Consolidated Formulas

This is the Ultimate Theory & Formula Handbook for Maths 1.

I have condensed 12 weeks of content into a high-density “Cheat Sheet” format. This covers every definition, formula, and theorem you need to solve the Qualifier and Quiz 3 questions.

---

📐 Part 1: Algebra & Geometry (Weeks 1-4)

Week 1: Sets & Relations

• Set Operations:
  • Union ($A\cup B$): Elements in A OR B.
  • Intersection ($A\cap B$): Elements in A AND B.
  • Difference ($A\setminus B$): In A but NOT in B.
  • Symmetric Difference ($A\Delta B$): $(A\setminus B)\cup (B\setminus A)$. (Everything except the middle).
• Cardinality Formula: $n(A\cup B)=n(A)+n(B)-n(A\cap B)$ $n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-\cdots +n(A\cap B\cap C)$
• Relations:
  • Reflexive: $(a,a)\in R$ for all $a$. (Every element relates to itself).
  • Symmetric: If $(a,b)\in R$, then $(b,a)\in R$.
  • Transitive: If $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.
  • Equivalence Relation: Reflexive + Symmetric + Transitive.

Week 2: Coordinate Geometry (Lines)

• Slope ($m$): $m=\tan \theta =\frac{y_2-y_1}{x_2-x_1}$.
• Equations of a Line:
  1. Slope-Intercept: $y=mx+c$.
  2. Point-Slope: $y-y_1=m(x-x_1)$.
  3. Two-Point: $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$.
• Relationship between Lines:
  • Parallel: $m_1=m_2$.
  • Perpendicular: $m_1\cdot m_2=-1$.
• Distance Formula: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
• Section Formula (Internal): $P=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$.

Week 3: Quadratic Functions (The High Yield Zone)

• Standard Form: $f(x)=a{x}^2+bx+c$.
• Vertex Form: $f(x)=a(x-h{)}^2+k$.
  • Vertex $(h,k)$: $h=\frac{-b}{2a}$, $k=f(h)=\frac{-D}{4a}$.
• Roots ($\alpha ,\beta$):
  • Sum: $\alpha +\beta =-\frac{b}{a}$.
  • Product: $\alpha \beta =\frac{c}{a}$.
• Discriminant ($D=b^2-4ac$):
  • $D>0$: 2 Distinct Real Roots.
  • $D=0$: 1 Real Root (Graph touches x-axis).
  • $D<0$: No Real Roots (Graph floats above/below axis).

Week 4: Polynomials

• Remainder Theorem: If $P(x)$ is divided by $(x-c)$, remainder is $P(c)$.
• Factor Theorem: $(x-c)$ is a factor if and only if $P(c)=0$.
• Multiplicity of Roots:
  • Odd Multiplicity: Graph crosses the x-axis.
  • Even Multiplicity: Graph touches/bounces off the x-axis.
• Turning Points: A polynomial of degree $n$ has at most $n-1$ turning points.

---

📈 Part 2: Functions & Calculus (Weeks 5-10)

Week 5: Functions

• One-to-One (Injective): Distinct inputs $\to$ Distinct outputs.
  • Test: Horizontal Line Test intersects at most once. Strictly increasing/decreasing functions are 1-1.
• Onto (Surjective): Range = Codomain.
  • Test: Horizontal Line Test intersects at least once everywhere.
• Bijective: Both 1-1 and Onto. (Invertible).
• Composite: $(f\circ g)(x)=f(g(x))$.
  • Domain = $\{x\in \text{Dom}(g)\mid g(x)\in \text{Dom}(f)\}$.
• Inverse: $f^{-1}(y)=x⟺f(x)=y$. Graph reflected across $y=x$.

Week 6: Exponents & Logarithms

• Log Rules (Memorize these):
  1. ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$
  2. ${\log }_a(x/y)={\log }_{a}x-{\log }_{a}y$
  3. ${\log }_a(x^n)=n{\log }_{a}x$
  4. Base Change: ${\log }_{a}b=\frac{\ln b}{\ln a}=\frac{1}{{\log }_{b}a}$.
• Inequalities:
  • If $a>1$: ${\log }_{a}x>y⟹x>a^y$. (Sign stays).
  • If $0<a<1$: ${\log }_{a}x>y⟹x<a^y$. (Sign FLIPS).

