Maths1_Ultimate_Pattern_Guide

Maths 1: Ultimate Pattern Recognition Guide 🎯

IMPORTANT

Purpose: This is your complete map of every question pattern in Maths 1. Organized by difficulty hierarchy, showing all variations, traps, and quick recognition strategies.

How to Use:

1. Pattern Families: Related questions are grouped together
2. Level 1 → 2 → 3: Basic → Intermediate → Advanced
3. Spot it Fast: Keywords to identify pattern during exam
4. Traps: Common mistakes to avoid

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📊 Pattern Hierarchy Overview

Week	Pattern Families	Total Patterns
1: Sets & Relations	Set Operations, Relations (RST), Domain	9
2: Coordinate Geometry	Lines, Optimization, Shapes	8
3: Quadratics	Vertex/Optimization, Intersections	7
4: Polynomials	Construction, Analysis, Remainder	6
5: Functions	Composition, Inverse, Properties	8
6: Logarithms	Equations, Inequalities, Change of Base	9
7: Sequences & Limits	AP/GP, Series, Limits	7
8: Derivatives	Basic Rules, Chain Rule, Optimization	8

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Week 1: Sets & Relations

Pattern Family 1.1: Set Operations (Cardinality Problems)

🟢 Level 1: Two-Set Inclusion-Exclusion

Spot it Fast: “both”, “neither”, “only A”, “only B”

Core Formula: $|A\cup B|=|A|+|B|-|A\cap B|$

Mental Algorithm:

1. Draw Venn diagram (2 circles)
2. Fill intersection first
3. Use formula for union
4. Neither = Total - Union

Example: 100 people. 60 like Football, 50 like Cricket, 20 like both. Find neither.

• $|F\cup C|=60+50-20=90$
• Neither = $100-90=10$

All Variations:

1. Find “neither”: Total - Union
2. Find “only A”: $|A|-|A\cap B|$
3. Find “at least one”: Union
4. Find “exactly one”: $(|A|-|A\cap B|)+(|B|-|A\cap B|)$

Common Traps:

• ❌ Forgetting to subtract intersection when calculating union
• ❌ Confusing “both” (intersection) with “either” (union)
• ❌ Not checking if numbers add up to total

🟡 Level 2: Three-Set Inclusion-Exclusion

Spot it Fast: Three categories (A, B, C) with intersections

Core Formula: $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$

Mental Algorithm:

1. Add all three sets
2. Subtract all pair-wise intersections
3. Add back triple intersection
4. Neither = Total - Union

Mnemonic: “Add singles, Subtract pairs, Add triple”

Common Traps:

• ❌ Missing the $+|A\cap B\cap C|$ term
• ❌ Sign errors (remember: subtract pairs, add triple)

🔴 Level 3: Complement & “At Least” Logic

Spot it Fast: “at least one”, “at least two”, “all three”

Mental Algorithm:

• At least one: Use union formula
• At least two: $|A\cap B|+|A\cap C|+|B\cap C|-2|A\cap B\cap C|$
• All three: $|A\cap B\cap C|$
• Exactly two: (At least two) - (All three)

Power Move: Use complement!

• “At least one” = Total - “None”
• “At least two” = Total - “None” - “Exactly one”

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Pattern Family 1.2: Relations (Checking RST Properties)

🟢 Level 1: Reflexivity

Spot it Fast: “Is every element related to itself?”

Formal Definition: $(a,a)\in R$ for all $a\in A$

Mental Algorithm:

1. Check if ALL diagonal pairs $(1,1),(2,2),...$ are present
2. Missing even ONE diagonal pair → Not reflexive

Example: $A=\{1,2,3\}$, $R=\{(1,1),(2,2),(3,3),(1,2)\}$

• ✅ Reflexive (all diagonals present)

Common Traps:

• ❌ Reflexive on SUBSET: If $R$ is defined on subset $B\subset A$, only check elements in $B$

🟡 Level 2: Symmetry

Spot it Fast: “If $(a,b)\in R$, is $(b,a)\in R$?”

Mental Algorithm:

1. For EACH pair $(a,b)$ where $a\ne b$
2. Check if reverse pair $(b,a)$ exists
3. Missing even ONE reverse → Not symmetric

Example: $R=\{(1,2),(2,1),(2,3)\}$

• $(2,3)$ exists but $(3,2)$ missing
• ❌ Not symmetric

Common Traps:

• ❌ Confusing with antisymmetry (if $(a,b)$ AND $(b,a)$, then $a=b$)

🔴 Level 3: Transitivity

Spot it Fast: “If $aRb$ and $bRc$, then $aRc$?”

Mental Algorithm:

1. Find all “chains”: $a\to b\to c$
2. Check if “shortcut” $a\to c$ exists
3. Missing even ONE shortcut → Not transitive

Proof Strategy (for defined relations):

1. Assume: $aRb$ and $bRc$
2. Use definition: Express what these mean
3. Prove: Show $aRc$ follows

Example: $R=\{(a,b):a-b\text{ is even}\}$

• Assume: $a-b$ even, $b-c$ even
• Sum: $(a-b)+(b-c)=a-c$ (sum of evens is even)
• ✅ Transitive

Common Traps:

• ❌ Not checking ALL possible chains
• ❌ Giving specific example instead of general proof

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Pattern Family 1.3: Domain of Functions

🟢 Level 1: Single Constraint

Spot it Fast: ONE restriction (denominator OR square root)

Types:

1. Denominator ≠ 0: $f(x)=\frac{1}{g(x)}⟹g(x)\ne 0$
2. Square root ≥ 0: $f(x)=\sqrt{g(x)}⟹g(x)\ge 0$
3. Log > 0: $f(x)=\log (g(x))⟹g(x)>0$

