Verified Ingestion Snapshot

Source	Files	Formats found	Extraction status
/content/	4	2 .txt, 2 .pdf	4/4 extracted
/Methods_wacevault_downloads/	80	53 .pdf, 27 .docx	80/80 extracted
Total	84	txt/pdf/docx	84/84 extracted, 0 skipped

All content below is based on ingested files only. OCR-distorted items were excluded if they were not reliably readable.

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Exam Structure Seen Repeatedly in Papers

Section	Reading	Working	Typical marks/weighting in ingested papers
Section One (Calculator-free)	5 min	50 min	~52 marks (~35%)
Section Two (Calculator-assumed)	10 min	100 min	~98 marks (~65%)

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High-Frequency Question Patterns (from ingested question snippets)

Pattern family	Count in extracted snippets*	Typical prompt style
Functions/relations	184	domain/range, function test, graph interpretation
Circle/geometry	134	circle equations, radius/centre, triangle geometry
Trig identities/equations	99	simplify/evaluate/solve trig expressions
Probability/sets	71	$P(A\cup B)$, conditional probability, independence
Line equations/coordinates	60	perpendicular/parallel line, midpoint, distance
Modelling/context	60	time-height models, applied probability contexts
Quadratic skills	44	turning point, factor form, min/max
Asymptotes/rational	25	hyperbola graph/asymptotes/intersections
Counting/binomial	9	combinations, expansion, coefficient meaning

*Counts include repeated exam+solution variants, so use as a priority signal only.

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Core Concepts + Formula Bank

1. Functions, relations, domain/range

• Function test: each $x$ maps to one $y$ (vertical line test on graphs).
• Mappings: One-to-One and Many-to-One are functions. One-to-Many is not a function.
• Domain: allowed $x$-values.
• Range: resulting $y$-values.
• Common exam move: state domain/range from graph/features first, then solve.

2. Lines and coordinate geometry

• Gradient from $y=mx+c$: $m$.
• Angle of Inclination: Gradient $m = \tan \theta$ ($\theta$ is the angle with the positive $x$-axis).
• Perpendicular gradients: $m_1m_2=-1$.
• Point-slope form: $y-y_1=m(x-x_1)$.
• Midpoint: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$, distance by Pythagoras.

3. Quadratics

• Turning point form: $y=a(x-h)^2+k$ (vertex $(h,k)$).
• Symmetry line: $x=h$.
• Min/max on restricted domain: check vertex + domain endpoints.
• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Discriminant ($\Delta = b^2 - 4ac$): $\Delta > 0$ (2 distinct real roots), $\Delta = 0$ (1 repeated real root/tangent to x-axis), $\Delta < 0$ (0 real roots).

4. Trigonometry

• Degree-radian: $\theta_{\text{rad}}=\theta^\circ\pi/180$, $\theta^\circ=\theta_{\text{rad}}180/\pi$.
• Identity used repeatedly: $\cos(A+B)=\cos A\cos B-\sin A\sin B$.
• Trig Graphs ($y = a \sin(b(x-h)) + k$): Amplitude = $|a|$, Period = $\frac{2\pi}{|b|}$ (for $\sin/\cos$) or $\frac{\pi}{|b|}$ (for $\tan$), Phase shift = $h$ (right).
• Solve trig equations by:
  1. isolate trig term,
  2. find reference angles,
  3. apply interval limits.

5. Probability

• Addition rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• Conditional: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$.
• Independence test: $P(A\cap B)=P(A)P(B)$.
• Complementary / Not A: $P(A') = 1 - P(A)$.
• Mutually Exclusive: $P(A\cap B) = 0$.

6. Counting & Set Theory

• Arrangement / Permutation ($P$): Order matters $\implies {}^nP_r=\frac{n!}{(n-r)!}$.
• Selection / Combination ($C$): Order doesn’t matter $\implies \binom{n}{r}=\frac{n!}{r!(n-r)!}$.
• Multiplication Principle: Multiply independent event outcomes (e.g. choosing shirt AND pants).
• Subsets & Binomial Terms: A set of $n$ elements has $2^n$ subsets. General binomial term in $(a+b)^n$ is $\binom{n}{r} a^{n-r} b^r$.
• Sets: $A \cap B$ (AND / intersection shape), $A \cup B$ (OR / union shape), $A'$ (NOT / complement).