Week 7: Limits

• Existence: ${\lim }_{x\to c}f(x)=L$ if and only if LHL = RHL.
  • LHL: ${\lim }_{x\to c^{-}}f(x)$.
  • RHL: ${\lim }_{x\to c^{+}}f(x)$.
• L’Hopital’s Rule (The Savior): If $\lim \frac{f(x)}{g(x)}$ results in $\frac{0}{0}$ or $\frac{\infty }{\infty }$, then: ${\lim }_{x\to c}\frac{f(x)}{g(x)}={\lim }_{x\to c}\frac{f^{\prime }(x)}{g^{\prime }(x)}$
• Growth Hierarchy (at $\infty$): $x^x\gg e^x\gg x^n\gg \ln x$.

Week 8: Continuity & Derivatives

• Continuity at $c$:
  1. Limit exists.
  2. $f(c)$ is defined.
  3. Limit = $f(c)$.
• Differentiability at $c$:
  1. Continuous at $c$.
  2. Left Derivative = Right Derivative. (No sharp corners).
• Equation of Tangent: At point $(x_1,y_1)$: $y-y_1=f^{\prime }(x_1)(x-x_1)$

Week 9: Differentiation Rules

• Chain Rule: $\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$.
• Product Rule: $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$.
• Quotient Rule: $(\frac{u}{v}{)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$.
• Standard Derivatives:
  • $(x^n{)}^{\prime }=n{x}^{n-1}$
  • $(e^x{)}^{\prime }=e^x$
  • $(\ln x{)}^{\prime }=1/x$
  • $(\sin x{)}^{\prime }=\cos x$, $(\cos x{)}^{\prime }=-\sin x$.
• Critical Points: Points where $f^{\prime }(x)=0$ or undefined.
• Max/Min:
  • $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)>0⟹$ Local Minimum.
  • $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)<0⟹$ Local Maximum.

Week 10: Integration

• Concept: Area under the curve $y=f(x)$ from $a$ to $b$.
• Fundamental Theorem of Calculus: ${\int }_a^{b}f(x)dx=F(b)-F(a)\text{where }{F}^{\prime }(x)=f(x)$
• Standard Integrals:
  • $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ (for $n\ne -1$).
  • $\int \frac{1}{x}dx=\ln |x|$.
  • $\int e^{ax}dx=\frac{1}{a}{e}^{ax}$.
• Substitution: If integrand has $g(x)$ and $g^{\prime }(x)$, set $u=g(x)$.

---

🕸️ Part 3: Discrete Math & Graphs (Weeks 11-12)

Week 11: Graph Theory Basics

• Components: $G=(V,E)$. Vertices (dots) and Edges (lines).
• Degree ($deg(v)$): Number of edges connected to vertex $v$.
• Handshaking Lemma: ${\sum }_{v\in V}deg(v)=2\times |E|$
  • Corollary: The number of odd-degree vertices must be Even.
• Adjacency Matrix:
  • $n\times n$ matrix. Entry $A_{ij}=1$ if connected, 0 if not.
  • Sum of row $i$ = Degree of vertex $i$.
  • Trace = $2\times$ Number of self-loops (usually 0).

Week 12: Algorithms & Special Graphs

• Path vs Walk:
  • Walk: Can repeat vertices/edges.
  • Path: No repeated vertices.
  • Trail: No repeated edges.
• Cycle: Path that starts and ends at the same vertex.
• Tree: Connected graph with No Cycles.
  • Edges in a tree with $n$ vertices = $n-1$.
• Spanning Tree: A subgraph that is a Tree and includes ALL vertices.
• BFS (Breadth-First Search):
  • Explores layer by layer (neighbors first).
  • Used for shortest path in unweighted graphs.
• DFS (Depth-First Search):
  • Goes deep into one branch before backtracking.
• Bipartite Graph: Vertices can be split into two sets $U,V$ such that edges only go between $U$ and $V$. (No odd cycles allowed).

