GATE DA 2026 — Solved Walkthrough

The 2026 paper rewarded the same habit every GATE DA paper does: recognise the kind of question, reach for the one formula or rule that cracks it, and substitute carefully. Below, the problems are grouped by subject so you can see how each area was tested.

Probability & Statistics

A disease affects 30% of a population. A test detects it correctly 80% of the time (sensitivity), with a 10% false-positive rate on healthy people. A person tests positive — what is the probability they actually have the disease?

This is a textbook Bayes inversion: the number you can measure is P(+ | disease), but the number you want is P(disease | +). Build the evidence denominator by total probability, then divide.

P(D) = 0.30,  P(+|D) = 0.80,  P(+|¬D) = 0.10,  P(¬D) = 0.70

P(D | +) =        0.80 · 0.30
            ──────────────────────────  =  0.24 / 0.31  ≈  0.77
            0.80·0.30 + 0.10·0.70

Answer: ≈ 0.77. Because the disease was common (30%) to begin with, a positive test is genuinely convincing — base rates dominate, and here the base rate is high.

→ Taught in Bayes’ Theorem

Linear Algebra

Let A be a symmetric matrix with eigenvalues 5 and 2. Find the maximum of the quadratic form xᵀAx over all unit vectors (‖x‖ = 1).

Write x in A’s orthonormal eigenbasis with coordinates y₁, y₂ where y₁² + y₂² = 1. Then xᵀAx = 5·y₁² + 2·y₂², a weighted average of the eigenvalues with weights summing to 1. That average is largest when all the weight sits on the biggest eigenvalue.

max ‖x‖=1  xᵀAx  =  largest eigenvalue  =  5
(the minimum is the smallest eigenvalue, 2)

Answer: 5. The maximum of a quadratic form on the unit sphere is always the largest eigenvalue — exactly the variance PCA maximises along its first component.

→ Taught in Quadratic Forms

Calculus & Optimization

Perform one step of gradient descent. The current weight is w = 10, the gradient of the loss at this point is 10, and the learning rate is η = 0.1. What is the updated weight?

The whole task is one substitution into the descent rule — subtract η times the gradient.

w_new = w − η · (∂L/∂w)
      = 10 − 0.1 × 10
      = 10 − 1
      = 9.0

Answer: 9.0. The minus sign is the trap: gradient descent subtracts the gradient (adding it is gradient ascent, which climbs the loss).

→ Taught in Gradient Descent (One Step)

Programming & DSA

Consider this function. What do f(1), then f(2), then f(3, []) return?

def f(val, lst=[]):
    lst.append(val)
    return lst

The default list is created once, when the def executes — not fresh on each call. So every call that omits lst shares the same list, and it accumulates:

• f(1) — appends 1 to the shared default → [1].
• f(2) — appends 2 to that same shared list, still holding 1 → [1, 2].
• f(3, []) — the caller passes its own fresh list, bypassing the default → [3].

Answer: [1], then [1, 2], then [3]. The jump from [1] to [1, 2] — with nothing seeming to carry the 1 forward — is the mutable-default trap.

→ Taught in Functions, Scope & the Mutable-Default Trap

Databases & Warehousing

A relation R is decomposed into fragments. One functional dependency now has its left-hand side in one fragment and its right-hand side in another. What is the consequence for enforcing that dependency?

When a single FD straddles two fragments, neither piece holds enough columns to check it on its own. To validate the dependency on every insert or update you must join the fragments back together first — the decomposition is not dependency-preserving.

Answer: the FD can no longer be checked on a single fragment; you must reconstruct the original relation by joining the pieces to enforce it. That join cost on every update is exactly why dependency preservation is a design goal, not a nicety.

→ Taught in Decomposition & Lossless Joins

Machine Learning

(1) PCA reduces 100 dimensions to 10. What is the angle (in degrees) between principal components PC1 and PC10?

Principal components are the eigenvectors of a symmetric covariance matrix, so they are mutually orthonormal — every distinct pair is perpendicular. No computation is needed.