Example: $f(x)=\frac{1}{x-2}$

• Domain: $x\ne 2$ → $(-\infty ,2)\cup (2,\infty )$

🟡 Level 2: Multiple Constraints (Intersection)

Spot it Fast: MULTIPLE restrictions (AND logic)

Mental Algorithm:

1. List ALL constraints separately
2. Find intersection (overlap) of valid regions

Example: $f(x)=\frac{\sqrt{x-1}}{x-3}$

• Constraint 1: $x-1\ge 0⟹x\ge 1$ → $[1,\infty )$
• Constraint 2: $x-3\ne 0⟹x\ne 3$
• Intersection: $[1,3)\cup (3,\infty )$

Common Traps:

• ❌ Taking UNION instead of INTERSECTION

🔴 Level 3: Fraction Under Square Root (Sign Analysis)

Spot it Fast: $\sqrt{\frac{f(x)}{g(x)}}$

Critical Insight: Fraction positive if BOTH same sign

Mental Algorithm:

1. Case 1 (+/+): $f(x)\ge 0$ AND $g(x)>0$
2. Case 2 (-/-): $f(x)\le 0$ AND $g(x)<0$
3. Union of both cases

Example: $f(x)=\sqrt{\frac{x-1}{x-2}}$

• Case 1: $x\ge 1$ AND $x>2$ → $x>2$ → $(2,\infty )$
• Case 2: $x\le 1$ AND $x<2$ → $x\le 1$ → $(-\infty ,1]$
• Domain: $(-\infty ,1]\cup (2,\infty )$

Common Traps:

• ❌ Forgetting Case 2 (both negative)
• ❌ Using $\ge$ for denominator (should be strict $>$)

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Week 2: Coordinate Geometry

Pattern Family 2.1: Lines (Equations & Slopes)

🟢 Level 1: Slope from Two Points

Spot it Fast: Given two points, find slope

Formula: $m=\frac{y_2-y_1}{x_2-x_1}$

Special Cases:

• Horizontal line: $m=0$
• Vertical line: $m=\text{undefined}$

🟡 Level 2: Equation from Point & Slope

Spot it Fast: “Line through $(x_1,y_1)$ with slope $m$”

Point-Slope Form: $y-y_1=m(x-x_1)$

Example: Line through $(2,3)$ with slope $4$

• $y-3=4(x-2)$
• $y=4x-5$

🔴 Level 3: Intersection of Two Lines

Spot it Fast: “Where do lines meet?”

Mental Algorithm:

1. Equate: Set equations equal
2. Solve for one variable
3. Find other variable: Plug back

Example: $y=2x+1$ and $y=-x+4$

• $2x+1=-x+4$
• $3x=3⟹x=1$
• $y=2(1)+1=3$
• Intersection: $(1,3)$

Special Cases:

• Parallel lines: No solution (same slope, different intercept)
• Coincident lines: Infinite solutions (same line)

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Pattern Family 2.2: Optimization (Shortest Paths)

🟢 Level 1: Distance Formula

Spot it Fast: Distance between two points

Formula: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

🟡 Level 2: Perpendicular Distance (Point to Line)

Spot it Fast: “Shortest distance from point to line”

Formula: For line $ax+by+c=0$ and point $(x_1,y_1)$: $d=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$

Example: Distance from $(1,2)$ to $3x+4y-5=0$

• $d=\frac{|3(1)+4(2)-5|}{\sqrt{9+16}}=\frac{6}{5}$

🔴 Level 3: Reflection Principle (Minimize Sum)

Spot it Fast: “Find point $P$ on line/axis such that $AP+PB$ is minimum”

Core Insight: Reflect one point, draw straight line

Mental Algorithm:

1. Reflect point $A$ across the line to get $A^{\prime }$
2. Line $A^{\prime }B$: This is the shortest path
3. Intersection: Where $A^{\prime }B$ meets the line is optimal $P$

Example: Minimize $AP+PB$, $A=(1,2)$, $B=(5,3)$, $P$ on x-axis

• Reflect $A$ across x-axis: $A^{\prime }=(1,-2)$
• Line $A^{\prime }B$: Slope $=\frac{3-(-2)}{5-1}=\frac{5}{4}$
• Equation: $y+2=\frac{5}{4}(x-1)$
• x-intercept (set $y=0$): $x=2.6$

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Pattern Family 2.3: Shapes (Parallelograms, Triangles)

🟢 Level 1: Midpoint Formula

Spot it Fast: “Find midpoint of segment”

Formula: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

🟡 Level 2: Parallelogram Fourth Vertex

Spot it Fast: “Three vertices given, find fourth”

Key Insight: Diagonals bisect each other

Mental Algorithm:

1. Identify which diagonal connects which points
2. Midpoint of one diagonal = Midpoint of other
3. Solve for unknown vertex

Shortcut Formula: If vertices are $A,B,C$, fourth vertex $D$: $D=A+C-B\text{(coordinate-wise)}$

Example: $A=(1,1)$, $B=(2,4)$, $C=(3,3)$

• $D=(1,1)+(3,3)-(2,4)=(2,0)$

Common Traps:

• ❌ Three possible answers depending on which points are adjacent!