---

⚡ The “Do Not Forget” List

1. Log Domain: $\log (f(x))$ exists only if $f(x)>0$.
2. Even/Odd Functions:
   • Even: $f(-x)=f(x)$ (Symmetric to y-axis, e.g., $x^2,\cos x$).
   • Odd: $f(-x)=-f(x)$ (Symmetric to origin, e.g., $x^3,\sin x$).
3. Modulus Derivative: $\frac{d}{dx}|x|=\frac{x}{|x|}$ (for $x\ne 0$).
4. Integration Area: If curve is below x-axis, integral is negative. Area = $|\text{Integral}|$.
5. Graph Degrees: Sum of degrees can NEVER be odd. If you calculate an odd sum, check your math.

This is everything. No fluff. Just the raw math you need to pass.

Here is the Ultimate Theory & Formula Handbook for Statistics 1.

I have condensed the entire 12-week syllabus into a high-density “Cheat Sheet” format. This covers every definition, formula, and concept you need for the Qualifier, Quiz 2, and End Term.

---

📊 Part 1: Descriptive Statistics (Weeks 1-4)

Week 1: Data Types & Scales

• Data Types:
  • Qualitative (Categorical): Labels/Attributes. (e.g., Color, Gender).
  • Quantitative (Numerical): Numbers.
    • Discrete: Countable integers. (e.g., Number of students).
    • Continuous: Measured values. (e.g., Height, Temperature).
• Scales of Measurement (Crucial):
  1. Nominal: Labels only. No order. (Red, Blue). Mode only.
  2. Ordinal: Ordered labels. Differences meaningless. (Rank 1, Rank 2). Median/Percentile.
  3. Interval: Ordered + Meaningful difference. No True Zero. (Temp in Celsius/Fahrenheit). Mean/SD. Addition/Subtraction valid.
  4. Ratio: True Zero exists. (Height, Weight, Salary). All stats valid. Mult/Div valid.

Week 2: Visualizing Data

• Categorical: Bar Chart (gaps), Pie Chart (parts of whole), Pareto Chart (sorted bar + cumulative line).
• Numerical: Histogram (no gaps), Box Plot (outliers), Scatter Plot (bivariate).
• Frequency Density (for unequal bins): $\text{Frequency Density}=\frac{\text{Frequency}}{\text{Class Width}}$

Week 3: Measures of Central Tendency & Dispersion

• Central Tendency:
  • Mean ($\bar{x}$): $\frac{\sum x_i}{n}$. Sensitive to outliers.
  • Median: Middle value (50th percentile). Robust to outliers.
  • Mode: Most frequent value.
• Dispersion (Spread):
  • Range: Max - Min.
  • IQR (Interquartile Range): $Q_3-Q_1$. (Middle 50%).
  • Variance (${\sigma }^2$): $\frac{1}{N}\sum (x_i-\mu {)}^2$.
  • Standard Deviation ($\sigma$): $\sqrt{\text{Variance}}$.
  • Coefficient of Variation (CV): $\frac{\sigma }{\mu }\times 100$. (Compare variability of different units).
• Linear Transformation ($Y=aX+b$):
  • $\bar{y}=a\bar{x}+b$
  • ${\sigma }_y^2=a^2{\sigma }_x^2$ (Ignore $b$, square $a$)
  • ${\sigma }_y=|a|{\sigma }_x$ (Ignore $b$, abs value $a$)

Week 4: Correlation & Bivariate Data

• Covariance: Direction of relationship. $\text{Cov}(X,Y)=\frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})$
• Correlation Coefficient ($r$): Strength and Direction (-1 to 1). $r=\frac{\text{Cov}(X,Y)}{{\sigma }_x{\sigma }_y}$
  • $r=1$: Perfect positive linear.
  • $r=0$: No linear correlation (could be curved).
  • $r=-1$: Perfect negative linear.

---

🎲 Part 2: Probability & Distributions (Weeks 5-8)

Week 5: Counting (Combinatorics)

• Fundamental Principle: $m$ ways $\times$ $n$ ways.
• Permutation (${}^n{P}_r$): Order Matters. (Ranking, Arrangement). ${}^n{P}_r=\frac{n!}{(n-r)!}$
• Combination (${}^n{C}_r$): Order Doesn’t Matter. (Selection, Groups). ${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Week 6: Probability Axioms

• Union Rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• Disjoint (Mutually Exclusive): $P(A\cap B)=0$.
• Independent: $P(A\cap B)=P(A)\cdot P(B)$.
• Conditional: $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Week 7: Bayes’ Theorem & Partition

• Law of Total Probability: $P(B)=P(B|A_1)P(A_1)+P(B|A_2)P(A_2)+\dots$
• Bayes’ Theorem (The Flip): $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$
  • Tip: Denominator is always the Total Probability.

