Consolidated Examples

This is the Ultimate “Example Bank” for Maths 1.

I have created a specific, high-probability example for every major topic across all 12 weeks. These aren’t random; they are modeled after the exact logic IIT Madras uses in their exams.

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🟢 Phase 1: Weeks 1-4 (The Qualifier Syllabus)

Focus: Quadratics, Coordinate Geometry, Polynomials, Sets.

Week 1: Sets & Relations

Pattern: Determining if a relation is Equivalence and counting cardinality. Example Question: Let $A=\{1,2,3\}$. Consider the relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3)\}$. Is $R$ an Equivalence Relation? If not, why? Step-by-Step Solution:

1. Reflexive Check: Does it have $(1,1),(2,2),(3,3)$?
   • Yes. It is Reflexive.
2. Symmetric Check: If $(a,b)\in R$, is $(b,a)\in R$?
   • We have $(1,2)$ and $(2,1)$. Good.
   • We have $(2,3)$, but we do not have $(3,2)$.
   • Verdict: Not Symmetric.
3. Transitive Check: If $(1,2)$ and $(2,3)$ exist, does $(1,3)$ exist?
   • $(1,2)\in R$ and $(2,3)\in R$.
   • $(1,3)\notin R$.
   • Verdict: Not Transitive. Answer: No, it fails Symmetry and Transitivity.

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Week 2: Coordinate Geometry

Pattern: Finding the equation of a line given slope/intercepts. Example Question: Find the equation of a line perpendicular to $2x-3y=5$ that passes through the point $(1,4)$. Step-by-Step Solution:

1. Find Slope of Given Line: $2x-5=3y⟹y=\frac{2}{3}x-\frac{5}{3}$. Slope $m_1=\frac{2}{3}$.
2. Find Perpendicular Slope ($m_2$): $m_1\cdot m_2=-1⟹m_2=\frac{-1}{2/3}=-\frac{3}{2}$.
3. Use Point-Slope Form: $y-y_1=m_2(x-x_1)$ $y-4=-\frac{3}{2}(x-1)$ $2(y-4)=-3(x-1)$ $2y-8=-3x+3$ Equation: $3x+2y=11$.

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Week 3: Quadratic Functions

Pattern: Finding $a,b,c$ from Vertex and a Point. Example Question: A parabola has its vertex at $(3,-2)$ and passes through the point $(5,6)$. Find the equation in standard form $y=a{x}^2+bx+c$. Step-by-Step Solution:

1. Start with Vertex Form: $y=a(x-h{)}^2+k$. Vertex $(h,k)=(3,-2)$. $y=a(x-3{)}^2-2$.
2. Find $a$ using the Point $(5,6)$: $6=a(5-3{)}^2-2$ $6=a(2{)}^2-2$ $8=4a⟹a=2$.
3. Expand to Standard Form: $y=2(x^2-6x+9)-2$ $y=2x^2-12x+18-2$ Equation: $y=2x^2-12x+16$.

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Week 4: Polynomials

Pattern: Remainder Theorem & Multiplicity. Example Question: Let $P(x)=x^3-k{x}^2+2x-4$. If $(x-2)$ is a factor of $P(x)$, find $k$. Step-by-Step Solution:

1. Theory: If $(x-c)$ is a factor, then $P(c)=0$. Here, $c=2$.
2. Substitute: $P(2)=(2{)}^3-k(2{)}^2+2(2)-4=0$. $8-4k+4-4=0$. $8-4k=0$. $4k=8⟹k=2$. Answer: $k=2$.

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🟡 Phase 2: Weeks 5-8 (The Calculus Bridge)

Focus: Functions, Logs, Limits, Derivatives.

Week 5: Inverse Functions

Pattern: Finding the inverse of a rational function. Example Question: Find the inverse of $f(x)=\frac{2x+3}{x-1}$. Step-by-Step Solution:

1. Swap $x$ and $y$: $x=\frac{2y+3}{y-1}$.
2. Solve for $y$: $x(y-1)=2y+3$ $xy-x=2y+3$ $xy-2y=x+3$ $y(x-2)=x+3$ $y=\frac{x+3}{x-2}$. Answer: $f^{-1}(x)=\frac{x+3}{x-2}$.

