Formula & Trap Revision Sheets
One compact, scannable formula-and-trap sheet per subject — the night-before-the-exam reference where the Traps lines flag exactly where marks leak.
12 min read
Intermediate GATE DA Lesson 122 of 122
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
Next, prove it under the clock: run the timed mocks in the Exam Lab, then use /gate-da/review to re-test whatever these sheets reminded you was shaky.