Maths1_Pattern_Bank

Maths 1: Pattern Bank & Mental Algorithms

IMPORTANT

What is this?: This is a collection of every “Question Type” you will face in assignments and exams. It gives you the Mental Algorithm to solve them step-by-step.

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Week 1: Sets & Relations

Pattern 1.1: Inclusion-Exclusion (Finding Missing Counts)

Question: “50 people total. 30 like A, 25 like B, 10 like both. How many like neither?” Algorithm:

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

Pattern 1.2: Equivalence Check (RST)

Question: “Is relation $R$ (where $a-b$ is even) an equivalence relation?” Algorithm:

1. Reflexive: Check $xRx$. (Is $x-x$ even? Yes).
2. Symmetric: Assume $xRy$. Check $yRx$. (If $x-y$ even, is $y-x$ even? Yes).
3. Transitive: Assume $xRy$ and $yRz$. Check $xRz$. (Sum them: $(x-y)+(y-z)=x-z$. Sum of evens is even? Yes).

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Week 2: Coordinate Geometry

Pattern 2.1: Parallelogram Vertex (Midpoint Hack)

Question: “Vertices are $A,B,C$. Find $D$.” Algorithm:

1. Formula: $D=A+C-B$ (Coordinate-wise).
2. Why: Diagonals bisect each other. Midpoint of $AC$ = Midpoint of $BD$.

Pattern 2.2: Reflection of Point

Question: “Reflect point $P$ about line $L$.” Algorithm:

1. Slope: Find slope of $L$ ($m$). Perpendicular slope is $-1/m$.
2. Line Eq: Write eq of line through $P$ with slope $-1/m$.
3. Intersection: Find where this line meets $L$ (Point $M$).
4. Midpoint: $M$ is midpoint of $P$ and $P^{\prime }$. Solve for $P^{\prime }$.

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Week 3: Quadratics

Pattern 3.1: Optimization (Max/Min)

Question: “Find max height of ball $h(t)=-5t^2+20t$.” Algorithm:

1. Vertex Time: $t=-b/2a$.
2. Max Value: Plug $t$ back into equation.

Pattern 3.2: Intersection

Question: “Where does line $y=x+2$ meet parabola $y=x^2$?” Algorithm:

1. Equate: $x^2=x+2$.
2. Solve: $x^2-x-2=0$. Find $x$.
3. Find Y: Plug $x$ into line equation.

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Week 4: Polynomials

Pattern 4.1: Construct from Roots

Question: “Polynomial with roots 1, 2, 3 passing through (0, 6).” Algorithm:

1. Template: $P(x)=A(x-1)(x-2)(x-3)$.
2. Find A: Plug in $(0,6)$ to solve for $A$.

Pattern 4.2: Remainder Theorem

Question: “Remainder of $P(x)$ divided by $x-2$.” Algorithm:

1. Plug in: Calculate $P(2)$. That’s the answer.

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Week 5: Functions

Pattern 5.1: Domain of Composite

Question: “Domain of $f(g(x))$.” Algorithm:

1. Inner: Domain of $g(x)$.
2. Outer: Solve $g(x)\in \text{Domain of }f$.
3. Intersect: Overlap of both regions.

Pattern 5.2: Inverse Function

Question: “Find inverse of $y=2x+1$.” Algorithm:

1. Swap: $x=2y+1$.
2. Isolate: $2y=x-1⟹y=(x-1)/2$.

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Week 6: Logarithms

Pattern 6.1: Exponential Equation

Question: “Solve $3^{x+1}=7$.” Algorithm:

1. Log: Take $\ln$ of both sides.
2. Power Down: $(x+1)\ln 3=\ln 7$.
3. Solve: $x=\frac{\ln 7}{\ln 3}-1$.

Pattern 6.2: Log Equation

Question: “Solve $\log x+\log (x-3)=1$.” Algorithm:

1. Combine: $\log (x(x-3))=1$.
2. Exponentiate: $x(x-3)=10^1$.
3. Solve: Quadratic $x^2-3x-10=0$.
4. Check: Discard negative arguments.

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Week 7: Limits

Pattern 7.1: Rational Limit ($\infty$)

Question: “Limit of $\frac{3n^2}{5n^2+1}$.” Algorithm:

1. Compare Degrees: Equal?
2. Ratio: Take coeffs $3/5$.

Pattern 7.2: Continuity Constant

Question: “Find $k$ for continuity at $x=2$.” Algorithm:

1. LHL: Plug $x=2$ into left formula.
2. RHL: Plug $x=2$ into right formula.
3. Equate: LHL = RHL. Solve for $k$.

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Week 8: Derivatives

Pattern 8.1: Tangent Line

Question: “Tangent to $y=x^2$ at $x=3$.” Algorithm:

1. Point: $y=3^2=9$. Point $(3,9)$.
2. Slope: $y^{\prime }=2x$. At $x=3$, slope $m=6$.
3. Line: $y-9=6(x-3)$.

Pattern 8.2: Chain Rule

Question: “Derivative of $(x^2+1{)}^5$.” Algorithm:

1. Outer: $5(\dots {)}^4$.
2. Inner: $2x$.
3. Multiply: $5(x^2+1{)}^4\cdot 2x$.