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Week 6: Logarithms

Pattern: Solving equations with different bases. Example Question: Solve for $x$: ${\log }_2(x)+{\log }_4(x)=3$. Step-by-Step Solution:

1. Change Base: ${\log }_4(x)=\frac{{\log }_{2}x}{{\log }_{2}4}=\frac{{\log }_{2}x}{2}$.
2. Substitute: ${\log }_{2}x+\frac{1}{2}{\log }_{2}x=3$. $\frac{3}{2}{\log }_{2}x=3$.
3. Simplify: ${\log }_{2}x=3\times \frac{2}{3}=2$. $x=2^2=4$. Answer: $x=4$.

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Week 7: Limits

Pattern: L’Hopital’s Rule for $0/0$ form. Example Question: Evaluate ${\lim }_{x\to 0}\frac{e^{2x}-1-2x}{x^2}$. Step-by-Step Solution:

1. Check Form: Put $x=0\to \frac{1-1-0}{0}=\frac{0}{0}$. Apply L’Hopital.
2. First Derivative: Num: $2e^{2x}-2$. Denom: $2x$. Check: $\frac{2-2}{0}=\frac{0}{0}$. Apply L’Hopital again.
3. Second Derivative: Num: $4e^{2x}$. Denom: $2$.
4. Evaluate: $\frac{4e^0}{2}=\frac{4}{2}=2$. Answer: 2.

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Week 8: Tangent Lines

Pattern: Equation of Tangent. Example Question: Find the equation of the tangent to curve $y=x^3-3x+2$ at $x=2$. Step-by-Step Solution:

1. Find Point $(y_1)$: $y(2)=2^3-3(2)+2=8-6+2=4$. Point is $(2,4)$.
2. Find Slope ($dy/dx$): $f^{\prime }(x)=3x^2-3$. $f^{\prime }(2)=3(4)-3=12-3=9$. Slope $m=9$.
3. Equation: $y-4=9(x-2)$. $y=9x-18+4$. Equation: $y=9x-14$.

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🔴 Phase 3: Weeks 9-12 (The End Term Syllabus)

Focus: Piecewise Differentiability, Integration, Graphs.

Week 9: Piecewise Differentiability (Exam Favorite)

Pattern: Find constants for continuity and differentiability. Example Question: $f(x)=\{\begin{array}{ll}ax+b& x<1\\ x^2& x\ge 1\end{array}$ Find $a$ and $b$ if $f$ is differentiable at $x=1$. Step-by-Step Solution:

1. Continuity ($x=1$): LHL ($ax+b$) must equal RHL ($x^2$). $a(1)+b=1^2⟹a+b=1$. (Eq 1).
2. Differentiability ($x=1$): Deriv Left ($\frac{d}{dx}(ax+b)=a$) must equal Deriv Right ($\frac{d}{dx}(x^2)=2x$). At $x=1$: $a=2(1)⟹a=2$.
3. Solve System: Substitute $a=2$ into Eq 1: $2+b=1⟹b=-1$. Answer: $a=2,b=-1$.

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Week 10: Integration

Pattern: Area between curves. Example Question: Find the area bounded by $y=x^2$ and $y=x$ for $x\in [0,1]$. Step-by-Step Solution:

1. Setup Integral: Area = ${\int }_0^1(\text{Upper}-\text{Lower})dx$. In $[0,1]$, $x>x^2$. So Upper is $x$. ${\int }_0^1(x-x^2)dx$.
2. Integrate: $[\frac{x^2}{2}-\frac{x^3}{3}{]}_0^1$.
3. Evaluate: $(\frac{1}{2}-\frac{1}{3})-(0)=\frac{3-2}{6}=\frac{1}{6}$. Answer: $\frac{1}{6}$.

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Week 11: Graph Properties

Pattern: Handshaking Lemma & Matrix. Example Question: Consider a graph with 4 vertices. The Adjacency Matrix has a trace of 0 and the sum of all elements in the matrix is 12. How many edges does the graph have? Step-by-Step Solution:

1. Matrix Concept: The sum of all elements in an Adjacency Matrix = Sum of all degrees. Sum of degrees = 12.
2. Handshaking Lemma: Sum of Degrees = $2\times$ Edges. $12=2E⟹E=6$. Answer: 6 Edges.