🔴 Level 3: Area of Triangle (Coordinate Method)

Spot it Fast: Three vertices, find area

Formula: For vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$: $A=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$

Alternative (if vertices form simple shape):

• Use base × height formula
• Break into simpler shapes

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Week 3: Quadratic Functions

Pattern Family 3.1: Vertex & Optimization

🟢 Level 1: Find Vertex

Spot it Fast: $y=a{x}^2+bx+c$, find max/min

Vertex Formula:

• x-coordinate: $x=-\frac{b}{2a}$
• y-coordinate: Plug $x$ back into equation

Type of Extremum:

• $a>0$: Parabola opens up → Minimum at vertex
• $a<0$: Parabola opens down → Maximum at vertex

Example: $y=-2x^2+8x-5$

• $x=-\frac{8}{2(-2)}=2$
• $y=-2(4)+8(2)-5=3$
• Vertex: $(2,3)$ (Maximum)

🟡 Level 2: Optimization with Linear Constraint

Spot it Fast: “Maximize $f(x,y)$ subject to $x+y=k$”

Mental Algorithm:

1. Express one variable: $y=k-x$
2. Substitute: $f(x)=f(x,k-x)$
3. Find vertex of resulting quadratic

Example: Maximize $xy$ subject to $x+y=10$

• $y=10-x$
• $f(x)=x(10-x)=10x-x^2$
• Vertex: $x=5$, $y=5$
• Max: $25$

Common Patterns:

• Max product with fixed sum → Both equal
• Min sum with fixed product → Both equal

🔴 Level 3: Perimeter/Area Problems

Spot it Fast: “Rectangle with perimeter/area constraint”

Mental Algorithm:

1. Setup equations: $P=2(l+w)$, $A=lw$
2. Eliminate variable using constraint
3. Optimize using vertex

Key Insight: For fixed perimeter, max area is square!

Example: Perimeter = 40, maximize area

• $l+w=20$
• $A=l(20-l)=20l-l^2$
• Vertex: $l=10$, $w=10$
• Max area: $100$

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Pattern Family 3.2: Roots & Discriminant

🟢 Level 1: Number of Real Roots

Spot it Fast: $a{x}^2+bx+c=0$, how many real solutions?

Discriminant: $\Delta =b^2-4ac$

• $\Delta >0$: 2 distinct real roots
• $\Delta =0$: 1 repeated real root
• $\Delta <0$: 0 real roots (2 complex)

🟡 Level 2: Sum & Product of Roots

Spot it Fast: Find sum/product without solving

Vieta’s Formulas: For $a{x}^2+bx+c=0$:

• Sum: $\alpha +\beta =-\frac{b}{a}$
• Product: $\alpha \beta =\frac{c}{a}$

Example: $x^2-5x+6=0$

• Sum: $5$
• Product: $6$
• Roots: $2,3$ (check: $2+3=5$, $2\times 3=6$)

🔴 Level 3: Common Roots Problem

Spot it Fast: “Find $k$ such that two equations share a root”

Mental Algorithm:

1. Solve the simpler equation
2. Substitute each root into other equation
3. Solve for $k$

Example: $x^2+kx+6=0$ and $x^2-5x+6=0$ share root

• Solve: $(x-2)(x-3)=0⟹x=2$ or $3$
• Try $x=2$: $4+2k+6=0⟹k=-5$
• Verify $x=3$: $9+3k+6=0⟹k=-5$ ✅

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Pattern Family 3.3: Intersections

🟢 Level 1: Line-Parabola

Spot it Fast: Where does line meet parabola?

Mental Algorithm:

1. Equate: $a{x}^2+bx+c=mx+d$
2. Rearrange: $a{x}^2+(b-m)x+(c-d)=0$
3. Solve quadratic

Number of intersections: Use discriminant

• 2 points: Line crosses parabola
• 1 point: Line is tangent
• 0 points: Line misses parabola

🟡 Level 2: Parabola-Parabola

Spot it Fast: Two parabolas intersecting

Mental Algorithm:

1. Equate: $a_1{x}^2+b_{1}x+c_1=a_2{x}^2+b_{2}x+c_2$
2. Simplify: $(a_1-a_2)x^2+(b_1-b_2)x+(c_1-c_2)=0$
3. Solve

Example: $y=x^2$ and $y=2-x^2$

• $x^2=2-x^2$
• $2x^2=2⟹x=\pm 1$
• Points: $(1,1)$ and $(-1,1)$

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Week 4: Polynomials

Pattern Family 4.1: Construction from Roots

🟢 Level 1: Basic Construction

Spot it Fast: “Polynomial with roots $r_1,r_2,r_3$”

Template: $P(x)=k(x-r_1)(x-r_2)(x-r_3)\cdots$

Example: Roots 1, 2, 3

• $P(x)=k(x-1)(x-2)(x-3)$

🟡 Level 2: Construction with Point Constraint

Spot it Fast: “…passing through point $(a,b)$”

Mental Algorithm:

1. Template: $P(x)=k(x-r_1)(x-r_2)\cdots$
2. Plug in point: $P(a)=b$
3. Solve for $k$

Example: Roots 1, 2, 3 passing through $(0,-6)$

• $P(x)=k(x-1)(x-2)(x-3)$
• $P(0)=k(-1)(-2)(-3)=-6k=-6$
• $k=1$
• $P(x)=(x-1)(x-2)(x-3)=x^3-6x^2+11x-6$

🔴 Level 3: Construct with Additional Constraints

Spot it Fast: Multiple conditions (roots + derivative at point)

Complexity: Need calculus to verify derivative constraint

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Pattern Family 4.2: Remainder Theorem

🟢 Level 1: Remainder when Divided by $(x-a)$

Spot it Fast: “Remainder of $P(x)÷(x-a)$”

Theorem: Remainder = $P(a)$

Example: Remainder of $x^3-2x+1$ divided by $(x-2)$

• $P(2)=8-4+1=5$

🟡 Level 2: Factor Theorem

Spot it Fast: “Is $(x-a)$ a factor of $P(x)$?”