Answer: 90°.

(2) A fully-connected MLP has architecture 30 → 4 → 3 → 1 with NO bias terms. How many learnable parameters does it have?

Without bias, a layer from a units to b units contributes a·b weights. Sum over the three layer transitions:

weights = 30·4 + 4·3 + 3·1
        = 120  + 12  + 3
        = 135

Answer: 135. Read the bias condition carefully — a with-bias version of a similar net would add one parameter per output unit.

(3) As the regularization coefficient λ in ridge regression increases, how do bias and variance change?

Larger λ squeezes the weights toward zero, making the model simpler and stiffer. A stiffer model varies less from one training sample to the next (variance down) but systematically misses structure (bias up).

Answer: bias increases, variance decreases.

(4) Points P1 = (2, 3, −1), P2 = (3, 1, 1), P3 = (5, −2, 3), P4 = (3, 3, 3). Using Manhattan (L1) distance, which pair merges first in agglomerative clustering?

With every point its own cluster, the first merge is simply the closest pair. Compute the L1 distances:

d(P2, P4) = |3−3| + |1−3| + |1−3| = 0 + 2 + 2 = 4   ← smallest
d(P1, P2) = |2−3| + |3−1| + |−1−1| = 1 + 2 + 2 = 5
d(P1, P4) = |2−3| + |3−3| + |−1−3| = 1 + 0 + 4 = 5
d(P3, P4) = |5−3| + |−2−3| + |3−3| = 2 + 5 + 0 = 7

Answer: P2 and P4 merge first (distance 4).

→ Taught in PCA & Dimensionality Reduction, Multi-Layer Perceptron & Activations, Ridge Regression & Regularization, and Hierarchical Clustering & Linkage

Artificial Intelligence

(1) MAX has three strategies. Each leads to a MIN node with three leaf utilities: strategy 1 → [8, 6, −1], strategy 2 → [1, 5, 7], strategy 3 → [−4, −3, −12]. Which strategy should MAX play?

Back up MIN at each strategy node (MIN takes the smallest leaf), then MAX picks the largest of those.

strategy 1: min(8, 6, −1) = −1
strategy 2: min(1, 5, 7)  =  1
strategy 3: min(−4, −3, −12) = −12

V(root) = max(−1, 1, −12) = 1   →  strategy 2

Answer: strategy 2 (value 1). Strategy 1 has the highest single leaf (8), but MIN will never let MAX reach it — it steers to the −1.

(2) Which first-order-logic formula correctly captures “Every king is a person”?

Under a universal quantifier, the connective is implication, not conjunction.

∀x  King(x) ⇒ Person(x)

The conjunction version ∀x King(x) ∧ Person(x) wrongly claims every object is both a king and a person.

Answer: ∀x King(x) ⇒ Person(x).

(3) Over a non-empty domain, which quantifier-implications are valid (true under every interpretation): (i) ∀x P(x) ⇒ ∃x P(x), (ii) ∃x P(x) ⇒ ∀x P(x), (iii) ∃x P(x) ⇔ ∀x P(x), (iv) ∀x P(x) ⇒ ∃x ¬P(x)?

Only (i) holds: if P is true for everything, pick any one object (the domain is non-empty) to witness “there exists.” (ii) and (iii) fail when some but not all objects satisfy P; (iv) fails precisely when ∀x P(x) is true, since then ∃x ¬P(x) is false.

Answer: only (i) is valid.

(4) Which statement is equivalent to “X entails Y” (X ⊨ Y)?

Entailment means every model of X is a model of Y — equivalently, no assignment makes X true and Y false at once.

X ⊨ Y   ⇔   X ∧ ¬Y is unsatisfiable   ⇔   X → Y is valid

Answer: X ⊨ Y if and only if X ∧ ¬Y is unsatisfiable.

→ Taught in Adversarial Search: Minimax, First-Order & Predicate Logic, and Propositional Logic