Maths 1: Formula Dissection & Memory Guide 🧠

IMPORTANT

Purpose: Break down EVERY formula piece by piece. Understand what each part means, when to use it, and how to remember it forever.

How to Use:

Anatomy: See each variable explained

Memory Trick: Mnemonic or visual to remember

When to Use: Pattern recognition

Common Mistakes: What NOT to do

📊 Formula Index by Week

Week	Key Formulas	Count
1: Sets & Relations	Inclusion-Exclusion, Cardinality	3
2: Coordinate Geometry	Distance, Slope, Midpoint, Area	7
3: Quadratics	Vertex, Discriminant, Vieta’s	5
4: Polynomials	Remainder Theorem, Degree	3
5: Functions	Composition, Inverse	2
6: Logarithms	Log Laws, Change of Base	8
7: Sequences & Limits	AP, GP, Infinite Sum	6
8: Derivatives	Power Rule, Product, Quotient, Chain	7

Week 1: Sets & Relations

Formula 1: Inclusion-Exclusion (2 Sets)

📐 The Formula

$∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣$

🔍 Anatomy (What Each Part Means)

$∣ A \cup B ∣$ : Size of UNION (everything in A OR B or both)
$∣ A ∣$ : Count of elements in set A
$∣ B ∣$ : Count of elements in set B
$∣ A \cap B ∣$ : Count in BOTH sets (intersection/overlap)
Why subtract?: We counted overlap TWICE (once in $∣ A ∣$ , once in $∣ B ∣$ ), so remove it once

🧠 Memory Trick

“Add both circles, subtract the lens”

Draw Venn diagram: A and B are circles, overlap is a lens shape
Add both circles, but the lens got counted twice, so subtract it once

✅ When to Use

Keywords: “both”, “neither”, “at least one”, “only A”

⚠️ Common Mistakes

❌ Forgetting to subtract intersection: $∣ A ∣ + ∣ B ∣$ ≠ $∣ A \cup B ∣$
❌ Using this for “neither”: Neither = Total - Union, NOT part of this formula

Formula 2: Inclusion-Exclusion (3 Sets)

📐 The Formula

$∣ A \cup B \cup C ∣ = ∣ A ∣ + ∣ B ∣ + ∣ C ∣ - ∣ A \cap B ∣ - ∣ A \cap C ∣ - ∣ B \cap C ∣ + ∣ A \cap B \cap C ∣$

🔍 Anatomy

Singles ( $∣ A ∣, ∣ B ∣, ∣ C ∣$ ): Add all three sets
Pairs ( $∣ A \cap B ∣, ...$ ): Subtract all pair overlaps (we overcounted these)
Triple ( $∣ A \cap B \cap C ∣$ ): Add back (we subtracted it 3 times, but should only subtract 2)

🧠 Memory Trick

“Add Singles, Subtract Pairs, Add Triple” (ASSP-AT)

Think of it like a wave: + + + - - - +

⚠️ Common Mistakes

❌ Wrong sign on triple intersection (it’s PLUS!)
❌ Missing one of the three pair terms

Week 2: Coordinate Geometry

Formula 3: Distance Between Two Points

📐 The Formula

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

🔍 Anatomy

$(x_{2} - x_{1})$ : Horizontal distance (run)
$(y_{2} - y_{1})$ : Vertical distance (rise)
Why square?: To make negatives positive (direction doesn’t matter for distance)
Why sqrt?: Pythagorean theorem! This IS the hypotenuse

🧠 Memory Trick

“Pythagorean theorem in disguise”

Draw right triangle with points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
Horizontal leg = $∣ x_{2} - x_{1} ∣$
Vertical leg = $∣ y_{2} - y_{1} ∣$
Hypotenuse = $d$
$d^{2} = horizontal^{2} + vertical^{2}$

⚠️ Common Mistakes

❌ Forgetting the square root: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ is NOT distance
❌ Subtracting wrong way: Doesn’t matter! $(x_{2} - x_{1})^{2} = (x_{1} - x_{2})^{2}$

Formula 4: Slope of a Line

📐 The Formula

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{rise}{run}$

🔍 Anatomy

Numerator: Change in $y$ (vertical change)
Denominator: Change in $x$ (horizontal change)
Meaning: “For every 1 unit right, go $m$ units up”

🧠 Memory Trick

“Rise over Run”

Rise = how much you go UP (y-direction)
Run = how much you go RIGHT (x-direction)
Mnemonic: “Ryan (rise) Runs (run)“

⚠️ Common Mistakes

❌ Flipping: $\frac{x _{2} - x _{1}}{y _{2} - y _{1}}$ is WRONG
❌ Dividing by zero: Vertical line has undefined slope (not zero!)