Theorem: $(x-a)$ is factor $⟺P(a)=0$

Use Case: Check if polynomial is divisible

🔴 Level 3: Remainder when Divided by Quadratic

Spot it Fast: “Remainder when divided by $(x-a)(x-b)$”

Mental Algorithm:

1. Remainder is linear: $R(x)=mx+c$
2. Use: $P(a)=R(a)$ and $P(b)=R(b)$
3. Solve system for $m,c$

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Pattern Family 4.3: Analysis (Turning Points, Intervals)

🟢 Level 1: Degree and Turning Points

Spot it Fast: “How many turning points can it have?”

Rule: Degree $n$ → at most $n-1$ turning points

Example: Cubic (degree 3) → at most 2 turning points

🟡 Level 2: Increasing/Decreasing Intervals

Spot it Fast: “Where is $P(x)$ increasing?”

Mental Algorithm:

1. Derivative: $P^{\prime }(x)$
2. Critical points: Solve $P^{\prime }(x)=0$
3. Test intervals: Sign of $P^{\prime }(x)$ in each region

Example: $P(x)=x^3-3x^2$

• $P^{\prime }(x)=3x^2-6x=3x(x-2)$
• Critical: $x=0,2$
• Test $x=-1$: $P^{\prime }(-1)=9>0$ (Increasing)
• Test $x=1$: $P^{\prime }(1)=-3<0$ (Decreasing)
• Test $x=3$: $P^{\prime }(3)=9>0$ (Increasing)
• Result: Increasing on $(-\infty ,0]\cup [2,\infty )$

🔴 Level 3: Asymptotic Behavior

Spot it Fast: “As $x\to \infty$, which term dominates?”

Rule: Highest degree term dominates

Example: $P(x)=2x^4-100x^3+5x$

• As $x\to \infty$: $2x^4$ dominates
• Behavior: $P(x)\to +\infty$

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Week 5: Functions

Pattern Family 5.1: Composition & Domain

🟢 Level 1: Function Composition

Spot it Fast: $f(g(x))$ or $(f\circ g)(x)$

Mental Algorithm:

1. Inside-out: Evaluate $g(x)$ first
2. Substitute: Plug result into $f$

Example: $f(x)=x^2$, $g(x)=x+1$

• $f(g(x))=f(x+1)=(x+1{)}^2$

🟡 Level 2: Domain of Composite

Spot it Fast: “Find domain of $f(g(x))$”

Mental Algorithm:

1. Inner domain: $x$ must be in domain of $g$
2. Range constraint: $g(x)$ must be in domain of $f$
3. Intersection: Both conditions

Example: $f(x)=\sqrt{x}$, $g(x)=x-4$

• Domain of $g$: All $\mathbb{R}$
• Need: $g(x)\ge 0$ → $x-4\ge 0$ → $x\ge 4$
• Domain of $f(g(x))$: $[4,\infty )$

🔴 Level 3: Multiple Compositions

Spot it Fast: $f(g(h(x)))$

Mental Algorithm: Work inside-out, check domain at each step

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Pattern Family 5.2: Inverse Functions

🟢 Level 1: Find Inverse

Spot it Fast: “Find $f^{-1}(x)$”

Mental Algorithm:

1. Replace: $y=f(x)$
2. Swap: $x=f(y)$
3. Solve for $y$
4. Result: $y=f^{-1}(x)$

Example: $f(x)=2x+1$

• $y=2x+1$
• $x=2y+1$
• $y=\frac{x-1}{2}$
• $f^{-1}(x)=\frac{x-1}{2}$

🟡 Level 2: Inverse with Restriction

Spot it Fast: “Inverse of $f(x)=x^2$ for $x\ge 0$”

Key: Restriction affects which branch to choose

Example: $f(x)=x^2+2x$ for $x\ge -1$

• Complete square: $y=(x+1{)}^2-1$
• $x+1=\pm \sqrt{y+1}$
• Since $x\ge -1$, $x+1\ge 0$, take positive
• $f^{-1}(x)=\sqrt{x+1}-1$

🔴 Level 3: Verify Injectivity/Surjectivity

Spot it Fast: “Is $f$ bijective?”

One-to-One (Injective):

• Algebraic: If $f(a)=f(b)$ then $a=b$
• Calculus: $f^{\prime }(x)>0$ always OR $f^{\prime }(x)<0$ always

Onto (Surjectivity):

• Range = Codomain

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Pattern Family 5.3: Even/Odd Functions

🟢 Level 1: Test for Even/Odd

Spot it Fast: “Is $f(x)$ even or odd?”

Definitions:

• Even: $f(-x)=f(x)$ (symmetric about y-axis)
• Odd: $f(-x)=-f(x)$ (symmetric about origin)

Mental Algorithm:

1. Calculate $f(-x)$
2. Compare with $f(x)$ and $-f(x)$

Example: $f(x)=x^2+1$

• $f(-x)=(-x{)}^2+1=x^2+1=f(x)$
• Even

🟡 Level 2: Properties

Spot it Fast: “Sum/product of even/odd functions”

Rules:

• Even + Even = Even
• Odd + Odd = Odd
• Even × Even = Even
• Odd × Odd = Even
• Even × Odd = Odd

🔴 Level 3: Decomposition

Spot it Fast: “Express as sum of even and odd parts”

Formula: $f(x)=\underset{\text{even}}{\underbrace{\frac{f(x)+f(-x)}{2}}}+\underset{\text{odd}}{\underbrace{\frac{f(x)-f(-x)}{2}}}$

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Week 6: Logarithms

Pattern Family 6.1: Solving Log Equations

🟢 Level 1: Single Log

Spot it Fast: ${\log }_b(x)=k$

Method: Exponentiate both sides

• $x=b^k$

🟡 Level 2: Combine Logs

Spot it Fast: $\log (x)+\log (y)=k$

Mental Algorithm:

1. Combine: $\log (xy)=k$
2. Exponentiate: $xy=10^k$
3. Solve

Example: $\log (x)+\log (x-3)=1$

• $\log (x(x-3))=1$
• $x(x-3)=10$
• $x^2-3x-10=0$
• $(x-5)(x+2)=0$
• Check domain: $x>3$
• Answer: $x=5$

🔴 Level 3: Hidden Domain Constraints

Spot it Fast: Solution fails domain check

Example: ${\log }_2(x-4)+{\log }_2(x-2)=3$

• Combined: ${\log }_2((x-4)(x-2))=3$
• $(x-4)(x-2)=8$
• $x^2-6x+8=8$
• $x(x-6)=0$ → $x=0$ or $x=6$
• Check: $x=0$ fails ($\log$ of negative)
• Answer: $x=6$

Common Traps:

• ❌ Not checking if solutions satisfy domain

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Pattern Family 6.2: Log Inequalities

🟢 Level 1: Same Base (Base > 1)

Spot it Fast: ${\log }_b(x)>{\log }_b(y)$ where $b>1$

Rule: Inequality preserves

• $x>y$

🟡 Level 2: Base Between 0 and 1

Spot it Fast: ${\log }_{0.5}(x)>{\log }_{0.5}(y)$

CRITICAL: Inequality flips

• $x<y$

Example: ${\log }_{0.5}(x^2-3)>{\log }_{0.5}(1)$

• Flip: $x^2-3<1$
• $x^2<4$ → $-2<x<2$
• Domain: $x^2-3>0$ → $|x|>\sqrt{3}$
• Intersection: $(\sqrt{3},2)\cup (-2,-\sqrt{3})$

🔴 Level 3: Change of Base

Spot it Fast: Different bases in same inequality

Formula: ${\log }_b(x)=\frac{{\log }_a(x)}{{\log }_a(b)}$

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Pattern Family 6.3: Exponential Equations

🟢 Level 1: Isolate Exponential

Spot it Fast: $3^x=7$

Mental Algorithm:

1. Log both sides: $\log (3^x)=\log (7)$
2. Power down: $x\log (3)=\log (7)$
3. Solve: $x=\frac{\log 7}{\log 3}$

🟡 Level 2: Same Base

Spot it Fast: $2^{x+1}=2^{3x}$

Mental Algorithm: Equate exponents

• $x+1=3x$
• $x=0.5$

🔴 Level 3: Substitution (Exponential Quadratic)

Spot it Fast: $4^x-5\cdot 2^x+4=0$

Mental Algorithm:

1. Substitute: Let $u=2^x$
2. Recognize: $4^x=(2^2{)}^x=u^2$
3. Quadratic: $u^2-5u+4=0$
4. Solve: $(u-1)(u-4)=0$ → $u=1$ or $4$
5. Back-substitute: $2^x=1$ → $x=0$ OR $2^x=4$ → $x=2$

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Week 7: Sequences & Limits

Pattern Family 7.1: AP & GP

🟢 Level 1: nth Term

Spot it Fast: “Find 10th term”

Formulas:

• AP: $a_n=a+(n-1)d$
• GP: $a_n=a{r}^{n-1}$

🟡 Level 2: Sum of Terms

Spot it Fast: “Find sum of first 20 terms”

Formulas:

• AP: $S_n=\frac{n}{2}(2a+(n-1)d)$ or $\frac{n}{2}(a+l)$
• GP: $S_n=a\frac{r^n-1}{r-1}$ (if $r\ne 1$)

🔴 Level 3: Mixed AP/GP Problems

Spot it Fast: “Numbers in AP, after transformation form GP”

Example: Three numbers in AP, sum = 15. Add 1, 4, 19 respectively → GP

• AP terms: $a-d,a,a+d$
• Sum: $3a=15$ → $a=5$
• After adding: $6-d,9,24+d$
• GP condition: $9^2=(6-d)(24+d)$
• $81=144-18d-d^2$
• $d^2+18d-63=0$
• $(d+21)(d-3)=0$
• $d=3$ → Numbers: 2, 5, 8

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Pattern Family 7.2: Infinite Series

🟢 Level 1: Infinite GP Sum

Spot it Fast: $1+\frac{1}{2}+\frac{1}{4}+\cdots$

Condition: $|r|<1$

Formula: $S=\frac{a}{1-r}$

Example: $5+\frac{5}{2}+\frac{5}{4}+\cdots$

• $a=5$, $r=0.5$
• $S=\frac{5}{1-0.5}=10$

🟡 Level 2: Recurring Decimals

Spot it Fast: Convert $\mathrm{0.333...}$ to fraction

Mental Algorithm: Express as GP sum

Example: $0.\overline{27}=\mathrm{0.272727...}$

• $=\frac{27}{100}+\frac{27}{10000}+\cdots$
• $=\frac{27}{100}(1+\frac{1}{100}+\frac{1}{10000}+\cdots )$
• $=\frac{27}{100}\cdot \frac{1}{1-0.01}=\frac{27}{99}=\frac{3}{11}$

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Pattern Family 7.3: Limits

🟢 Level 1: Rational Function Limits (at Infinity)

Spot it Fast: ${\lim }_{x\to \infty }\frac{3x^2+1}{5x^2-2}$

Mental Algorithm: Compare degrees

• Same degree: Ratio of leading coefficients
• Top higher: $\pm \infty$
• Bottom higher: $0$

Example: ${\lim }_{x\to \infty }\frac{3x^2}{5x^2+1}=\frac{3}{5}$

🟡 Level 2: Indeterminate Form $\frac{0}{0}$

Spot it Fast: Direct substitution gives $\frac{0}{0}$

Techniques:

1. Factor and cancel
2. Rationalize
3. L’Hôpital’s Rule (if learned)

Example: ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$

• Factor: $\frac{(x-2)(x+2)}{x-2}=x+2$
• Limit: $2+2=4$

🔴 Level 3: Rationalization

Spot it Fast: Square root in denominator causing $\frac{0}{0}$

Example: ${\lim }_{x\to 2}\frac{x^2-4}{\sqrt{x+2}-2}$

• Multiply by $\frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$
• Denominator: $(x+2)-4=x-2$
• Numerator: $(x-2)(x+2)(\sqrt{x+2}+2)$
• Cancel: $(x+2)(\sqrt{x+2}+2)$
• Plug $x=2$: $4(2+2)=16$

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Week 8: Derivatives

Pattern Family 8.1: Basic Differentiation

🟢 Level 1: Power Rule

Spot it Fast: $x^n$

Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$

Example: $\frac{d}{dx}(x^5)=5x^4$

🟡 Level 2: Sum/Difference Rule

Spot it Fast: Polynomial

Mental Algorithm: Differentiate each term

Example: $y=3x^4-2x^2+5$

• $y^{\prime }=12x^3-4x$

🔴 Level 3: Product/Quotient Rule

Spot it Fast: Multiplication/division of functions

Product Rule: $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$ Quotient Rule: $(\frac{u}{v}{)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

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Pattern Family 8.2: Chain Rule

🟢 Level 1: Simple Composition

Spot it Fast: $(ax+b{)}^n$

Mental Algorithm:

1. Outer: $n(...{)}^{n-1}$
2. Inner: $a$
3. Multiply: $na(ax+b{)}^{n-1}$

Example: $(2x+1{)}^3$

• $3(2x+1{)}^2\cdot 2=6(2x+1{)}^2$

🟡 Level 2: Nested Functions

Spot it Fast: $f(g(x))$

Mental Algorithm: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Example: $\sqrt{x^2+1}$

• Outer: $\frac{1}{2\sqrt{...}}$
• Inner: $2x$
• Result: $\frac{2x}{2\sqrt{x^2+1}}=\frac{x}{\sqrt{x^2+1}}$

🔴 Level 3: Multiple Layers

Spot it Fast: ${\sin }^3(2x^2+1)$

Mental Algorithm: Work outside-in

1. Layer 1: Power $({)}^3$ → $3({)}^2$
2. Layer 2: Sine → $\cos (...)$
3. Layer 3: Inside → $4x$
4. Result: $3{\sin }^2(2x^2+1)\cdot \cos (2x^2+1)\cdot 4x$

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Pattern Family 8.3: Applications

🟢 Level 1: Tangent Line

Spot it Fast: “Equation of tangent at $x=a$”

Mental Algorithm:

1. Point: $(a,f(a))$
2. Slope: $m=f^{\prime }(a)$
3. Line: $y-f(a)=m(x-a)$

🟡 Level 2: Horizontal Tangent

Spot it Fast: “Where is tangent horizontal?”

Mental Algorithm:

1. Condition: $f^{\prime }(x)=0$
2. Solve for $x$

Example: $y=x^3-3x$

• $y^{\prime }=3x^2-3=0$
• $x=\pm 1$
• Points: $(1,-2)$ and $(-1,2)$

🔴 Level 3: Optimization

Spot it Fast: “Maximize/minimize”

Mental Algorithm:

1. Function: Express what you want to optimize
2. Derivative: $f^{\prime }(x)$
3. Critical points: $f^{\prime }(x)=0$
4. Verify: Second derivative test OR endpoints

Example: Box from 12×12 sheet, cut squares of side $x$

• Volume: $V(x)=x(12-2x{)}^2$
• $V^{\prime }(x)=(12-2x{)}^2-4x(12-2x)=0$
• Factor: $(12-2x)(12-2x-4x)=0$
• $(12-2x)(12-6x)=0$
• $x=6$ or $x=2$
• $x=6$ gives $V=0$ (min)
• $x=2$ gives $V=128$ (max)

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🎯 Quick Reference: Pattern Recognition Cheat Sheet

Week 1: Sets & Relations

• “both”, “neither” → Inclusion-Exclusion
• Check RST → Systematically test R, S, T
• Domain with $\frac{...}{\sqrt{...}}$ → Sign analysis (both +/+ OR both -/-)

Week 2: Coordinate Geometry

• “minimize $AP+PB$” → Reflection principle
• Three parallelogram vertices → Midpoint formula or $D=A+C-B$

Week 3: Quadratics

• “maximize/minimize” → Vertex formula $x=-\frac{b}{2a}$
• “common root” → Solve simpler, substitute into other

Week 4: Polynomials

• “remainder when divided by” → Remainder theorem $P(a)$
• “increasing/decreasing” → Find $P^{\prime }(x)$, test intervals

Week 5: Functions

• “domain of $f(g(x))$” → Inner domain AND range constraint
• “find inverse” → Swap and solve

Week 6: Logarithms

• Base $0<b<1$ → FLIP inequality
• $4^x-5\cdot 2^x+4$ → Substitute $u=2^x$

Week 7: Sequences & Limits

• Infinite GP → $|r|<1$, use $S=\frac{a}{1-r}$
• $\frac{0}{0}$ → Factor OR rationalize

Week 8: Derivatives

• Chain rule → Outside-in: deriv of outer × deriv of inner
• Optimization → $f^{\prime }(x)=0$, check endpoints

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📝 Cross-Reference to Weekly Notes

Each pattern links to detailed examples in weekly notes:

• Week 1: Maths1_Week1_Sets_Relations.md
• Week 2: Maths1_Week2_Coordinate_Geometry.md
• Week 3: Maths1_Week3_Quadratic_Functions.md
• Week 4: Maths1_Week4_Polynomials.md
• Week 5: Maths1_Week5_Functions.md
• Week 6: Maths1_Week6_Logarithms.md
• Week 7: Maths1_Week7_Sequences_Limits.md
• Week 8: Maths1_Week8_Derivatives.md

