring hangs highest”. “First reaches maximum” = solve where the trig argument $= 0$ (cos) or $\tfrac{\pi}{2}$ (sin) — includes the leading coefficient sign.

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12. Probability: Rules + Applications (Section One)

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Venn: union is the whole shaded shape; intersection the overlap.
• $P(A') = 1 - P(A)$; $P(A \cap B') = P(A) - P(A \cap B)$.
• Conditional probability from a small table or given numbers: identify the reduction of the sample space first, then divide.
• “Of the people who have the virus, what fraction test positive” type questions → reverse conditional, build a tree/table with a total of 1, then $\dfrac{P(A \cap B)}{P(B)}$.
• Independent: multiply. Not independent: must use conditional — don’t multiply.
• Give probabilities as decimals/fractions, check $0 \leq P \leq 1$, and state the answer rounded as the paper requests (often 3 dp).

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13. Rectilinear Motion + Antidifferentiation (Unit 2 tail)

• Displacement $x(t)$ → velocity $v(t) = \dfrac{dx}{dt}$; acceleration $a(t) = \dfrac{dv}{dt}$. Reverse: antidifferentiate to go down the chain.
• $v = 0$ → turning point of motion (instantly at rest). Positive $v$ = moving forward/up; negative $v$ = moving backward/down.
• Antidifferentiate: $\displaystyle \int x^n\, dx = \frac{x^{\,n+1}}{n+1} + c$; add the constant $c$ and find it from an initial condition.
• General solution at a given time, then use $x$ at $t=0$ (or a stated position) to fix $c$ — otherwise the graph is wrong.
• RED TRAP: “find displacement after $t$ seconds” sometimes wants net displacement ($x(t) - x(0)$) and sometimes total distance — calculate both, read which is asked.

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Extended-Answer Scenarios (the 5 shapes that repeat)

1. A word problem → an exponential/geometric model. Write model → answer two numeric parts (given a value find the time, or given time find value) → then one interpretation sentence (“this is the amount remaining when…”). Read off answers rounded to 3 dp with units.
2. Optimisation with a real container/field: full 5-step recipe above. Setup in one variable is worth the most marks.
3. Trigonometric model (wheel/swinging weight): write amplitude/period/max/min; solve “first time the object is above a height”; sketch one cycle with labelled intercepts/max/min.
4. Two-way table + conditioning/probability: fill the table, answer 3–4 probability questions, then an independence/conditional “is $P(A \mid B)$ equal to $P(A)$?” verification — conclude with “…therefore the events are independent” only if both values agree.
5. Line tangent to a curve (hyperbola/parabola): substitute, discriminant, tangency $= \Delta = 0$, find the tangent equation / no-tangency range for $m$.

Marking hints: 1 mark = correct model/equation written; 1 mark = solving done (calculator accepted); 1 mark = the answer stated with correct units; context sentences earn the last mark in “interpret” questions — always write one complete sentence.

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Final Checklist Before the Test

• Sequences: AP $u_n = a + (n-1)d$, GP $u_n = ar^{n-1}$; $S_\infty$ only if $|r| < 1$; use the $r$-value that survives the $|r| < 1$ test.
• Exponent rules: negatives to reciprocals, fractional = root, log power-move-down, change of base in a calculator.
• Exact trig values & radians: segment = sector − triangle; $\theta$ in radians for $\tfrac12 r^2\theta$.
• Two-way tables: fill totals first; conditional denominator = the row/column total of the given; independent ⇔ $P(A \cap B) = P(A)P(B)$.
• Counting: “order matters” → $^nP_r$; “choose/groups” → $\binom{n}{r}$; “at least one” → $1 - P(\text{none})$.
• Difference quotient: average over an interval uses the interval width; instantaneous uses the smallest $h$; brackets on the calculator.
• Differentiation: power rule + chain − always keep the inner-derivative factor; tangent vs normal gradients are negative reciprocals; double root of $f'$ = stationary point of inflection (sign doesn’t change).
• Optimisation: define → one variable → $f' = 0$ → test nature → answer with units and domain.
• Trig models: amplitude $a$, period $\dfrac{2\pi}{b}$, centre $c$; “rising vs falling” branch matters for first-time heights.
• Discriminant: $\Delta > 0$ two intersections, $\Delta = 0$ tangent, $\Delta < 0$ none — answer in those words.
• Motion/antidiff: always $+c$ and fix it; net displacement vs total distance is a real question you must notice.
• Section One is a no-calculator marathon: exact surds, first-principles limits, exact trig, and the definitions above are your marks. Section Two: set up, let the calculator solve, write units.