Formula & Trap Revision Sheets

This is that page — or rather, eight of them, one per subject, set end to end. Everything here earns its place by being the kind of thing that vanishes under exam stress and costs a mark when it does: a formula you only half-remember, a definition whose direction you flip, a trap you have walked into before and will again. Nothing on this page is new. It exists to do one thing the drills and mocks cannot — re-fire, in a single unbroken scan, everything months of study already built into you.

Probability & Statistics

Formulas to know cold

Idea	Formula
Union	P(A∪B) = P(A) + P(B) − P(A∩B)
Bayes	`P(H
Variance	Var(X) = E[X²] − (E[X])²
Covariance	Cov(X,Y) = E[XY] − E[X]·E[Y]
Bernoulli(p)	mean p, variance p(1−p)
Binomial(n,p)	mean np, variance np(1−p)
Uniform(a,b)	mean (a+b)/2, variance (b−a)²/12
Exponential(λ)	mean 1/λ, variance 1/λ²
Poisson(λ)	mean λ, variance λ (equal)
Normal(μ,σ²)	mean μ, variance σ²
Standardize	z = (x − μ)/σ (divide by σ, the SD)
CLT	sample-mean SD = σ/√n (the standard error)

Traps

Linear Algebra

Formulas to know cold

Idea	Fact
Determinant	det(A) = Πλᵢ (product of eigenvalues)
Trace	trace(A) = Σλᵢ (sum of eigenvalues)
Rank–nullity	rank(A) + nullity(A) = number of COLUMNS
Invertible	det ≠ 0 ⇔ full rank ⇔ 0 is NOT an eigenvalue
Symmetric	real eigenvalues, orthogonal eigenvectors
Idempotent	P² = P ⇒ eigenvalues are 0 or 1 only
Orthogonal Q	QᵀQ = I, det(Q) = ±1, columns orthonormal
Singular values	σᵢ = √(eigenvalues of AᵀA), always ≥ 0
Quadratic form max	max xᵀAx on ‖x‖ = 1 is the largest eigenvalue (min is the smallest)

Traps

Calculus & Optimization

Formulas to know cold

Idea	Fact
Standard limits	sin x / x → 1 and (eˣ − 1)/x → 1 as x → 0
L’Hôpital	for 0/0 or ∞/∞ only, differentiate top and bottom
2nd-derivative test	f'' > 0 ⇒ local min; f'' < 0 ⇒ local max; f'' = 0 inconclusive
Closed-interval optimisation	compare critical points AND both endpoints
Convexity	convex ⇒ any local min is the global min
Taylor coefficient	f⁽ⁿ⁾(0) = n! · (coefficient of xⁿ) in the series
Maclaurin	eˣ = Σ xⁿ/n!; sin x = x − x³/3! + …; cos x = 1 − x²/2! + …

Traps

Programming & DSA

Formulas to know cold

Algorithm	Time	Notes
Bubble / Insertion / Selection	Θ(n²)	bubble & insertion stable; selection NOT stable
Mergesort	Θ(n log n) always	stable; needs O(n) extra space
Quicksort	avg Θ(n log n), worst Θ(n²)	in place, NOT stable
Binary search	Θ(log n)	requires a SORTED array
Hashing (open addressing)	expected probes ≈ 1/(1 − α)	α = n/m load factor
BFS / DFS	O(V+E)	BFS uses a queue, DFS uses a stack
Tree reconstruction	—	inorder + (pre or post) rebuilds a binary tree uniquely

Traps

Databases & Warehousing

Formulas to know cold

Idea	Fact
Candidate key	a minimal attribute set whose closure is all attributes
Attribute closure	repeatedly add RHS of any FD whose LHS is already in the set
2NF	no non-prime attribute depends on part of a candidate key
3NF	for every FD X → A: X is a superkey OR A is prime
BCNF	for every non-trivial FD X → A: X must be a superkey (no exception)
Lossless-join	the common attribute set R1 ∩ R2 is a superkey of one piece
Index choice	hash index for equality; B+-tree for range / ordered scans
Scaling	min-max maps to [0,1]; z-score gives mean 0, SD 1
Schema	star = denormalised dimensions; snowflake = normalised (more joins)
Measures	distributive (SUM, COUNT, MIN, MAX); algebraic (AVG, stddev); holistic (MEDIAN, MODE)

Traps

Machine Learning

Formulas to know cold

Idea	Formula
Least squares	minimize Σ(yᵢ − ŷᵢ)²; normal equation w = (XᵀX)⁻¹Xᵀy
Ridge	Σ(yᵢ − ŷᵢ)² + λ·‖w‖² (L2 penalty); ↑λ ⇒ ↑bias, ↓variance
Precision / Recall	precision = TP/(TP+FP), recall = TP/(TP+FN)
F1	F1 = 2·precision·recall / (precision + recall)
Sigmoid	σ(z) = 1/(1 + e^−z) maps to (0,1)
Naive Bayes	posterior ∝ prior · Π likelihoods
k-means	alternate assign-to-nearest-centroid, then update centroid to the mean
MLP params	per layer a·b weights + b biases; total summed over layers
ReLU	max(0, x) — continuous, NOT differentiable at 0
PCA	components orthogonal; variance explained = eigenvalue / sum of eigenvalues
LOOCV	number of folds = n (leave one out each time)

Traps

Artificial Intelligence

Formulas to know cold

Idea	Fact
Uninformed search	BFS complete + optimal (unit costs); DFS neither; UCS optimal on costs; IDDFS combines BFS-optimality with DFS-space
Admissible heuristic	h(n) ≤ true cost, i.e. never overestimates
A*	expand by f = g + h; admissible h ⇒ optimal
Minimax	back values up: MAX takes the max of children, MIN the min
Alpha-beta	prunes branches but returns the SAME value as full minimax
Entailment	X ⊨ Y if and only if X ∧ ¬Y is unsatisfiable
FOL translation	”all A are B” is ∀x A(x) ⇒ B(x); “some A is B” is ∃x A(x) ∧ B(x)
Bayes net	joint `= Π P(node
Inference	variable elimination is EXACT; sampling is APPROXIMATE

Traps

General Aptitude

Formulas to know cold

Idea	Fact
Successive percentages	MULTIPLY the factors: +20% then −10% is ×1.20×0.90, not +10%
Percentage change	always measured over the BASE (the original value)
Ratios	scale every part by the same multiplier; compare like-for-like
The four GA buckets	quantitative, verbal, analytical reasoning, spatial reasoning

Traps

---

And that is the whole of it. Every formula above was earned a lesson at a time; every trap was paid for once so that it need not cost you again — and this single sheet is what all of that study compresses to on the last night. Nothing remains to be learned now, only kept sharp: skim it, then prove it one final time under the clock with the timed mocks, and use the Review & Mastery Checks to re-test whatever a line here reminded you was shaky. When this page reads like old news instead of new information, you are ready — walk in, read each question for the kind it is, reach for the one rule that cracks it, and substitute with care. That is the habit every paper in this course rewarded. Good luck.