Formula 5: Midpoint

📐 The Formula

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

🔍 Anatomy

Average of x-coordinates: Middle x-value
Average of y-coordinates: Middle y-value

🧠 Memory Trick

“Just average both coordinates”

To find middle of two numbers: add and divide by 2
Do this for BOTH x and y

⚠️ Common Mistakes

❌ Forgetting to divide by 2: $(\frac{x _{1} + x _{2}}{2}, y_{1} + y_{2})$ is wrong

Formula 6: Point-Slope Form

📐 The Formula

$y - y_{1} = m (x - x_{1})$

🔍 Anatomy

$(x_{1}, y_{1})$ : A point ON the line
$m$ : Slope of the line
Why this form?: Rearranging $m = \frac{y - y _{1}}{x - x _{1}}$

🧠 Memory Trick

“y minus y-one equals slope times (x minus x-one)”

Think: “The change in y equals slope times change in x”

✅ When to Use

“Line through point $(x_{1}, y_{1})$ with slope $m$ “

Formula 7: Area of Triangle (Coordinate Method)

📐 The Formula

$A = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$

🔍 Anatomy

Each term: $x_{i}$ times (difference of OTHER two y-values)
Absolute value: Area is always positive
Divide by 2: Standard triangle formula

🧠 Memory Trick

“Shoelace method” or use base × height instead

Honestly, this formula is hard to memorize
Better: Find base and height from coordinates

⚠️ Common Mistakes

❌ Forgetting absolute value
❌ Wrong pairing of coordinates

Week 3: Quadratic Functions

Formula 8: Vertex Formula

📐 The Formula

For $y = a x^{2} + b x + c$ , vertex x-coordinate: $x = - \frac{b}{2 a}$

🔍 Anatomy

$a$ : Coefficient of $x^{2}$ (determines if parabola opens up/down)
$b$ : Coefficient of $x$ (determines horizontal shift)
$- \frac{b}{2 a}$ : The “balancing point” of the parabola

🧠 Memory Trick

“Negative bee over two-ay”

Visual: Parabola is symmetric, this is the axis of symmetry
Why negative?: Completes the square: $a (x + \frac{b}{2 a})^{2} + ...$

✅ When to Use

“Find maximum/minimum”, “vertex”, “optimize”

⚠️ Common Mistakes

❌ Forgetting negative sign: $\frac{b}{2 a}$ is WRONG
❌ Using this for y-coordinate: This only gives x! Plug back in for y

Formula 9: Discriminant

📐 The Formula

For $a x^{2} + b x + c = 0$ : $Δ = b^{2} - 4 a c$

🔍 Anatomy

$b^{2}$ : Square of linear coefficient
$4 a c$ : Four times product of $a$ and $c$
What it tells:
- $Δ > 0$ : 2 distinct real roots
- $Δ = 0$ : 1 repeated real root (perfect square)
- $Δ < 0$ : 0 real roots (2 complex)

🧠 Memory Trick

“Bee-squared minus four-ay-see”

Visual: This is the part under the square root in quadratic formula
Mnemonic: “Bee Building 4 Apartments Complex” (b² - 4ac)

✅ When to Use

“How many real solutions?”, “Does line intersect parabola?”

Formula 10: Quadratic Formula

📐 The Formula

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

🔍 Anatomy

$- b$ : Opposite of linear coefficient
$\pm$ : TWO solutions (one with +, one with -)
$b^{2} - 4 a c$ : The discriminant (see above)
$2 a$ : Twice the leading coefficient

🧠 Memory Trick

“x equals negative bee, plus or minus ROOT bee squared minus four ay see, ALL over two ay”

Song rhythm: “Pop Goes the Weasel” tune

⚠️ Common Mistakes

❌ Forgetting $\pm$ : You get TWO roots!
❌ Only dividing part by $2 a$ : ENTIRE numerator divided by $2 a$