Week 8: Random Variables (Discrete)

• PMF (Probability Mass Function): $P(X=x)$. Sum = 1.
• CDF (Cumulative Distribution Function): $F(x)=P(X\le x)$. Step function.
• Expectation ($E[X]$): Mean $\mu$. $E[X]=\sum x\cdot P(X=x)$
• Variance ($Var(X)$): $Var(X)=E[X^2]-(E[X]{)}^2$ $E[X^2]=\sum x^2\cdot P(X=x)$
• Linear Rules:
  • $E[aX+b]=aE[X]+b$
  • $Var(aX+b)=a^{2}Var(X)$

---

📉 Part 3: Standard Distributions (Weeks 9-12)

Week 9: Discrete Distributions

1. Bernoulli($p$): Single trial (Success/Fail).
   • Mean = $p$, Var = $p(1-p)$.
2. Binomial($n,p$): Number of successes in $n$ trials.
   • P(X=k) = \,^nC_k \, p^k (1-p)^{n-k}.
   • Mean = $np$, Var = $np(1-p)$.
3. Geometric($p$): Trials until first success.
   • $P(X=k)=(1-p{)}^{k-1}p$.
   • Mean = $1/p$.
4. Poisson($\lambda$): Rare events in interval.
   • $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$.
   • Mean = $\lambda$, Var = $\lambda$.

Week 10: Continuous Random Variables

• PDF (Probability Density Function): $f(x)$. Area = 1.
  • $P(a<X<b)={\int }_a^{b}f(x)dx$.
  • Note: $P(X=c)=0$ for continuous variables.
• CDF: $F(x)={\int }_{-\infty }^{x}f(t)dt$.
  • To get PDF from CDF: $f(x)=F^{\prime }(x)$.

Week 11: Standard Continuous Distributions

1. Uniform($a,b$): Flat PDF.
   • $f(x)=\frac{1}{b-a}$ for $a\le x\le b$.
   • Mean = $\frac{a+b}{2}$, Var = $\frac{(b-a{)}^2}{12}$.
2. Exponential($\lambda$): Time until event. (Memoryless).
   • $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$.
   • CDF $F(x)=1-e^{-\lambda x}$.
   • Mean = $1/\lambda$.
3. Normal($\mu ,{\sigma }^2$): Bell curve.
   • Standard Normal ($Z$): $\mu =0,\sigma =1$.
   • Z-Score: $Z=\frac{X-\mu }{\sigma }$.
   • Symmetry: $P(Z<-x)=P(Z>x)=1-P(Z<x)$.

Week 12: Joint Distributions & CLT

• Joint PMF: $P(X=x,Y=y)$.
• Marginal PMF: Sum rows/cols. $P(X=x)={\sum }_{y}P(X=x,Y=y)$.
• Independence: $P(X,Y)=P(X)P(Y)$.
• Expectation of Sum: $E[X+Y]=E[X]+E[Y]$ (Always true).
• Variance of Sum: $Var(X+Y)=Var(X)+Var(Y)$ (Only if Independent).
  • General: $Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$.
• Central Limit Theorem (CLT): For large $n$ ($n\ge 30$), the sum (or mean) of i.i.d. random variables follows a Normal Distribution.
  • Sum $S_n\sim N(n\mu ,n{\sigma }^2)$.
  • Mean $\bar{X}\sim N(\mu ,\frac{{\sigma }^2}{n})$.

---

⚡ The “Do Not Forget” List

1. Variance is Squared: Standard deviation is $\sigma$, Variance is ${\sigma }^2$. Don’t mix them up.
2. Probability bounds: $0\le P\le 1$. If you get 1.2, you are wrong.
3. Variance Properties: $Var(Constant)=0$. $Var(2X)=4Var(X)$.
4. Independent Intersection: Intersection implies Multiply ($P(A)\times P(B)$) ONLY if independent.
5. At Least One: Always calculate as $1-P(\text{None})$.

This handbook covers every formula needed for the Stats 1 exam. Keep your calculator ready.

Here is the Ultimate Theory & Formula Handbook for Computational Thinking (CT).