7. Circular/triangle geometry

• Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (Ambiguous case: Given two sides and a non-included angle ($\text{SSA}$), the unknown angle can be $\theta$ or $180^\circ - \theta$).
• Cosine rule: $c^2=a^2+b^2-2ab\cos C$.
• Arcs & Sectors (radians): Arc length $L=r\theta$, sector area $A=\frac12 r^2\theta$.
• Triangle area: $A=\frac12ab\sin C$.
• Circular Segment area: $A=\frac12 r^2(\theta - \sin\theta)$.
• Circles: Center $(h,k)$, radius $r \implies (x-h)^2 + (y-k)^2 = r^2$.

8. Polynomials & Division

• Remainder Theorem: If a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$.
• Factor Theorem: $(x-a)$ is a factor of $P(x)$ if and only if $P(a) = 0$.
• Synthetic Division / Equating Coefficients: Use to completely factorise cubics once one root is found.

9. General Transformations ($y=af(b(x-h))+k$)

• Parent functions: Hyperbola $y=1/x$ (asymptotes $x=0, y=0$), Square root $y=\sqrt{x}$, Circle $x^2+y^2=r^2$.
• $a$: Vertical dilation by factor $a$. If negative, reflection in the $x$-axis.
• $b$: Horizontal dilation by factor $\frac{1}{b}$. If negative, reflection in the $y$-axis.
• $h$: Horizontal translation ($h>0$ moves right).
• $k$: Vertical translation ($k>0$ moves up).

10. Abstract Algebra & Proportion

• Sum of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
• Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
• Direct Proportion: $y \propto x \implies y = kx$.
• Inverse Proportion: $y \propto \frac{1}{x} \implies y = \frac{k}{x}$.

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Blueprint Solving Methods (Pattern → Method)

If you see…	Do this immediately
“perpendicular line through point”	get given slope → invert/negate for perpendicular slope → point-slope form
“radian/degree conversion”	apply conversion factor first; avoid calculator rounding until end
trig equation with interval	isolate trig ratio → solve base angles → list all interval-valid solutions
hyperbola/rational with asymptotes	rewrite to expose denominator zero and horizontal trend
“min/max on interval”	convert to turning-point form, then test both endpoints
$P(A\cup B)$, $P(A\mid B)$, independence	write the correct rule first, then substitute values
combinations selection constraints	split into cases (e.g., 4+2, 5+1, 6+0), sum each case
model $h(t)=a+b\cos(kt)$	baseline $a$, amplitude $

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Marking Insights Seen in Solutions

• Solutions repeatedly allocate marks for method milestones (“Specific behaviours”), not just the last line.
• In exam instructions across papers: for multi-mark parts, unsupported final answers lose marks.
• Practical rule: every part should show:
  1. formula/rule selected,
  2. substitution,
  3. simplified result with units/context.

Use this for partial-mark maximization.

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Exam Strategy Layer

Time structure

• Section One: reading 5 + working 50.
• Section Two: reading 10 + working 100.

Fast pattern-recognition workflow

1. Circle trigger words (“perpendicular”, “independent”, “amplitude”, “at least one”, “asymptote”).
2. Map to one blueprint method (table above).
3. Write first method line immediately (locks in method marks).

Partial-mark tactics

• If stuck, still write the governing formula.
• For graph-heavy questions, state key features (asymptotes/intercepts/turning point) even before full sketch.
• In probability/combinatorics, define the case split explicitly before arithmetic.

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Special Consideration Mode: “I did not prepare. I have 2 days.”

Last-resort strategy (Replication > Understanding)

1. Memorize 10 templates only (line, trig solve, trig identity, rational asymptote, quadratic min/max, $P(A\cup B)$, conditional probability, combinations case split, binomial coefficient meaning, cosine model threshold time).
2. Drill the Master Reference Exam below twice:
   • Pass 1: copy method lines exactly.
   • Pass 2: reproduce from memory with timing.
3. Exam-day rule: if a question matches a template, replicate the structure first, then compute.

This is a pattern-matching system under pressure, optimized for marks.

