Can you solve a whole family of questions rather than memorizing one answer?
Layer J — Independent retrieval
You solve it without my help.
That last layer matters. I don't want to turn you into someone who can recognize my explanations but cannot solve the question themselves.
2. Canonical Quiz 2 syllabus
A. Mathematics for Data Science I
The official syllabus gives Weeks 1–8 as follows. turn1view0
Concepts
Polynomial expressions
Addition
Subtraction
Multiplication
Division
Polynomial algorithms
Roots / x-intercepts
Multiplicity
End behaviour
Turning points
Polynomial graphing
Constructing polynomials
Pattern families
Easy
Add/subtract polynomials.
Multiply.
Evaluate polynomial.
Identify degree.
Identify leading coefficient.
Medium
Polynomial division.
Factorization.
Roots → polynomial.
Polynomial → graph characteristics.
Hard
Multiplicity + graph behaviour.
Determine unknown coefficients from constraints.
Reconstruct polynomial from roots/points.
Reason about end behaviour without fully expanding.
Important mental model
A polynomial isn't merely:
"some algebraic expression."
Think of it simultaneously as:
expression↔equation↔function↔graph
Quiz questions can move between those representations.
5. Maths Week 5 — Functions
Official topics include horizontal/vertical line tests, exponential functions, composite functions and inverse functions. turn1view0
Concepts
Definition:
logbx=y⟺by=x
Properties:
logb(xy)=logbx+logby
logb(yx)=logbx−logby
logb(xk)=klogbx
Change of base:
logbx=logablogax
Patterns
Evaluate logs.
Simplify logarithmic expressions.
Convert log ↔ exponential form.
Solve exponential equations.
Solve logarithmic equations.
Determine whether a solution is valid.
Interpret logarithmic graphs.
Major trap
Whenever you manipulate logarithmic equations, domain restrictions matter.
You cannot blindly accept algebraic solutions.
7. Maths Week 7 — Sequences, Limits & Continuity
Officially this week introduces functions of one variable, graphs/tangents, limits for sequences and functions, and continuity. turn1view0
Core object
f′(x)
Interpretations:
instantaneous rate of change
slope of tangent
local behaviour
Patterns
Derivative computation
Apply appropriate rules.
Tangent
At x=a:
y−f(a)=f′(a)(x−a)
Linear approximation
f(x)≈f(a)+f′(a)(x−a)
Critical points
Typically solve:
f′(x)=0
and also consider where f′ is undefined.
Local extrema
Use derivative behaviour around critical points.
L'Hôpital
For appropriate indeterminate forms:
=
\lim_{x\to a}\frac{f'(x)}{g'(x)}$$
when the conditions for the rule hold.
---
# 9. Computational Thinking
This course is particularly important because IITM explicitly describes CT as learning programming concepts through **manual execution**, rather than simply writing code. turn1view2
Patterns:
- Variable tracing
- State changes
- Initialization
- Iterating through data
- Filtering
- Datatype identification
- Flowchart interpretation
- Detecting invalid/insane input
Example abstract pattern:
```text
value ← initial_value
for each item:
update value
output value
```
You should eventually be able to look at this and immediately ask:
> What is the state?
> What changes it?
> What remains invariant?
> What is the final state?
---
# 10. CT Week 2
- Iteration
- Filtering
- Selection
- Pseudocode
- Finding maximum/minimum
- AND
Patterns:
### Accumulation
```text
total ← 0
for each x:
total ← total + x
```
### Maximum
```text
best ← first item
for each x:
if x > best:
best ← x
```
### Filtering
```text
if condition:
keep/process x
```
### Conjunction
$$A\land B$$
Both must be true.
---
# 11. CT Week 3
- Multiple non-nested iterations
- Three-prizes problem
- Procedures
- Parameters
- Side effects
- OR
This is where we start thinking in **reusable computational units**.
Pattern:
$$\text{input}\rightarrow\text{procedure}\rightarrow\text{output}$$
We'll distinguish:
- parameter
- argument
- local state
- returned result
- side effect
And:
$$A\lor B$$
means at least one condition is true.
---
# 12. CT Week 4
- Nested iterations
- Birthday paradox
- Binning
This is a major pattern jump.
Nested loop structure:
```text
for each x:
for each y:
do something
```
Conceptually:
$$n\times n$$
potential pairwise interactions.
### Birthday-paradox pattern
Not merely a probability question.
It's a **pair-generation / collision-detection** pattern.
### Binning
Map continuous/discrete values into categories:
$$x\rightarrow\text{bin}(x)$$
This idea becomes useful throughout data science.
