Pattern_Analysis_Consolidated

Maths 1: Comprehensive Pattern Analysis & Solutions

Complete Qualifier Prep Guide - All 12 Weeks

Exam in 6 hours. Lock in. No distractions.

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🎯 MASTER PATTERN CHEAT SHEET

Week 1-4: HIGH WEIGHT in Qualifier (60-70%)

Pattern 1: Sets & Cardinality

Type: Find $|A\setminus B|$, $|A\cap B|$, $|A\cup B|$ Method:

1. List elements explicitly
2. Apply formula: $|A\cup B|=|A|+|B|-|A\cap B|$
3. For “Only One Set”: $(A\setminus B)\cup (B\setminus A)=A\Delta B$

Example (W1Q7):

• $A=\{2,4,6,8,10\}$, $B=\{10,15,20,25\}$, $C=\{7,14,21,28\}$
• $A\cap B=\{10\}$. Only unique: $4+3+4=11$.

Pattern 2: Relations (Reflexive, Symmetric, Transitive)

Quick Tests:

Property	Test
Reflexive	$(a,a)\in R$ for ALL $a$?
Symmetric	$(a,b)\in R\Rightarrow (b,a)\in R$?
Transitive	$(a,b),(b,c)\in R\Rightarrow (a,c)\in R$?

Trap (W1Q5): “Both have property P” - NOT reflexive on universe if not ALL elements have P.

Pattern 3: Domain “Holes”

Types of Holes:

1. Denominator = 0: Exclude those values.
2. $\sqrt{\text{negative}}$: $x^2-16\ge 0\Rightarrow x\le -4$ or $x\ge 4$.
3. $\log (\le 0)$: Inside must be $>0$.

Example (W1Q2): $f(x)=\frac{\sqrt{x^2-16}}{x+4}$

• Root: $|x|\ge 4$.
• Denom: $x\ne -4$.
• Bad integers: $\{-4,-3,-2,-1,0,1,2,3\}$. Count = 8.

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Week 2: Coordinate Geometry

Pattern 4: Line Equations

Forms:

• Slope-Intercept: $y=mx+c$
• Two-Point: $\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$
• Intercept: $\frac{x}{a}+\frac{y}{b}=1$

Parallel: $m_1=m_2$. Perpendicular: $m_1\times m_2=-1$.

Example (W2Q1): Line $2y-6x=6\Rightarrow y=3x+3$. Slope = 3. Plane passes through $(4,12)$: $y-12=3(x-4)\Rightarrow y=3x$. Do they intersect? Set $3x+3=3x$. No. They are parallel. Answer: 0.

Pattern 5: Distance Point-to-Line

$d=\frac{|a{x}_0+b{y}_0+c|}{\sqrt{a^2+b^2}}$

Example (W2Q7-8): Find shortest distance from B(5,7) to highway. Highway: $(2,1)$ to $(10,7)$. Slope = $\frac{6}{8}=0.75$. Line: $3x-4y-2=0$. $d=\frac{|15-28-2|}{5}=\frac{15}{5}=3$ units = 300m.

Pattern 6: SSE (Sum Squared Error)

$SSE=\sum (y_{actual}-y_{predicted}{)}^2$

Example (W2Q18): Fitted line $y=4x+2$. Data: (1,6), (3,14), (5,24)…

• Predicted: $y(1)=6$, $y(3)=14$, $y(5)=22$, $y(7)=30$, $y(9)=38$, $y(11)=46$.
• Errors: $0,0,2,-1,1,-1$.
• $SSE=4+1+1+1=7$.

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Week 3: Quadratic Functions

Pattern 7: Vertex of Parabola

Vertex Form: $y=a(x-h{)}^2+k$ where Vertex = $(h,k)$. From Standard Form: Vertex $=\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$.

Example (W3Q7): Max value = $-3$, axis $x=2$, $f(0)=-9$. $f(x)=a(x-2{)}^2-3$. $f(0)=4a-3=-9\Rightarrow a=-1.5$.

Pattern 8: Slope of Parabola (Derivative)

The Slope at point $(x_0,y_0)$ on $y=a{x}^2+bx+c$ is: $\text{Slope}=2a{x}_0+b$

Example (W3Q1): Slope at $(3,2)$ is 32, at $(2,3)$ is 2. $6a+b=32$ and $4a+b=2$. Subtract: $2a=30\Rightarrow a=15$.

Pattern 9: Projectile/Height Questions

$h(t)=a{t}^2+bt+c$. Max height at $t=-b/2a$.