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What to Put on Your Notes Sheet

(2 double-sided A4, calculator section only, formula sheet also available)

Sheet 1 (front): Trigger → First line

• Perpendicular line: $m_\perp=-1/m$, then $y-y_1=m_\perp(x-x_1)$
• Trig solve template with interval line-by-line
• $P(A\cup B)$, $P(A\mid B)$, independence check template
• Hyperbola checklist: vertical asymptote from denominator $=0$, horizontal asymptote from end behaviour

Sheet 1 (back): Exact-value + conversion block

• Degree↔radian conversions
• $\cos(A\pm B)$, $\sin(A\pm B)$
• Exact values table:

  $\theta$	$0$	$\frac{\pi}{6}$ ($30^\circ$)	$\frac{\pi}{4}$ ($45^\circ$)	$\frac{\pi}{3}$ ($60^\circ$)	$\frac{\pi}{2}$ ($90^\circ$)
  $\sin\theta$	$0$	$1/2$	$1/\sqrt{2}$	$\sqrt{3}/2$	$1$
  $\cos\theta$	$1$	$\sqrt{3}/2$	$1/\sqrt{2}$	$1/2$	$0$
  $\tan\theta$	$0$	$1/\sqrt{3}$	$1$	$\sqrt{3}$	Undefined

Sheet 2 (front): Counting + modelling block

• $\binom{n}{r}$ setup template
• “At least one” = total − forbidden cases
• Binomial expansion skeleton $(a+b)^6$
• $h(t)=a+b\cos(kt)$: min/max/time threshold solving steps

Sheet 2 (back): Mini worked prototypes

• One full line-equation example
• One trig-equation interval example
• One probability conditional example
• One combinations case-split example

Keep this side dense and procedural: method lines over theory paragraphs.

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Master Reference Exam (Print on Notes)

1. Questions Section

Q1 (Source: content/SEM1 2025 YR11 METH U1 S1)

1. Through $(2,-7)$, find the line perpendicular to $y=-\frac14x+1$.
2. Solve $(x-7)^2=3$.

Q2 (Source: content/SEM1 2025 YR11 METH U1 S1)

1. Convert $3^\circ$ to radians.
2. Convert $\frac{11\pi}{18}$ to degrees.
3. Solve $2\sin(2x)-\sqrt2=0$, for $0\le x\le\pi$.

Q3 (Source: content/SEM1 2025 YR11 METH U1 S1)

1. For $f(t)=5+3\cos(2t)$, state amplitude and period.
2. Evaluate $\cos\!\left(\frac{2\pi}{3}+\frac{\pi}{4}\right)$.

Q4 (Source: content/SEM1 2025 YR11 METH U1 S1) Given $y=\frac{c}{2-x}$ and $x+y+3=0$, and the hyperbola crosses the $y$-axis at $y=3$:

1. Find the line slope.
2. Find $c$.
3. State both asymptotes.
4. Find the $x$-coordinates of intersections.

Q5 (Source: content/SEM1 2025 YR11 METH U1 S1) Square has vertices $(0,a+6)$ and $(a,0)$, with $-7\le a\le4$:

1. Find area when $a=-7$.
2. Show $A=2a^2+12a+36$.
3. Find turning point of $A(a)$.
4. Find min and max area on the given domain.

Q6 (Source: Rossmoyne Methods 11 2024 Sem 1 CA) Given $P(A)=0.3,\;P(B)=0.5$, find $P(A\cup B)$ when:

1. $A,B$ mutually exclusive.
2. $P(A\mid B)=0.8$.
3. $A,B$ independent.

Q7 (Source: Rossmoyne Methods 11 2024 Sem 1 CA) Choose 6 dancers from 7 Year 11 and 6 Year 12 students:

1. equal numbers from each year.
2. at least one from each year.
3. exactly one choreographer included (one of two choreographers, not both).
4. more Year 11 than Year 12.
5. Expand $(a+b)^6$.
6. Interpret coefficient of $a^2b^4$ in context.

Q8 (Source: Rossmoyne Methods 11 2024 Sem 1 CA)

1. Using $\cos36^\circ=\frac45,\sin36^\circ=\frac35,\cos55^\circ=\frac47,\sin55^\circ=\frac56$, approximate $\cos19^\circ$.
2. Peter sees Ray 450 m away on bearing $110^\circ$T. Mary is 200 m south of Peter. Find Ray’s bearing to Mary.