---
# 13. CT Week 5 — Lists
Officially: lists and insertion sort. turn1view2
Patterns:
### Table
Think:
$$\text{row}\times\text{column}$$
### Dictionary
Think:
$$\text{key}\rightarrow\text{value}$$
Typical questions:
- Lookup
- Update
- Count frequencies
- Map identifiers to information
- Represent relationships
- Translate table ↔ dictionary representation
Frequency counting is a particularly important reusable pattern:
```text
for each item:
frequency[item] += 1
```
---
# 15. CT Week 7 — Graphs & Matrices
Officially:
- Graphs
- Matrices. turn1view3turn0search5
And here's where I want to be particularly strict:
**CT pattern → Python implementation → Python-specific behaviour.**
---
## Python W1 — Algorithms
Concepts:
- Python execution model
- Variables
- Expressions
- Values
- Types
- Assignment
- Input/output
- Basic algorithmic thinking
Patterns:
```python
x = ...
y = ...
z = ...
```
Trace the state after every line.
---
# 18. Python W2–3 — Conditionals
Core structures:
```python
if condition:
...
elif condition:
...
else:
...
```
Patterns:
### Binary decision
$$P\rightarrow A/B$$
### Multiple mutually exclusive cases
$$P_1,P_2,\ldots,P_n$$
### Compound condition
```python
if A and B:
```
### Alternative condition
```python
if A or B:
```
Important distinction:
```python
and
```
versus
```python
or
```
and nested conditionals.
We'll also train **truth-table thinking**, not just syntax.
---
# 19. Python W4–5 — Iterations & Ranges
Core structures:
```python
for x in ...:
```
and
```python
while condition:
```
Patterns:
- fixed repetition
- condition-controlled repetition
- counting
- accumulation
- filtering
- searching
- maximum/minimum
- nested iteration
- range generation
Important:
```python
range(start, stop, step)
```
has an **exclusive stop**.
This is one of the classic Python traps.
---
# 20. Python W6–8 — Basic Collections
This is a large chunk.
### Lists
```python
[x1, x2, x3]
```
Patterns:
- indexing
- traversal
- modification
- append
- insertion
- deletion
- membership
- slicing
- aggregation
### Tuples
```python
(x, y)
```
Think:
> ordered collection with different mutability semantics.
### Dictionaries
```python
{
key: value
}
```
Think:
$$key\rightarrow value$$
### Collection patterns
We'll train:
- frequency counting
- lookup
- grouping
- transformation
- filtering
- aggregation
- nested collections
- iteration over collection structures
---
# 21. Statistics for Data Science I
Stats is probably the course where **interpretation traps** will matter most.
The official Weeks 1–8 are clearly defined. turn1view1
### Categorical × categorical
Contingency tables.
### Numerical × numerical
Scatterplot.
### Covariance
Conceptually:
> Do two variables tend to move together?
### Pearson correlation
$$-1\le r\le1$$
Interpret:
- sign → direction
- magnitude → strength of linear association
### Point-biserial correlation
Useful when:
- one variable is binary/categorical
- one is numerical
### Major trap
**Correlation ≠ causation.**
And:
> $r\approx0$ does not mean "no relationship whatsoever"; it primarily indicates little/no **linear** association.
---
# 25. Stats W5 — Counting
Concepts:
- Addition rule
- Multiplication rule
- Factorials
Factorial:
$$n!=n(n-1)(n-2)\cdots1$$
### Addition rule
Use when alternatives are mutually exclusive:
$$N=N_1+N_2+\cdots$$
### Multiplication rule
Use for sequential choices:
$$N=N_1N_2\cdots$$
The real skill is **recognizing whether a problem is additive or multiplicative**.
---
# 26. Stats W6 — Permutations & Combinations
Permutation:
$$P(n,r)=\frac{n!}{(n-r)!}$$
Combination:
$$C(n,r)=\binom nr
=\frac{n!}{r!(n-r)!}$$
### Recognition rule
Ask:
> **Does order matter?**
If yes → permutation-type reasoning.
If no → combination-type reasoning.
This becomes a major pattern classifier.
---
# 27. Stats W7 — Probability
Concepts:
- Random experiment
- Sample space
- Event
- Probability
- Probability properties
Core axioms:
$$0\le P(A)\le1$$
$$P(S)=1$$
For disjoint events:
$$P(A\cup B)=P(A)+P(B)$$
General addition:
$$P(A\cup B)
=
P(A)+P(B)-P(A\cap B)
Complement:
P(Ac)=1−P(A)
Pattern families
direct probability
complement
union
intersection
mutually exclusive events
sample-space counting
probability from combinatorics
28. Stats W8 — Conditional Probability
This is probably one of the highest-value Quiz 2 areas.
Concepts:
Conditional probability
Multiplication rule
Independence
Law of total probability
Bayes' theorem
Definition:
P(A∣B)=P(B)P(A∩B)
Multiplication:
P(A∩B)=P(A∣B)P(B)
Independence:
P(A∩B)=P(A)P(B)
or equivalently:
P(A∣B)=P(A)
when applicable.
Bayes:
P(A∣B)=P(B)P(B∣A)P(A)
The pattern we're going to hammer
A question gives you:
evidence B
and asks:
probability of underlying condition A.
That is often a Bayesian inversion problem.
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