Formula 11: Vieta’s Formulas

📐 The Formula

For $a x^{2} + b x + c = 0$ with roots $α, β$ :

Sum: $α + β = - \frac{b}{a}$
Product: $α β = \frac{c}{a}$

🔍 Anatomy

Sum: Negative of (middle coef / leading coef)
Product: Constant / leading coef

🧠 Memory Trick

“Sum is negative b/a, Product is c/a”

Pattern: If $a = 1$ : sum = $- b$ , product = $c$
Example: $x^{2} - 5 x + 6 = 0$ → sum = 5, product = 6 → roots are 2, 3

✅ When to Use

“Find sum/product without solving”

Week 6: Logarithms

Formula 12: Change of Base

📐 The Formula

$lo g_{b} (x) = \frac{l o g _{a} ( x )}{l o g _{a} ( b )}$

🔍 Anatomy

$lo g_{b} (x)$ : What you want (log base $b$ of $x$ )
$lo g_{a} (x)$ : Log of $x$ in new base $a$
$lo g_{a} (b)$ : Log of OLD base in new base
Common choice: $a = 10$ or $a = e$

🧠 Memory Trick

“New base on bottom”

To change from base $b$ to base $a$ : divide by $lo g_{a} (b)$
Visual: $\frac{log of argument}{log of old base}$

✅ When to Use

Calculator only has log/ln, but you need $lo g_{2}$ or $lo g_{5}$

Formula 13: Product Rule

📐 The Formula

$lo g_{b} (x y) = lo g_{b} (x) + lo g_{b} (y)$

🧠 Memory Trick

“Multiplication inside becomes addition outside”

$2 \times 3 = 6$ but $lo g (2 \times 3) = lo g 2 + lo g 3$

Formula 14: Quotient Rule

📐 The Formula

$lo g_{b} (\frac{x}{y}) = lo g_{b} (x) - lo g_{b} (y)$

🧠 Memory Trick

“Division inside becomes subtraction outside”

Formula 15: Power Rule

📐 The Formula

$lo g_{b} (x^{n}) = n lo g_{b} (x)$

🔍 Anatomy

$n$ : The exponent comes OUT front as a multiplier

🧠 Memory Trick

“Power comes down”

Exponent “drops down” in front of log

Week 7: Sequences & Limits

Formula 16: AP nth Term

📐 The Formula

$a_{n} = a + (n - 1) d$

🔍 Anatomy

$a$ : First term
$d$ : Common difference (what you ADD each time)
$(n - 1)$ : Number of “jumps” from first term to nth term
Why $n - 1$ ?: To get to 2nd term, you jump ONCE (not twice)

🧠 Memory Trick

“Start plus (steps minus one) times diff”

From term 1 to term 5: you take 4 steps
Position 10 = First + 9 jumps

Formula 17: AP Sum

📐 The Formula

$S_{n} = \frac{n}{2} (2 a + (n - 1) d) OR S_{n} = \frac{n}{2} (a + l)$

🔍 Anatomy (Second form)

$n$ : Number of terms
$a$ : First term
$l$ : Last term
Average: $(a + l) /2$ is the average
Logic: Average × count

🧠 Memory Trick

“Number of terms times average of first and last, divided by 2”

Trapezoid area formula!

Formula 18: GP nth Term

📐 The Formula

$a_{n} = a r^{n - 1}$

🔍 Anatomy

$a$ : First term
$r$ : Common ratio (what you MULTIPLY each time)
$r^{n - 1}$ : Multiply first term by $r$ exactly $(n - 1)$ times

🧠 Memory Trick

“First times ratio to the (n minus one)”

To get 3rd term: multiply by $r$ TWICE (not 3 times)

Formula 19: Infinite GP Sum

📐 The Formula

$S = \frac{a}{1 - r} (only if ∣ r ∣ < 1)$

🔍 Anatomy

Condition $∣ r ∣ < 1$ : Terms must get smaller and approach zero
$a$ : First term
$1 - r$ : Denominator is “one minus ratio”

🧠 Memory Trick

“First over one-minus-ratio”

If $∣ r ∣ \geq 1$ : series DIVERGES (no sum!)

⚠️ Common Mistakes

❌ Using this when $∣ r ∣ \geq 1$ : Divergent series!