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End of Maths 1 Ultimate Pattern Guide ✅

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How to Use This Guide

1. Learn the Pattern Hierarchy: Start with Level 1 (Basic) → Level 2 (Intermediate) → Level 3 (Advanced)
2. Pattern Families: Related patterns are grouped together
3. Quick Recognition: Use the “Spot it Fast” section to identify patterns during exams
4. Trap Awareness: Each pattern lists common mistakes

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Week 1: Sets & Relations

Pattern Family 1: Set Operations (Finding Missing Pieces)

Level 1: Direct Inclusion-Exclusion

Spot it Fast: Keywords “both”, “neither”, “at least one” Pattern: Given two sets with overlap, find missing counts.

Mental Algorithm:

1. Union: $|A\cup B|=|A|+|B|-|A\cap B|$
2. Neither: $|U|-|A\cup B|$

Example: 50 people. 30 like A, 25 like B, 10 like both. Find neither.

• $|A\cup B|=30+25-10=45$
• Neither = $50-45=5$

Common Traps:

• ❌ Forgetting to subtract the intersection
• ❌ Confusing “both” with “either”

Level 2: Three-Set Inclusion-Exclusion

Spot it Fast: Three sets (A, B, C) with various intersections

Mental Algorithm: $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$

Common Traps:

• ❌ Missing the triple intersection term
• ❌ Sign errors (subtract pairs, add triple)

Level 3: Complement with Multiple Conditions

Spot it Fast: “At least”, “Exactly”, combined with “not”

Mental Algorithm:

1. Find “Total - None”
2. Use complement: $|A^c|=|U|-|A|$

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Pattern Family 2: Relations (RST Properties)

Level 1: Reflexivity Check

Spot it Fast: “Is every element related to itself?”

Mental Algorithm:

• Check if $(a,a)\in R$ for all $a\in A$

Common Traps:

• ❌ Reflexive on subset vs full set

Level 2: Symmetry Check

Spot it Fast: “If $aRb$, then $bRa$?”

Mental Algorithm:

• For each pair $(a,b)$, check if $(b,a)$ exists

Common Traps:

• ❌ Confusing with antisymmetry

Level 3: Transitivity Proof

Spot it Fast: “If $aRb$ and $bRc$, then $aRc$?”

Mental Algorithm:

1. Assume $aRb$ and $bRc$
2. Use relation definition to prove $aRc$

Example: $R=\{(a,b):a-b\text{ is even}\}$

• $a-b$ even, $b-c$ even
• Sum: $(a-b)+(b-c)=a-c$ (even)
• ✅ Transitive

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Pattern Family 3: Domain of Functions

Level 1: Single Constraint (Denominator or Square Root)

Spot it Fast: One restriction (denominator ≠ 0 OR inside √ ≥ 0)

Mental Algorithm:

1. Denominator: Set $\ne 0$
2. Square Root: Set $\ge 0$

Example: $f(x)=\frac{1}{x-2}$

• Domain: $x\ne 2$

Level 2: Multiple Constraints (AND Logic)

Spot it Fast: Multiple restrictions that must ALL be satisfied

Mental Algorithm:

1. List all constraints
2. Find intersection of valid regions

Example: $f(x)=\frac{\sqrt{x-1}}{x-3}$

• Constraint 1: $x-1\ge 0⟹x\ge 1$
• Constraint 2: $x-3\ne 0⟹x\ne 3$
• Domain: $[1,3)\cup (3,\infty )$

Common Traps:

• ❌ Taking union instead of intersection

Level 3: Fraction Under Square Root (Sign Analysis)

Spot it Fast: $\sqrt{\frac{f(x)}{g(x)}}$

Mental Algorithm:

1. $\frac{f(x)}{g(x)}\ge 0$ means BOTH positive OR BOTH negative
2. Case 1: $f(x)\ge 0$ AND $g(x)>0$
3. Case 2: $f(x)\le 0$ AND $g(x)<0$
4. Union of both cases

Example: $f(x)=\sqrt{\frac{x-1}{x-2}}$

• Case 1: $x\ge 1$ AND $x>2$ → $x>2$
• Case 2: $x\le 1$ AND $x<2$ → $x\le 1$
• Domain: $(-\infty ,1]\cup (2,\infty )$

Common Traps:

• ❌ Forgetting the “BOTH negative” case

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Week 2: Coordinate Geometry

Pattern Family 1: Lines (Equations & Intersections)

Level 1: Slope-Intercept Form

Spot it Fast: Given slope $m$ and y-intercept $c$

Mental Algorithm:

• $y=mx+c$

Level 2: Point-Slope Form

Spot it Fast: Given point $(x_1,y_1)$ and slope $m$

Mental Algorithm:

• $y-y_1=m(x-x_1)$

Level 3: Intersection of Two Lines

Spot it Fast: “Find where lines meet”

Mental Algorithm:

1. Equate: Set $y$ equal from both equations
2. Solve for x
3. Find y: Plug $x$ into either equation

Example: $y=2x+1$ and $y=-x+4$

• $2x+1=-x+4$
• $3x=3⟹x=1$
• $y=2(1)+1=3$
• Intersection: $(1,3)$

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Pattern Family 2: Optimization (Shortest Path)

Level 1: Direct Distance

Spot it Fast: Distance between two points

Mental Algorithm: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Level 2: Perpendicular Distance (Point to Line)

Spot it Fast: “Shortest distance from point to line”