Q9 (Source: Rossmoyne Methods 11 2024 Sem 1 CA) Malaria test: prevalence $2\%$, sensitivity $87\%$, specificity $94\%$.

1. Find $P(\text{malaria and test positive})$.
2. Find $P(\text{malaria}\mid\text{test positive})$.

Q10 (Source: Rossmoyne Methods 11 2024 Sem 1 CA)

$$h(t)=5.4+1.7\cos\!\left(\frac{\pi t}{6}\right),\quad 0\le t\le24$$

$t$ is hours after 5 am.

1. Find clearance at 5:00 am and 8:45 am.
2. First minimum time and value.
3. Latest morning time for clearance $\ge 4.6$ m.

Q11 (Source: Rossmoyne Methods 11 2024 Sem 1 CA) Voucher table (total 400):

Event	1 person	2 people	3 people	4 people
Basketball	50	35	25	40
Football	27	17	21	45
Soccer	48	20	42	30

Find probabilities that a random voucher:

1. is soccer + admits 2.
2. is basketball.
3. admits no more than 3.
4. admits 1 person or is basketball.
5. admits an odd number given soccer is excluded.

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2. Full Worked Solutions Section

Q1

1. Given line slope is $-\frac14$, so perpendicular slope $m=4$.
   $y+7=4(x-2)\Rightarrow y=4x-15$.
   $y=4x-15$
2. $(x-7)^2=3\Rightarrow x-7=\pm\sqrt3\Rightarrow x=7\pm\sqrt3$.
   $x=7\pm\sqrt3$

Q2

1. $3^\circ\times\frac{\pi}{180}=\frac{\pi}{60}$.
   $\frac{\pi}{60}$ rad
2. $\frac{11\pi}{18}\times\frac{180}{\pi}=110^\circ$.
   $110^\circ$
3. $2\sin(2x)-\sqrt2=0\Rightarrow \sin(2x)=\frac{\sqrt2}{2}$.
   $2x=\frac\pi4,\frac{3\pi}4$ in $[0,2\pi]\Rightarrow x=\frac\pi8,\frac{3\pi}8$.
   $x=\frac\pi8,\frac{3\pi}8$

Q3

1. For $f(t)=5+3\cos(2t)$: amplitude $=3$, period $=\frac{2\pi}{2}=\pi$.
   Amplitude $3$, period $\pi$

$$

\cos\!\left(\frac{2\pi}{3}+\frac{\pi}{4}\right)
=\cos\frac{2\pi}{3}\cos\frac\pi4-\sin\frac{2\pi}{3}\sin\frac\pi4
=-\frac12\frac{\sqrt2}{2}-\frac{\sqrt3}{2}\frac{\sqrt2}{2}
=-\frac{\sqrt2+\sqrt6}{4}.
$$

<span class="highlight">$-\frac{\sqrt2+\sqrt6}{4}$</span>

Q4

1. $x+y+3=0\Rightarrow y=-x-3$, slope $=-1$.
   $-1$
2. At $x=0,\;y=3$: $3=\frac{c}{2-0}\Rightarrow c=6$.
   $c=6$
3. For $y=\frac{6}{2-x}$: horizontal asymptote $y=0$, vertical asymptote $x=2$.
   $y=0,\;x=2$
4. Intersections: $\frac{6}{2-x}=-x-3$.
   $6=(2-x)(-x-3)=x^2+x-6\Rightarrow x^2+x-12=0\Rightarrow(x+4)(x-3)=0$.
   $x=-4,\;x=3$

Q5

1. At $a=-7$, vertices $(0,-1)$ and $(-7,0)$: side $l=\sqrt{7^2+1^2}=\sqrt{50}$.
   Area $A=l^2=50$.
   $50$
2. $l^2=a^2+(a+6)^2\Rightarrow A=2a^2+12a+36$.
   $A=2a^2+12a+36$
3. $A=2(a^2+6a+18)=2(a+3)^2+18\Rightarrow$ turning point $(-3,18)$.
   $(-3,18)$
4. Minimum at vertex: $18$.
   Endpoints: $A(-7)=50,\;A(4)=116\Rightarrow$ maximum $116$.
   Minimum $18$, maximum $116$