CT is different from Math/Stats. It’s not about memorizing formulas; it’s about tracing logic. This guide gives you the “Mental Models” to run code in your head without crashing.

---

💻 Part 1: The Foundation (Weeks 1-4)

Focus: Flowcharts, Pseudocode, Operators, Datasets.

1. Standard Datasets (Know these cold)

• Scores: Student info.
  • Fields: SeqNo, Name, Gender, City, Marks (Math, Physics, Chem).
  • Pattern: Calculating Averages, Filtering by City, Finding Toppers.
• Words: Text analysis.
  • Fields: Word, Part of Speech (Noun/Verb), Letter Count.
  • Pattern: Counting vowels, finding palindromes, sentence structure.
• Shopping Bills: Transactions.
  • Fields: Shop, Customer, Item, Category, Price, Quantity.
  • Pattern: Total bill amount, filtering items by category.

2. Logical Operators (The Truth Tables)

• AND (A and B): True only if BOTH are True. (Strict).
  • Stop Condition: If A is False, B is ignored (Short-circuit).
• OR (A or B): True if AT LEAST ONE is True. (Generous).
  • Stop Condition: If A is True, B is ignored.
• NOT (not A): Flips the boolean.

3. Iteration (Loops)

• while(Table 1 has more rows): The standard engine.
  • Step 1: Read row X.
  • Step 2: Process X (Filter/Count/Sum).
  • Step 3: Move X to Table 2 (or discard).
• Initialization Trap:
  • Count = 0 Outside loop $\to$ Global count.
  • Count = 0 Inside loop $\to$ Resets for every card (useless for global counting).

---

📦 Part 2: Data Structures (Weeks 5-8)

Focus: Lists, Dictionaries, Nested Loops.

1. Lists (L = [])

• Concept: Ordered collection of items.
• Operations:
  • L = L ++ [x] (Append x to end).
  • L = [x] ++ L (Prepend x to start).
  • Length(L): Number of items.
  • L[i]: Access item at index $i$.

2. Dictionaries (D = {})

• Concept: Key-Value pairs. Keys must be unique.
• The “Histogram” Pattern:
  • Goal: Count how many students are in each City.
  • Code:

    if (isKey(D, X.City)):
        D[X.City] = D[X.City] + 1
    else:
        D[X.City] = 1

• The “Group By” Pattern:
  • Goal: List all students in each City.
  • Code:

    if (not isKey(D, X.City)):
        D[X.City] = []
    D[X.City] = D[X.City] ++ [X.Name]

3. Nested Iteration (The “Pairs” Trap)

• Concept: Comparing every card with every other card.
• Complexity: $O(N^2)$.
• Structure:

  foreach x in List:
      foreach y in List:
          if (x != y and condition):
              Count++

• Double Counting Warning: If you compare (A, B) and (B, A) separately, your count will be 2x the actual number of pairs. Divide by 2 at the end if order doesn’t matter.

---

🌳 Part 3: Algorithms & Graphs (Weeks 9-12)

Focus: Recursion, Searching, Graphs, Trees.

1. Recursion (Function calling itself)

• Base Case: The condition where recursion stops (e.g., if list is empty, return 0). Crucial. Without this, infinite loop.
• Recursive Step: Reducing the problem size (e.g., return 1 + Count(Rest of List)).
• Trace Strategy:
  • Don’t try to hold it in your head. Write down the stack:
  • Func([1,2,3]) calls 1 + Func([2,3])
  • Func([2,3]) calls 1 + Func([3]) …

2. Binary Search

• Pre-requisite: List must be SORTED.
• Logic: Check middle. If target < middle, discard right half. If target > middle, discard left half.
• Complexity: $O({\log }_{2}N)$. (Very fast).
• Max Iterations: For $N=1000$, max steps $\approx 10$.

3. Graphs (G = V, E)

• Adjacency Matrix:
  • Square matrix. A[i][j] = 1 if edge exists.
  • Degree: Sum of row i = Out-degree. Sum of col j = In-degree.
  • Symmetric: If undirected, matrix is symmetric across diagonal.
• Paths & Cycles:
  • Path: Sequence of non-repeating vertices connected by edges.
  • Cycle: Path starting and ending at same vertex.
• Handshaking Lemma (CT version):
  • Sum of degrees = $2\times$ Number of Edges.