Mental Algorithm: $d=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$

Example: Distance from $(1,2)$ to line $3x+4y-5=0$

• $d=\frac{|3(1)+4(2)-5|}{\sqrt{9+16}}=\frac{6}{5}$

Level 3: Reflection Principle (Minimize Sum)

Spot it Fast: “Find point $P$ on line such that $AP+PB$ is minimum”

Mental Algorithm:

1. Reflect one point across the line
2. Draw straight line from reflected point to other point
3. Intersection with the line is the optimal point

Example: Minimize $AP+PB$ where $A=(1,2)$, $B=(5,3)$, $P$ on x-axis

• Reflect $A$ across x-axis: $A^{\prime }=(1,-2)$
• Line $A^{\prime }B$: Find x-intercept
• Slope: $\frac{3-(-2)}{5-1}=\frac{5}{4}$
• Eq: $y-3=\frac{5}{4}(x-5)$
• Set $y=0$: $x=2.6$

Common Traps:

• ❌ Trying to minimize algebraically without reflection

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Week 3: Quadratic Functions

Pattern Family 1: Vertex & Optimization

Level 1: Standard Form to Vertex

Spot it Fast: $y=a{x}^2+bx+c$, find max/min

Mental Algorithm:

1. Vertex x-coordinate: $x=-\frac{b}{2a}$
2. Plug in to find y-coordinate

Example: $y=-2x^2+8x-5$

• $x=-\frac{8}{2(-2)}=2$
• $y=-2(4)+8(2)-5=3$
• Vertex: $(2,3)$ (Maximum)

Common Traps:

• ❌ Forgetting to check if $a>0$ (min) or $a<0$ (max)

Level 2: Optimization with Constraints

Spot it Fast: “Maximize/minimize $f(x,y)$ subject to constraint”

Mental Algorithm:

1. Express one variable in terms of the other using constraint
2. Substitute into function
3. Find vertex of resulting quadratic

Example: Maximize $A=xy$ subject to $x+y=10$

• $y=10-x$
• $A(x)=x(10-x)=10x-x^2$
• Vertex: $x=5$, $y=5$
• Max Area: $25$

Level 3: Constrained Optimization (Perimeter/Area Problems)

Spot it Fast: “Maximize area given fixed perimeter” or vice versa

Mental Algorithm:

1. Setup equations: Area and Perimeter
2. Eliminate one variable
3. Maximize/minimize using vertex formula

Example: Rectangle perimeter 40. Maximize area.

• $2(l+w)=40⟹l+w=20$
• $A=lw=l(20-l)=20l-l^2$
• Vertex: $l=10$, $w=10$
• Max Area: $100$ (Square!)

Common Traps:

• ❌ Not recognizing that max area for fixed perimeter is always a square

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Pattern Family 2: Intersections & Solutions

Level 1: Line-Parabola Intersection

Spot it Fast: Find where line meets parabola

Mental Algorithm:

1. Equate: $a{x}^2+bx+c=mx+d$
2. Solve quadratic
3. Find y-values

Level 2: Parabola-Parabola Intersection

Spot it Fast: Two parabolas intersecting

Mental Algorithm:

1. Equate: $y_1=y_2$
2. Simplify: $a{x}^2+bx+c=0$
3. Check discriminant for number of solutions

Level 3: Common Roots Condition

Spot it Fast: “Find $k$ such that two equations have a common root”

Mental Algorithm:

1. Solve the simpler equation
2. Substitute each root into the other equation
3. Solve for $k$

Example: $x^2+kx+6=0$ and $x^2-5x+6=0$ have common root

• Solve: $x^2-5x+6=(x-2)(x-3)=0$
• Roots: $2,3$
• Try $x=2$: $4+2k+6=0⟹k=-5$
• Verify with $x=3$: Same result

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Week 4: Polynomials

Pattern Family 1: Construction & Factorization

Level 1: Construct from Roots

Spot it Fast: “Polynomial with roots $r_1,r_2,r_3$”

Mental Algorithm:

• $P(x)=k(x-r_1)(x-r_2)(x-r_3)$

Level 2: Construct with Additional Constraint

Spot it Fast: “Polynomial with roots … passing through point $(a,b)$”

Mental Algorithm:

1. Template: $P(x)=k(x-r_1)(x-r_2)\cdots$
2. Plug in point: $P(a)=b$
3. Solve for $k$

Level 3: Remainder Theorem Applications

Spot it Fast: “Remainder when $P(x)$ divided by $(x-a)$”

Mental Algorithm:

• $P(a)$ is the remainder

Advanced: “Remainder when divided by $(x-a)(x-b)$”

• Remainder is linear: $R(x)=mx+c$
• Use $P(a)=R(a)$ and $P(b)=R(b)$ to find $m,c$

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Pattern Family 2: Analysis (Turning Points, Intervals)

Level 1: Number of Turning Points

Spot it Fast: “How many peaks/valleys?”

Mental Algorithm:

• Degree $n$ polynomial has at most $n-1$ turning points

Level 2: Increasing/Decreasing Intervals

Spot it Fast: “Where is $P(x)$ increasing?”

Mental Algorithm:

1. Find derivative: $P^{\prime }(x)$
2. Critical points: Solve $P^{\prime }(x)=0$
3. Test intervals: Pick test points between critical points

Example: $P(x)=x^3-3x^2$

• $P^{\prime }(x)=3x^2-6x=3x(x-2)=0$
• Critical: $x=0,2$
• Test $x=-1$: $P^{\prime }(-1)=9>0$ (Increasing)
• Test $x=1$: $P^{\prime }(1)=-3<0$ (Decreasing)
• Test $x=3$: $P^{\prime }(3)=9>0$ (Increasing)

Common Traps:

• ❌ Not including critical points in the final intervals

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[Continues for Weeks 5-8…]