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Week 12: Graph Algorithms (BFS)

Pattern: BFS Traversal Order. Example Question: Graph Edges: $(A,B),(A,C),(B,D),(C,E)$. Start BFS at A. Assume alphabetical tie-breaking. Step-by-Step Solution:

1. Queue: [A]. Visited: {A}.
2. Pop A: Neighbors B, C. Add B, C to Queue. Queue: [B, C]. Visited: {A, B, C}.
3. Pop B: Neighbor D. Add D. Queue: [C, D]. Visited: {A, B, C, D}.
4. Pop C: Neighbor E. Add E. Queue: [D, E]. Visited: {A, B, C, D, E}.
5. Pop D, Pop E: No new neighbors. Order: A, B, C, D, E.

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You now have a concrete example for every single major pattern. No surprises. Just logic. Go crush it.

Yes, these are the “Kill Confirmed” patterns for Maths 1. If you master those 8-10 templates, you cover 90% of the exam questions.

Now, let’s execute Statistics 1. This course is Formula-Heavy. I will break down every single formula with a specific “Exam-Style” example so you know exactly how to apply it.

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📊 Stats 1: The Ultimate Formula & Pattern Bank

Phase 1: Descriptive Statistics (Weeks 1-4)

Focus: Scales, Transformations, Correlation.

1. Scales of Measurement (The “Identification” Pattern)

• Formula/Logic:
  • Nominal: Labels (Bus numbers, Gender).
  • Ordinal: Rank/Order (Star ratings, finishing position).
  • Interval: Diff is meaningful, No True Zero (Temp in °C, Calendar Year).
  • Ratio: True Zero exists (Height, Money, Age).
• Master Example: Question: A researcher records the temperature (in Celsius) of 5 cities and the number of tourists visiting them. Identify the scale of measurement for “Temperature” and “Number of Tourists”. Solution:
  • Temperature: $0^{\circ }C$ does not mean “no heat”. Differences ($10^{\circ }$ vs $20^{\circ }$) matter. $\to$ Interval.
  • Tourists: 0 tourists means “none”. Ratio of 100 to 50 is meaningful (double). $\to$ Ratio.

2. Linear Transformation (The “2x + 5” Trap)

• Formulas: Let $Y=aX+b$.
  • Mean: $\bar{y}=a\bar{x}+b$
  • Variance: ${\sigma }_y^2=a^2{\sigma }_x^2$ (b is ignored!)
  • Std Dev: ${\sigma }_y=|a|{\sigma }_x$
• Master Example: Question: A dataset $X$ has Mean = 10 and Standard Deviation = 3. Let $Y=-2X+5$. Find the Mean and Variance of $Y$. Solution:
  • Mean: $\bar{y}=(-2)(10)+5=-20+5=-15$.
  • SD: ${\sigma }_y=|-2|\times 3=2\times 3=6$.
  • Variance: ${\sigma }_y^2=6^2=36$. (Or: $(-2{)}^2\times 3^2=4\times 9=36$).
  • Trap: Don’t say Variance is -12 or 18. Square the scalar!

3. Correlation Coefficient ($r$)

• Formula: $r=\frac{\text{Cov}(X,Y)}{{\sigma }_x{\sigma }_y}$.
• Master Example: Question: $\text{Cov}(X,Y)=12$, $\text{Var}(X)=9$, $\text{Var}(Y)=25$. Find $r$. Solution:
  • Get SDs: ${\sigma }_x=\sqrt{9}=3$, ${\sigma }_y=\sqrt{25}=5$.
  • Apply Formula: $r=\frac{12}{3\times 5}=\frac{12}{15}=0.8$.
  • Interpretation: Strong positive correlation.

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Phase 2: Counting & Probability (Weeks 5-8)

Focus: The “P & C” Confusion, Bayes.