Week 8: Derivatives

Formula 20: Power Rule

📐 The Formula

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

🔍 Anatomy

Exponent comes down: $n$ becomes coefficient
Reduce exponent by 1: $x^{n - 1}$

🧠 Memory Trick

“Bring down the power, reduce power by one”

$x^{5} \to 5 x^{4}$
$x^{2} \to 2 x^{1} = 2 x$
$x^{1} \to 1 x^{0} = 1$

Formula 21: Product Rule

📐 The Formula

$(uv)^{'} = u^{'} v + u v^{'}$

🧠 Memory Trick

“First D-second plus First D-second”

Derivative of first × second + first × derivative of second
Mnemonic: “You Drive, I Watch (uv’ + vu’)” wait that’s backwards…
Better: “Left-D-Right plus Left-Right-D”

Formula 22: Quotient Rule

📐 The Formula

$(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$

🧠 Memory Trick

“Low-D-High minus High-D-Low, over Low-squared”

Low = denominator ( $v$ )
High = numerator ( $u$ )
Bottom (v) DERIV top (u) MINUS top DERIV bottom, ALL OVER bottom squared

Formula 23: Chain Rule

📐 The Formula

$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$

🔍 Anatomy

Outside derivative: Derivative of $f$ , evaluated at $g (x)$
Times inside derivative: Derivative of $g (x)$

🧠 Memory Trick

“Outside times inside”

Derivative of outer function × derivative of inner function
Work outside-in: differentiate outer layer first, then inner

🎯 Quick Formula Reference Card

Most Used Formulas (Memorize These First!)

Coordinate Geometry

Distance: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Slope: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Midpoint: $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

Quadratics

Vertex x: $x = - \frac{b}{2 a}$
Discriminant: $Δ = b^{2} - 4 a c$
Roots: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Sequences

AP: $a_{n} = a + (n - 1) d$
GP: $a_{n} = a r^{n - 1}$
Infinite GP: $S = \frac{a}{1 - r}$ (if $∣ r ∣ < 1$ )

Derivatives

Power: $(x^{n})^{'} = n x^{n - 1}$
Product: $(uv)^{'} = u^{'} v + u v^{'}$
Quotient: $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Chain: $[f (g)]^{'} = f^{'} (g) \cdot g^{'}$

📝 Memory Palace Technique

Group formulas by “room”:

Room 1: Distance & Location (Coordinate Geometry)

Distance, Slope, Midpoint all involve two points

Room 2: The Parabola Room (Quadratics)

Vertex, Discriminant, Roots all about $a x^{2} + b x + c$

Room 3: The Sequence Hall (AP & GP)

Two parallel corridors: AP on left, GP on right

Room 4: The Calculus Lab (Derivatives)

Power Rule is basic tool, others are combinations

End of Formula Dissection Guide ✅

🧠 IIT Madras Notes

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Maths1_Formula_Dissection_Guide

Maths 1: Formula Dissection & Memory Guide 🧠

📊 Formula Index by Week

Week 1: Sets & Relations

Formula 1: Inclusion-Exclusion (2 Sets)

📐 The Formula

🔍 Anatomy (What Each Part Means)

🧠 Memory Trick

✅ When to Use

⚠️ Common Mistakes

Formula 2: Inclusion-Exclusion (3 Sets)

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Week 2: Coordinate Geometry

Formula 3: Distance Between Two Points

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Formula 4: Slope of a Line

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Formula 5: Midpoint

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Formula 6: Point-Slope Form

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

✅ When to Use

Formula 7: Area of Triangle (Coordinate Method)

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Week 3: Quadratic Functions

Formula 8: Vertex Formula

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

✅ When to Use

⚠️ Common Mistakes

Formula 9: Discriminant

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

✅ When to Use

Formula 10: Quadratic Formula

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

⚠️ Common Mistakes

Formula 11: Vieta’s Formulas

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

✅ When to Use

Week 6: Logarithms

Formula 12: Change of Base

📐 The Formula

🔍 Anatomy

🧠 Memory Trick

✅ When to Use

Formula 13: Product Rule

📐 The Formula

🧠 Memory Trick

Formula 14: Quotient Rule

📐 The Formula

🧠 Memory Trick

Formula 15: Power Rule

📐 The Formula

🔍 Anatomy