4. Trees

• Root: Top node.
• Leaf: Node with no children.
• Binary Tree: Max 2 children per node.
• Binary Search Tree (BST) Property:
  • Left Child < Parent.
  • Right Child > Parent.
  • In-order Traversal of a BST gives sorted output.

---

🛠️ The “Cheat Codes” for Exam Solving

1. The “Init” Check

• Look at where variables are initialized.
• If Max = 0 is inside the loop, Max will never actually find the maximum of the whole list (it resets). This is a common bug in “Debug the code” questions.

2. De Morgan’s Law (Logic Simplified)

• Sometimes questions give complex NOT conditions. Simplify them:
  • not (A and B) $\to$ (not A) or (not B)
  • not (A or B) $\to$ (not A) and (not B)
  • Example: “Not (Tall and Smart)” means “Short OR Dumb”.

3. The “Trace Table” Technique

• Never dry run in your head. Draw a table:

Iteration	Variable X	Variable Y	Condition (T/F)	Action
1	5	0	True	Sum = 5
2	3	5	True	Sum = 8

4. Graph Distance (BFS Logic)

• If asked for “shortest path” or “degrees of separation”:
  • Start Node = Level 0.
  • Neighbors = Level 1.
  • Neighbors of Neighbors = Level 2.
  • Don’t count nodes you’ve already visited.

5. Scoring Strategy

• CT is precise. One missed = or one swapped > can change the answer.
• MSQ (Multiple Select): In CT, usually, logical equivalents are correct.
  • e.g., if (A and B) is logically same as if A { if B { ... } }. Both options would be correct.

You are ready. CT is about process. Trust the trace table.

Here is the Ultimate Theory & Formula Handbook for English 1.

English 1 in this program is technical. It is not just about “knowing English”; it is about knowing Linguistics and Grammar Rules. You cannot guess the Phonetics questions; you have to apply the rules below.

---

🗣️ Part 1: Phonetics & Sounds (The “Math” of English)

This is where 90% of students lose marks. Memorize these rules.

1. Plural Markers (/s/, /z/, /iz/)

• Rule 1: /iz/ (The Hissing Sounds)
  • If the word ends in a Sibilant sound: s, z, sh ($\int$), ch ($t\int$), j ($d\text{ʒ}$), zh ($\text{ʒ}$).
  • Examples: Bus$\to$Buses, Judge$\to$Judges, Dish$\to$Dishes.
• Rule 2: /s/ (The Quiet Sounds)
  • If the word ends in a Voiceless non-sibilant: p, t, k, f, th ($\theta$).
  • Examples: Cat$\to$Cats, Lip$\to$Lips, Cliff$\to$Cliffs.
• Rule 3: /z/ (The Noisy Sounds)
  • If the word ends in a Voiced sound (b, d, g, v, th ($ð$), l, m, n, r, or ANY Vowel).
  • Examples: Dog$\to$Dogz, Boy$\to$Boyz, Car$\to$Carz.

2. Word Stress Rules

• Noun vs. Verb Rule:
  • Noun/Adjective: Stress the 1st syllable. (OB-ject, PRE-sent).
  • Verb: Stress the 2nd syllable. (ob-JECT, pre-SENT).
• Suffix Rule 1 (-ic, -tion, -sion):
  • Stress the syllable immediately before the suffix.
  • Eco-NOM-ic, Gra-DA-tion.
• Suffix Rule 2 (-cy, -ty, -phy, -gy, -al):
  • Stress the 3rd syllable from the end (Antepenultimate).
  • De-MOC-ra-cy, Ge-OL-o-gy, Pho-TOG-ra-phy.

3. Aspiration (The “Puh” Sound)

• The sounds /p/, /t/, /k/ are Aspirated (explode with air) IF:
  1. They are at the start of a syllable.
  2. That syllable is stressed.
  • Examples: Pot, Top, Kit.
• Exception: If they follow ‘s’, they are NOT aspirated.
  • Examples: Spot, Stop, Skit.

4. Vowel Sounds

• Monophthong: Single vowel sound (e.g., “Sit” /I/).
• Diphthong: Two vowel sounds gliding together.
  • Examples: “Buy” (/aI/), “Cow” (/aU/), “Go” (/oU/).
• Schwa (/ə/): The weak, unstressed “uh” sound.
  • Examples: About, Teacher. (Most common sound in English).