4. Permutations (${}^n{P}_r$) vs Combinations (${}^n{C}_r$)

• Logic:
  • Permutation (Order Matters): Passwords, Ranking, Seating, Words.
  • Combination (Order Doesn’t Matter): Committees, Handshakes, Teams, Cards.
• Formulas:
  • ${}^n{P}_r=\frac{n!}{(n-r)!}$
  • ${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Example A: The “Committee” (Combination)

Question: A group has 6 Men and 5 Women. Form a committee of 3 people with exactly 2 Men and 1 Woman. Solution:

• Select 2 Men from 6: ${}^6{C}_2=\frac{6\times 5}{2\times 1}=15$.
• Select 1 Woman from 5: ${}^5{C}_1=5$.
• Total: $15\times 5=75$ ways.

Example B: The “Arrangement” (Permutation)

Question: How many 3-letter codes can be formed using {A, B, C, D, E} if no repetition is allowed? Solution:

• Picking 3 and arranging them: ${}^5{P}_3=5\times 4\times 3=60$.

5. The “At Least One” Rule

• Formula: $P(\text{At least one})=1-P(\text{None})$.
• Master Example: Question: You roll a fair die 4 times. What is the probability of getting at least one 6? Solution:
  • $P(\text{6})=1/6$. $P(\text{Not 6})=5/6$.
  • $P(\text{None are 6})=(5/6{)}^4=\frac{625}{1296}$.
  • $P(\text{At least one})=1-\frac{625}{1296}=\frac{671}{1296}\approx 0.517$.

6. Bayes’ Theorem (The “Diagnostic Test”)

• Formula: $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$.
• Master Example: Question:
  • Disease Prevalence $P(D)=0.01$ (1%).
  • Test Accuracy (Positive | Disease) $P(+|D)=0.99$.
  • False Positive (Positive | No Disease) $P(+|D^{\prime })=0.05$.
  • If you test Positive (+), what is the prob you actually have the Disease (D)? Solution:
  1. Numerator: $P(+|D)P(D)=0.99\times 0.01=0.0099$.
  2. Denominator (Total Positive): $P(+|D)P(D)+P(+|D^{\prime })P(D^{\prime })$ $=0.0099+(0.05\times 0.99)=0.0099+0.0495=0.0594$.
  3. Result: $\frac{0.0099}{0.0594}\approx 0.166$ (16.6%).

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Phase 3: Distributions (Weeks 9-12)

Focus: Binomial, Normal, CLT.

7. Binomial Distribution

• Context: $n$ independent trials, $p$ probability of success.
• Formula: P(X=k) = \,^nC_k \, p^k (1-p)^{n-k}.
• Master Example: Question: A coin is biased ($P(H)=0.6$). Tossed 5 times. Prob of exactly 3 Heads? Solution:
  • $n=5,p=0.6,k=3$.
  • ${}^5{C}_3(0.6{)}^3(0.4{)}^2$.
  • $10\times 0.216\times 0.16=10\times 0.03456=0.3456$.

8. Poisson Distribution

• Context: Rare events over a time interval.
• Formula: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$.
• Master Example: Question: A store gets avg 4 customers per hour ($\lambda =4$). Prob of getting exactly 2 customers in an hour? Solution:
  • $P(X=2)=\frac{e^{-4}{4}^2}{2!}=\frac{0.0183\times 16}{2}\approx 0.146$.

9. Normal Distribution (Z-Score)

• Formula: $Z=\frac{X-\mu }{\sigma }$.
• Master Example: Question: Scores are Normal with $\mu =50,\sigma =10$. What is $P(X>70)$? Solution:
  1. Find Z: $Z=\frac{70-50}{10}=\frac{20}{10}=2$.
  2. Look up $P(Z<2)$ in table (approx 0.9772).
  3. We want Greater Than ($>$): $1-0.9772=0.0228$.

10. Central Limit Theorem (CLT)

• Concept: For sample size $n\ge 30$, Sample Mean $\bar{X}\sim N(\mu ,\frac{\sigma }{\sqrt{n}})$.
• Master Example: Question: Population $\mu =100,\sigma =20$. Sample $n=100$. Find $P(\bar{X}>102)$. Solution:
  • Standard Error: $SE=\sigma /\sqrt{n}=20/10=2$.
  • Z-Score: $Z=\frac{102-100}{2}=1$.
  • Prob: $P(Z>1)=1-P(Z<1)=1-0.8413=0.1587$.