Example (W3Q11): $h(t)=-8t^2+32t+40$. Max at $t=32/16=2$. Max height: $h(2)=-32+64+40=72$m. Hits ground: $h(t)=0\Rightarrow -8(t^2-4t-5)=0\Rightarrow t=5$ or $t=-1$.

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Week 4: Polynomials

Pattern 10: Multiplicity & Behavior

Multiplicity	X-axis Behavior
Odd (1, 3…)	Crosses
Even (2, 4…)	Bounces (Touches)

Pattern 11: End Behavior

Degree	Leading Coeff	$x\to \infty$	$x\to -\infty$
Even	$+$	$\uparrow$	$\uparrow$
Even	$-$	$\downarrow$	$\downarrow$
Odd	$+$	$\uparrow$	$\downarrow$
Odd	$-$	$\downarrow$	$\uparrow$

Pattern 12: Polynomial Intersections

Set $p(x)=q(x)$ and solve.

Example (W4Q5): $f(x)=(x-9)(x-7{)}^2$, $g(x)=-(x-9)(x-7)$. Set equal: $(x-9)(x-7{)}^2=-(x-9)(x-7)$. $(x-9)(x-7)[(x-7)+1]=0\Rightarrow x=9,7,6$.

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Week 5-6: Functions & Logs/Exponents

Pattern 13: Inverse Functions

• Graph of $f^{-1}$ is reflection of $f$ over line $y=x$.
• Intersection points: Solve $f(x)=x$.

Pattern 14: Composite Domain

Domain of $g\circ f=\{x\in \text{Dom}(f):f(x)\in \text{Dom}(g)\}$.

Pattern 15: Log Rules

${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$ ${\log }_b(x^n)=n{\log }_{b}x$ $a^{{\log }_{a}x}=x$

Pattern 16: Population/Half-Life

Growth: $P(t)=P_0(1+r{)}^t$ Decay: $A(t)=A_0{\left(\frac{1}{2}\right)}^{t/\gamma }$ ($\gamma$ = half-life)

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Week 7-8: Limits & Derivatives

Pattern 17: Key Limits

${\lim }_{x\to 0}\frac{\sin x}{x}=1$ ${\lim }_{x\to 0}\frac{e^x-1}{x}=1$ ${\lim }_{x\to 0}\frac{\ln (1+x)}{x}=1$

Pattern 18: Continuity Check

$f$ continuous at $a$ if:

1. $f(a)$ exists.
2. ${\lim }_{x\to a}f(x)$ exists.
3. ${\lim }_{x\to a}f(x)=f(a)$.

Pattern 19: Differentiability

Differentiable $\Rightarrow$ Continuous. NOT vice versa. $|x|$ is continuous but NOT differentiable at $x=0$ (cusp).

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Week 9: Optimization & Integration

Pattern 20: Critical Points

Set $f^{\prime }(x)=0$. Solve for $x$. Check endpoints for global max/min.

Pattern 21: Area Under Curve

$\text{Area}={\int }_a^{b}f(x)dx$ Negative area if curve is below x-axis.

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Week 10-12: Graphs & Algorithms

Pattern 22: Handshaking Lemma

$\sum \text{degrees}=2\times |E|$

Pattern 23: Adjacency Matrix

• $A_{ij}=1$ if edge exists.
• $A_{ij}^k$ = number of paths of length $k$ from $i$ to $j$.

Pattern 24: BFS vs DFS

BFS	DFS
Queue	Stack
Shortest Path (unweighted)	Cycle Detection
Level Order	Topological Sort

Pattern 25: Dijkstra vs Bellman-Ford

Algorithm	Handles Negative Edges?	Time
Dijkstra	NO	$O(V^2)$ or $O(E\log V)$
Bellman-Ford	YES (no neg cycles)	$O(VE)$

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🚨 TOP 10 EXAM TRAPS

1. Sign Errors: Double-check negative signs in quadratics.
2. Domain Holes: Don’t forget $x\ne$ values from denominator.
3. Reflexive Trap: “Both have P” ≠ reflexive on universe.
4. Cardinality of Relations: Count $(a,b)$ AND $(b,a)$ as 2.
5. Parallel Lines: Same slope = No intersection.
6. Max vs Local Max: Check endpoints for global.
7. Even Multiplicity: Bounces, doesn’t cross.
8. $\log$ Base < 1: Inequality flips!
9. BFS Tree Edges: Only edges between adjacent levels.
10. Negative Cycle: Bellman-Ford fails.

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Time to crush it. You got this, dawg. 🔥