---

✍️ Part 2: Grammar Mechanics (Weeks 1-12)

1. Subject-Verb Agreement

• The “Neither/Nor” Rule: The verb agrees with the closest subject.
  • Neither the teacher nor the students are here.
• The “Uncountable” Rule: Water, Information, Furniture, Advice $\to$ Always Singular.
• The “Percentage” Rule:
  • “The percentage of…” $\to$ Singular. (The percentage of water is low).
  • “X percent of [Plural Noun]” $\to$ Plural. (50% of the students are passed).

2. Advanced Tenses

• Present Perfect: Action happened in past, relevant now. (I have eaten).
• Past Perfect: Action happened before another past action. (I had eaten before he arrived).
• Future Perfect Progressive: Action continuing up to a point in future.
  • Structure: Will have been + Verb-ing.
  • Example: “By 2026, I will have been studying here for 4 years.”

3. Modals (Politeness & Probability)

• Permission:
  • Can: Casual. (Can I go?)
  • May: Formal. (May I go?) → Preferred in Exam.
• Probability:
  • Might: Weak possibility. (It might rain).
  • May: Stronger possibility.
  • Must: Deduction/Certainty. (He is late; he must be stuck in traffic).
• Advice vs Obligation:
  • Should: Advice.
  • Must/Have to: Obligation/Rule.

4. Conditionals

• Zero: Fact. (If you heat ice, it melts.)
• First: Real possibility. (If it rains, I will stay home.)
• Second: Hypothetical/Unreal. (If I were you, I would go.) → Subjunctive “Were”.
• Third: Past Regret. (If I had known, I would have come.)

---

📝 Part 3: Sentence Structure & Punctuation

1. Sentence Types

• Simple: One independent clause. (The dog ran.)
• Compound: Two independent clauses joined by FANBOYS (For, And, Nor, But, Or, Yet, So).
  • The dog ran*, and** the cat slept.*
• Complex: Independent + Dependent clause (starts with Because, Although, If, When).
  • Although the dog ran, the cat slept.

2. Punctuation Rules

• The Oxford Comma: Used before ‘and’ in a list. (Apples, oranges*, and** bananas*). IITM prefers this.
• Restrictive vs Non-Restrictive Clauses:
  • Restrictive (No Commas): Defines the noun. Essential info.
    • The car which was parked there is mine. (Specific car).
  • Non-Restrictive (Commas): Extra info.
    • My car, which is red, is parked there. (I only have one car, color is extra info).
• Apostrophes:
  • Possession: The boy’s bag (Singular). The boys’ bags (Plural ending in s). The children’s toys (Plural not ending in s).
  • Contraction: It’s = It is. Its = Possession.

---

📖 Part 4: Reading Comprehension Strategy

1. Matching Questions (Adjective-Noun)

• These are “Collocations” (words that naturally go together).
• Strategy: Don’t guess. Scan the text. The text will explicitly say “stale fumes” or “bronchial rattles”. Match exactly what is written.

2. Tone & Inference

• Tone Words:
  • Melancholic/Somber: Sad, gloomy.
  • Nostalgic: Longing for the past.
  • Critical: Judging, finding faults.
  • Optimistic: Hopeful.
• Strategy: Look for adjectives. If the author uses words like “miserable”, “woes”, “poverty” $\to$ Tone is Misery/Somber.

3. True/False Logic

• If the text implies something but doesn’t say it, be careful.
• Example: Text says “Limerick gained a reputation for piety, but we knew it was only the rain.”
  • Question: Is Limerick truly pious?
  • Answer: False. (The “but” negates the reputation).

---

🚀 Final Exam “Cheat Codes”

1. Question Tags:
   • Positive Sentence $\to$ Negative Tag. (He is here, isn’t he?)
   • Negative Sentence $\to$ Positive Tag. (He isn’t here, is he?)
   • Past Tense: (He went there, didn’t he?)
2. Embedded Questions:
   • Wrong: Do you know where is the station?
   • Right: Do you know where the station is? (Subject before Verb).
3. Spelling:
   • Receive (i before e except after c).
   • Accommodation (double c, double m).

You are now fully armed for English 1. Don’t let the “easy” look fool you; stick to the Phonetic and Grammar rules. Good luck.