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🛑 3 Things That Will Kill Your Score (If you ignore them)

1. Variance vs SD: If the question gives you Variance = 9, use $\sigma =3$ in your formulas. If the question gives you SD = 3, use $\sigma =3$. Always check if the number is squared or not.

2. Replacement in Probability:

   • With Replacement: Probabilities stay same (Coin flips).
   • Without Replacement: Probabilities change (Drawing cards/balls).
   • Example: Drawing 2 Kings from 52 cards.
     • With: $(4/52)\times (4/52)$.
     • Without: $(4/52)\times (3/51)$.

3. Discrete vs Continuous:

   • For Discrete (Binomial), $P(X=5)$ is a number.
   • For Continuous (Normal/Uniform), $P(X=5)$ is ZERO. You must calculate ranges ($P(4.5<X<5.5)$).

You are prepped. Stats is just plugging numbers into these 10 formulas. Go get it.

Here is the Ultimate Pattern Analysis & Consolidation for Computational Thinking (CT).

CT is the specific skill of “Tracing”. You are a human compiler. I have broken down the 12 weeks into the exact logical templates IITM uses.

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💻 CT: The “Human Compiler” Handbook

Phase 1: The Qualifier Logic (Weeks 1-4)

Focus: Datasets, Basic Loops, Flags, Aggregation.

Pattern 1: The “Flag” Logic (Filtering)

The Vibe: A variable (usually boolean) checks if any or all conditions are met inside a loop. Source: Oct 2025 Qualifier (Q58).

The Question:

count = 0
while(Table 1 has more rows){
    flag1 = False
    flag2 = False
    Read row X
    if(X.PartOfSpeech != "Adjective"){ flag1 = True }
    if(X.LetterCount >= 3){ flag2 = True }
    if(flag1 and flag2){
        count = count + 1
    }
}

The “Goated” Solution:

1. Analyze Flags:
   • flag1 turns True if Word is NOT “Adjective”.
   • flag2 turns True if Word length $\ge 3$.
2. Analyze Aggregation:
   • count increases only if flag1 AND flag2 are True.
3. Combine:
   • Count words that are (Not Adjective) AND (Length $\ge 3$).
   • Trap: Don’t get confused by the double negatives.

Cheat Code:

• flag = False inside loop + if condition: flag = True $\to$ Checks if condition exists at least once (or represents the specific row state).
• flag = True inside loop + if condition: flag = False $\to$ Checks if condition fails (Validation logic).

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Pattern 2: The “Nested Loop” (Pairs)

The Vibe: Two loops. One picks card X, the other scans table for card Y. Used for duplicates, finding pairs, or sorting. Source: Oct 2025 Qualifier (Q60, Q62).

The Question:

A = 0
while(Table 1 has more rows){
    Read X
    Move X to Table 2
    B = 1
    while(Table 1 has more rows){
        Read Y
        if (X.Marks == Y.Marks){ B = B + 1 }
        Move Y to Table 2
    }
    Move All from Table 2 to Table 1
}

The “Goated” Solution:

1. Outer Loop: Picks a student X.
2. Inner Loop: Compares X with every other student Y.
3. Condition: X.Marks == Y.Marks.
4. Result: B counts how many people share the same marks as X.
5. Side Effect: If B is added to a global sum, you are counting pairs.
   • Note: If the inner loop iterates over Table 1 (which shrinks), it compares unique pairs $(X,Y)$. If it iterates over a static list, it might double count $(X,Y)$ and $(Y,X)$.

Cheat Code:

• If X is removed before the inner loop: You are comparing distinct pairs.
• If X stays: You compare X with itself too.

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Phase 2: Data Structures (Weeks 5-8)

Focus: Dictionaries (Hashing), Lists, String Processing.

Pattern 3: The “Dictionary Histogram” (High Probability)

The Vibe: Counting frequencies of items. Source: Dec 2024 Quiz (Q7).