Q6

1. Mutually exclusive $\Rightarrow P(A\cap B)=0$:
   $P(A\cup B)=0.3+0.5=0.8$.
   $0.8$
2. $P(A\mid B)=0.8\Rightarrow P(A\cap B)=0.8\times0.5=0.4$.
   $P(A\cup B)=0.3+0.5-0.4=0.4$.
   $0.4$
3. Independent $\Rightarrow P(A\cap B)=0.3\times0.5=0.15$.
   $P(A\cup B)=0.3+0.5-0.15=0.65$.
   $0.65$

Q7

1. Equal split $3+3$: $\binom73\binom63=700$.
   $700$
2. At least one from each year: $\binom{13}6-\binom70\binom66-\binom76\binom60=1708$.
   $1708$
3. Exactly one choreographer: $\binom21\binom{11}5=924$.
   $924$
4. More Year 11: $$$$

\binom74\binom62+\binom75\binom61+\binom76\binom60 =525+126+7=658.

<span class="highlight">$658$</span> 5. $$ (a+b)^6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6.

$(a+b)^6$ expansion shown above 6. Coefficient of $a^2b^4$ is $15$: 15 ways to allocate 2 dancers to group $a$ and 4 to group $b$.
15 allocations

Q8

1. $$$$

\cos19^\circ=\cos(55^\circ-36^\circ) =\cos55^\circ\cos36^\circ+\sin55^\circ\sin36^\circ =\frac47\cdot\frac45+\frac56\cdot\frac35 =\frac{67}{70}\approx0.9571.

<span class="highlight">$\cos19^\circ\approx0.9571$</span> 2. Triangle $PMR$: $PR=450,\;PM=200,\;\angle RPM=70^\circ$. $$ MR^2=200^2+450^2-2(200)(450)\cos70^\circ\Rightarrow MR\approx425.366.

Then cosine rule gives angle at $R$: $\theta\approx26.22^\circ$.
Bearing from $R$ to $M$: $360^\circ-70^\circ-\theta\approx263.78^\circ$.
$264^\circ\text{T (nearest degree)}$

Q9

1. $P(M\cap +)=0.02\times0.87=0.0174$.
   $0.0174$
2. False positive $=0.98\times0.06=0.0588$.
   $P(+)=0.0174+0.0588=0.0762$. $$$$

P(M\mid +)=\frac{0.0174}{0.0762}=0.2283.

<span class="highlight">$0.2283$</span> ### Q10 1. $h(0)=5.4+1.7=7.1$ m. For 8:45 am, $t=3.75$: $h(3.75)=5.4+1.7\cos\!\left(\frac{\pi(3.75)}6\right)\approx4.75$ m. <span class="highlight">$7.1$ m and $4.75$ m</span> 2. Minimum when $\cos\left(\frac{\pi t}{6}\right)=-1\Rightarrow t=6$. Time $=11{:}00$ am, clearance $=5.4-1.7=3.7$ m. <span class="highlight">$11{:}00$ am, $3.7$ m</span> 3. Solve $5.4+1.7\cos\!\left(\frac{\pi t}{6}\right)=4.6\Rightarrow t\approx3.9357$ h after 5 am. $5{:}00+3{:}56\approx8{:}56$ am. <span class="highlight">$8{:}56$ am (nearest minute)</span> ### Q11 1. Soccer + 2 people: $\frac{20}{400}=0.05$. <span class="highlight">$0.05$</span> 2. Basketball total: $\frac{50+35+25+40}{400}=\frac{150}{400}=\frac38=0.375$. <span class="highlight">$\frac38$</span> 3. No more than 3 people: $\frac{400-(40+45+30)}{400}=\frac{285}{400}=\frac{57}{80}=0.7125$. <span class="highlight">$\frac{57}{80}$</span> 4. One person or basketball: $$ \frac{(50+27+48)+(50+35+25+40)-50}{400} =\frac{225}{400}=\frac9{16}=0.5625.

$\frac9{16}$ 5. Odd number given soccer excluded:
Non-soccer total $=400-(48+20+42+30)=260$.
Odd-count non-soccer $=(50+25)+(27+21)=123$.
$$ P=\frac{123}{260}\approx0.473.

<span class="highlight">$\frac{123}{260}\approx0.473$</span> --- ## Final Use Rule Before each exam question, identify the template first. Then copy the matching method structure from this sheet. That is the fastest path to <span class="highlight">replicable marks under pressure</span>.