Study_Strategy

Study Strategy & Pattern Analysis - IIT Madras BS DS Foundational Level

Overview

This document maps the syllabus topics to specific question patterns identified from previous years’ questions (PYQs) and assignments. It serves as a blueprint for creating “Goated” notes and mental models.

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Mathematics 1

Week 1: Sets, Relations, and Functions

Core Concepts:

• Number Systems (Rational vs Irrational)
• Set Operations (Union, Intersection, Difference)
• Relations (Reflexive, Symmetric, Transitive, Equivalence)
• Functions (Domain, Codomain, Injective, Surjective, Bijective)

Recurring Question Patterns:

1. Irrationality Check: Given an expression involving roots, determine if it’s rational or irrational.
   • Mental Model: Simplify terms first. $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$ (Rational).
2. Cardinality Calculation: Find $|A\cup B|$ or $|A\setminus B|$ given set definitions.
   • Mental Model: Use Venn diagrams. $|A\cup B|=|A|+|B|-|A\cap B|$.
3. Relation Properties: Given a set of pairs or a definition, check R/S/T properties.
   • Mental Model:
     • Reflexive: Is $(a,a)$ always present?
     • Symmetric: If $(a,b)$ exists, does $(b,a)$ exist?
     • Transitive: If $(a,b)$ and $(b,c)$ exist, does $(a,c)$ exist?
4. Function Type: Check if $f(x)$ is 1-1 or Onto.
   • Mental Model:
     • 1-1: $f(x)=f(y)⟹x=y$. (Horizontal Line Test)
     • Onto: Range = Codomain. (Is every $y$ reachable?)

Week 2: Coordinate Geometry & Straight Lines

Core Concepts:

• Slope, Equations of Lines
• Parallel/Perpendicular Lines
• Distance Formula, Section Formula
• Intersection of Lines

Recurring Question Patterns:

1. Intersection Point: Find where two lines meet.
   • Mental Model: Solve system of linear equations.
2. Geometry Problems: Area of triangle given vertices, or finding a 4th vertex of a parallelogram.
   • Mental Model: Parallelogram diagonals bisect each other. Midpoint of AC = Midpoint of BD.
3. Word Problems (Optimization): Shortest path connecting points to a line.
   • Mental Model: Reflection principle. Reflect point A across the line to A’, then connect A’ to B.

Week 3: Quadratic Functions

Core Concepts:

• Parabola ($y=a{x}^2+bx+c$)
• Vertex, Axis of Symmetry
• Roots (Quadratic Formula)
• Maxima/Minima

Recurring Question Patterns:

1. Vertex & Extrema: Find max/min height or value.
   • Mental Model: Vertex $x=-b/2a$. Substitute back for $y$.
2. Projectile Motion: Height vs Time problems.
   • Mental Model: It’s just a downward parabola. Max height at vertex. Hit ground when $h(t)=0$.
3. Tangent/Slope: Slope of parabola at a point.
   • Mental Model: Derivative $y^{\prime }=2ax+b$.

Week 4: Polynomials

Core Concepts:

• Degree, Roots, Turning Points
• End Behavior ($x\to \pm \infty$)
• Graphing

Recurring Question Patterns:

1. Graph Matching: Match equation to graph based on roots and end behavior.
   • Mental Model: Check roots (x-intercepts). Check sign of leading coefficient (End behavior).
2. Turning Points: Number of local max/min.
   • Mental Model: Degree $n$ polynomial has at most $n-1$ turning points.

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Statistics 1

Week 1: Introduction & Data Types

Core Concepts:

• Sample vs Population
• Variables (Categorical vs Numerical)
• Scales (Nominal, Ordinal, Interval, Ratio)

Recurring Question Patterns:

1. Identify Scale: Given a variable (e.g., “Temperature”, “Rank”), identify the scale.
   • Mental Model:
     • Named categories? → Nominal.
     • Ordered? → Ordinal.
     • Differences meaningful but no true zero? → Interval.
     • True zero (ratios work)? → Ratio.
2. Sample/Population: Identify from a scenario.
   • Mental Model: Population = Everyone of interest. Sample = The subset actually measured.

Week 2: Categorical Data

Core Concepts:

• Frequency Tables
• Bar Charts, Pie Charts, Pareto Charts
• Mode (for categorical)

Recurring Question Patterns:

1. Chart Interpretation: Read values from Bar/Pie charts.
2. Misleading Graphs: Identify why a graph is bad (e.g., Pie chart not summing to 100%, truncated y-axis).

Week 3: Numerical Data

Core Concepts:

• Mean, Median, Mode
• Variance, Standard Deviation, IQR
• Outliers

Recurring Question Patterns:

1. Calculate Stats: Given a list of numbers, find Mean/Median/IQR.
   • Mental Model: Sort data first for Median/IQR.
2. Effect of Operations: If all data points $+k$ or $\times k$, what happens to Mean/SD?
   • Mental Model:
     • $+k$: Mean $+k$, SD unchanged.
     • $\times k$: Mean $\times k$, SD $\times |k|$.
3. Outlier Check: Is $x$ an outlier?
   • Mental Model: Check if $x<Q1-1.5\ast IQR$ or $x>Q3+1.5\ast IQR$.

Week 4: Association

Core Concepts:

• Scatterplots
• Covariance, Correlation ($r$)
• Causation vs Correlation

Recurring Question Patterns:

1. Correlation Strength: Interpret $r$ (Strong/Weak, Positive/Negative).
   • Mental Model: $r$ is between -1 and 1. Near 0 = weak.
2. Linear Relationship: Does the scatterplot show a line?

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Computational Thinking

Week 1: Pseudocode & Flowcharts

Core Concepts:

• Variables, Initialization
• Step-by-step execution
• Flowchart symbols

Recurring Question Patterns:

1. Trace Execution: Given a dataset and pseudocode, what are the final values of variables?
   • Mental Model: Create a “Trace Table”. Columns = Variables. Rows = Steps/Iterations. Update values line by line.

Week 2: Conditionals & Logic

Core Concepts:

• If/Else
• Boolean Operators (AND, OR, NOT)
• Filtering data

Recurring Question Patterns:

1. Logic Debugging: “This code is supposed to count X, but fails. Why?”
   • Mental Model: Check edge cases. Check initialization (inside vs outside loop). Check condition logic (AND vs OR).

Week 3: Iteration

Core Concepts:

• While loops
• “Pile 1 to Pile 2” metaphor
• Aggregation (Sum, Count, Min, Max)

Recurring Question Patterns:

1. Find Min/Max: Code to find the student with highest marks.
   • Mental Model: Initialize MaxVal to -1 (or very low). If Current > MaxVal, update MaxVal.

Week 4: Lists & Complex Algorithms

Core Concepts:

• Iterating over lists
• Nested logic

Recurring Question Patterns:

1. Complex Filtering: Count items meeting multiple criteria.