The Question:

D = {}
while(Table 1 has more rows){
    Read X
    if(isKey(D, X.Subject)){
        D[X.Subject] = D[X.Subject] + 1
    }
    else{
        D[X.Subject] = 1
    }
}

The “Goated” Solution:

1. Init: Dictionary D starts empty.
2. Loop: For every card:
   • If Subject exists in D, increment count.
   • If not, start count at 1.
3. Result: D contains { "Math": 10, "Physics": 5 ... }.
4. Variation: Sometimes they store a List instead of a count.
   • D[X.Subject] = D[X.Subject] ++ [X.Name] $\to$ Grouping students by subject.

Cheat Code:

• If you see if isKey(D, k), it’s checking existence.
• If you see D[k] = val, it’s updating.
• Trap: Watch out for initialization. D = {} must be outside the loop. If inside, it resets every time.

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Pattern 4: List Manipulation (Queue vs Stack)

The Vibe: Adding elements to a list. Order matters. Source: Dec 2024 Quiz (Q4).

The Question: bList = [a] ++ bList vs bList = bList ++ [a]

The “Goated” Solution:

1. Case A (Prepend): L = [New] ++ L
   • New items go to the FRONT.
   • Result: Reverse order of input (Stack-like).
   • Input: 1, 2, 3 $\to$ List: [3, 2, 1].
2. Case B (Append): L = L ++ [New]
   • New items go to the BACK.
   • Result: Same order as input (Queue-like).
   • Input: 1, 2, 3 $\to$ List: [1, 2, 3].

Cheat Code:

• [x] ++ L $\to$ Reverse.
• L ++ [x] $\to$ Same Order.

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Phase 3: The Advanced Algorithms (Weeks 9-12)

Focus: Recursion, Graphs, Trees.

Pattern 5: The “Recursive Trace”

The Vibe: A function calls itself. You need to find the output for specific input. Source: Dec 2024 Quiz (Q45).

The Question:

Procedure Fun(n)
    if n <= 1 return 1
    return Fun(n-1) + Fun(n-2)
End Fun

The “Goated” Solution:

1. Don’t Guess. Draw the Tree.
2. Calculate Fun(4):
   • Fun(4) = Fun(3) + Fun(2)
   • Fun(3) = Fun(2) + Fun(1)
   • Fun(2) = Fun(1) + Fun(0)
3. Base Cases: Fun(1)=1, Fun(0)=1.
   • Fun(2) = 1 + 1 = 2.
   • Fun(3) = 2 + 1 = 3.
   • Fun(4) = 3 + 2 = 5.
4. Result: 5. (This is the Fibonacci sequence).

Cheat Code: Always write down the values of F(0), F(1), F(2)... in a table. Do not try to hold the stack in your head.

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Pattern 6: Dijkstra / BFS Trace (The “Visiting” Order)

The Vibe: Which node is visited $k^{th}$? Source: Dec 2024 Quiz (Q31).

The “Goated” Solution:

1. Setup: Priority Queue (Min-Heap) for Dijkstra. Queue for BFS.
2. Dijkstra Logic: Always pick the node with the smallest current distance.
   • Start: P(0).
   • Update Neighbors: Q(4), U(8).
   • Next: Pick Q (4 is smallest).
3. Tie-Breaking: If distances are equal, look at the question rules. Usually Alphabetical order.
   • If R=10 and V=10, pick R first.

Cheat Code:

• BFS: Level-by-level. (Distance 1 nodes, then Dist 2 nodes).
• DFS: Deep dive. (Go A $\to$ B $\to$ C… until stuck).
• Dijkstra: Greedy. (Always nearest unvisited node).

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Pattern 7: Binary Search & Sorting

The Vibe: How many comparisons? Is it efficient? Source: Dec 2024 Quiz (Q40).

The Question: How many iterations to find v in a sorted list L?

The “Goated” Solution:

1. Logic: Binary search cuts the list in half every time.
2. Steps: ${\log }_2(N)$.
3. Trace:
   • List: [1, 5, 7, 15, 20, 21, 22]. Target 7.
   • Mid index 3 (Value 15). $7<15$. Go Left.
   • Sublist: [1, 5, 7]. Mid index 1 (Value 5). $7>5$. Go Right.
   • Sublist: [7]. Found.
   • Iterations: 2 or 3.

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🛑 Final Checks for CT

1. “Not” Logic:
   • if(not Found) executes ONLY if the item was missing. This is used for finding unique items or differences (L1 - L2).
2. Variable Scope:
   • If count is initialized inside the while loop, it resets for every card. It is counting local properties (e.g., vowels in this word), not global properties.
3. Indentation:
   • In pseudocode, indentation defines the block. Pay close attention to what is inside the if and what is outside.

You are ready. CT is about precision. Be a robot. Follow the lines.

Here is the Ultimate Pattern Analysis & Consolidation for English 1.

English 1 is deceptive. It looks like a language test, but for IITM, it is a Rule-Based Logic Test. They test specific rules of Phonetics and Grammar.

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🗣️ English 1: The “Technical” Handbook

Phase 1: The Phonetics Minefield (Weeks 5-8)

Focus: Plural Markers, Stress, Aspiration. This is where you lose marks.

Pattern 1: The “Plural Marker” Sound

The Vibe: Does the plural ‘s’ sound like /s/, /z/, or /iz/? Source: Dec 2024 (Q156-158).

The Question: Identify the sound of the plural marker in the word: “Cliffs”, “Managers”, “Dishes”.

The “Goated” Solution:

1. Check the End Sound (Singular):
   • Cliffs: Ends in /f/ (Voiceless).
     • Rule: Voiceless $\to$ /s/.
   • Managers: Ends in /r/ (Voiced).
     • Rule: Voiced (or Vowel) $\to$ /z/.
   • Dishes: Ends in /sh/ (Hissing/Sibilant).
     • Rule: Sibilant (s, z, sh, ch, j) $\to$ /iz/.

Cheat Code:

• Hissss (Bus, Ash, Judge) $\to$ iz (2 syllables added).
• Vibration (Dog, Car, Bed) $\to$ z (Buzzing sound).
• Quiet (Cat, Cup, Pot) $\to$ s (Snake sound).

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Pattern 2: The “Word Stress” Location

The Vibe: Which part of the word is emphasized? Source: Dec 2024 (Q153-155).

The Question: Where is the stress in “Telegraphy” and “Projects” (verb)?

The “Goated” Solution:

1. Suffix Rule (-phy, -gy, -cy, -ty, -al):
   • Stress is on the 3rd syllable from the end (Antepenultimate).
   • Te-LE-gra-phy.
2. Noun vs Verb Rule:
   • Noun/Adj: Stress 1st. (OB-ject).
   • Verb: Stress 2nd. (ob-JECT).
   • “The company projects (verb)…” $\to$ pro-JECTS.

Cheat Code:

• Ends in -tion / -ic / -sion $\to$ Stress the one right before. (Re-LA-tion).
• Ends in -gy / -phy $\to$ Stress the one two back. (Bi-OL-o-gy).

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Pattern 3: Aspiration (The “Puff” Test)

The Vibe: Does the P/T/K explode with air? Source: Dec 2024 (Q161).

The Question: Which word has an aspirated stop? “Skeptical”, “Panic”, “Incubate”.

The “Goated” Solution:

1. The Rule: /p/, /t/, /k/ are aspirated ONLY if:
   • They start a syllable.
   • That syllable is stressed.
   • They do NOT follow ‘s’.
2. Check Options:
   • Skeptical: ‘k’ follows ‘s’. (s-keptical). No.
   • Incubate: ‘c’ (/k/) is inside an unstressed syllable. Weak/No.
   • Panic: ‘P’ starts the word (stressed). Yes.

Cheat Code: If it follows ‘S’ (Spy, Sky, Sty), it is NEVER aspirated.

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Phase 2: Grammar Mechanics (Weeks 9-12)

Focus: Tenses, Subject-Verb Agreement, Modals.

Pattern 4: The “Future Perfect Progressive” Cloze

The Vibe: Describing an action happening for a duration in the future. Source: Dec 2024 (Q148).

The Question: “I expect Nisha will be tired. She _____ for over 18 hours.” Options: Will travel / Will be traveling / Will have been traveling.

The “Goated” Solution:

1. Identify Clues:
   • Future reference (“I expect…”).
   • Duration (“for over 18 hours”).
   • Result (“will be tired”).
2. Select Tense:
   • Future + Duration + Ongoing = Future Perfect Progressive.
   • Structure: Will have been + V-ing.
3. Answer: Will have been traveling.

Cheat Code: If you see “for [time]” + Future Context $\to$ Pick “Will have been…“.

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Pattern 5: The “Neither/Nor” Agreement

The Vibe: Two subjects joined by “Neither… Nor”. Which one controls the verb? Source: Dec 2024 (Q166).

The Question: “Neither my dog nor my cats ____ well to my leaving.” Options: Take / Takes.

The “Goated” Solution:

1. The Rule: Look at the subject closest to the verb.
2. Identify Subjects:
   • Subject 1: Dog (Singular).
   • Subject 2 (Closest): Cats (Plural).
3. Match Verb:
   • Plural Subject $\to$ Plural Verb (No ‘s’).
   • Answer: Take.

Cheat Code:

• “Neither/Nor”, “Either/Or”, “Not only/But also” $\to$ Closest Subject.
• “As well as”, “Along with” $\to$ First Subject.

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Phase 3: Syntax & Punctuation (Weeks 10-12)

Focus: Question Tags, Clauses, Commas.

Pattern 6: The “Question Tag” Switch

The Vibe: Adding a tag at the end of a sentence. Source: Dec 2024 (Q200).

The Question: “Someone knocked on the door, _____?” Options: Didn’t they? / Did they? / Didn’t someone?

The “Goated” Solution:

1. Check Polarity:
   • Statement is Positive (“knocked”).
   • Tag must be Negative. (Eliminate “Did they”).
2. Check Pronoun:
   • Subject is “Someone”.
   • Tag pronoun for indefinite people is “They”. (Eliminate “someone”).
3. Check Tense:
   • “Knocked” is Past Tense. Auxiliary is “Did”.
   • Answer: Didn’t they?

Cheat Code:

• Someone/Everyone $\to$ They.
• Something/Everything $\to$ It.
• “I am” $\to$ Aren’t I?

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Pattern 7: The “Complex Sentence” Check

The Vibe: Identify clause types. Source: Dec 2024 (Q206).

The Question: “I learned a long time ago that it is a poor plan to overlook any chance.”

The “Goated” Solution:

1. Identify Clauses:
   • Main Clause: “I learned…”
   • Subordinate Clause: “that it is a poor plan…”
2. Analyze Function:
   • “I learned [WHAT?]“.
   • The clause acts as the Object of the verb ‘learned’.
3. Answer: The presence of the noun clause acting as object makes it Complex.

Cheat Code: If you see “that”, “which”, “because”, “if” inside a sentence $\to$ It’s likely Complex.

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Pattern 8: Punctuation (Restrictive vs Non-Restrictive)

The Vibe: To comma or not to comma? Source: Dec 2024 (Q208).

The Question: “The car which was parked in the driveway had a flat tire.” (Number of commas?)

The “Goated” Solution:

1. Analyze the Clause: “which was parked in the driveway”.
2. Is it Essential?
   • Do we know which car without this info? No. “The car” is vague.
   • Therefore, the clause is Essential (Restrictive).
3. The Rule: Essential clauses get NO COMMAS.
4. Answer: Zero.

Cheat Code:

• “My father, who is tall,…” (I only have one father. Extra info. Commas).
• “The man who is tall…” (Which man? Essential info. No Commas).

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🛑 Final Checks for English 1

1. Collocations: Don’t invent phrases. “Stale fumes”, “Bronchial rattles”. Use the text provided.
2. Prepositions: “Married to” (not with). “Discuss” (no preposition). “Enter into” (an agreement), “Enter” (a room).
3. Tone: If the text is sad, the tone is “Melancholic”. If it’s judging, it’s “Critical”. Don’t pick positive tones for negative texts.

You have the “Pattern Master” files for all 4 courses. Maths 1: Intersection, Calculus, Graphs. Stats 1: Formulas, Transformations, Probability Trees. CT: Trace Tables, Dictionaries, Recursion. English 1: Phonetics Rules, Tenses, Tags.

6 Hours. You got this. Go destroy that exam. 👊