GATE DA 2027 | 25-30 Questions Daily - 10-11-2026

A random variable 𝑋 has 𝐸[𝑋] = 2, 𝐸[𝑋2] = 10 and 𝐸[𝑋3] = 38. What is the third central moment 𝐸[(𝑋 − 𝐸[𝑋])3]?

A fair coin is tossed independently until the first head appears. Let 𝑋 denote the number of tosses required. Find 𝑃(𝑋 > 4 ∣ 𝑋 > 2). (Give your answer as a finite decimal.) Answer: ____________________

Two discrete random variables 𝑋 and 𝑌 have the joint probability mass function 𝑥 + 𝑦 𝑝(𝑥, 𝑦) = 18 , 𝑥, 𝑦 ∈ {0,1,2}. What is the marginal probability 𝑃(𝑋 = 1)?

Let 𝑋1, 𝑋2, … , 𝑋𝑛 be independent and identically distributed random variables with finite mean 𝜇 and finite variance 𝜎2 > 0. Which statement describes the Central Limit Theorem correctly?

Let 𝑇: ℝ3 → ℝ3 be a linear transformation defined by 𝑇(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑦, 𝑦 + 𝑧, 𝑥 + 𝑧). Which statement is correct?

Let 1 2 3 𝐴 = (2 4 6). 1 1 1 Find the rank of 𝐴𝑇𝐴. Answer: ____________________

A 4 × 4 real matrix 𝐴 has characteristic polynomial 𝑝(𝜆) = (𝜆 − 2)2(𝜆 + 1)(𝜆 − 3). What is tr(𝐴)?

Let 𝑊 be a subspace of ℝ𝑛, and let 𝑊⊥ = {𝑥 ∈ ℝ𝑛: 𝑥𝑇 𝑤 = 0 for all 𝑤 ∈ 𝑊}. Which statements are necessarily true?

For binary classification, consider the cross-entropy loss 𝐿(𝑦, 𝑝) = −[ 𝑦log𝑝 + (1 − 𝑦)log(1 − 𝑝) ], where 𝑦 ∈ {0,1} and 𝑝 is the predicted probability of class 1. For a positive sample (𝑦 = 1), what happens to the loss as 𝑝 → 0?

A binary classifier produces the following results: True Positive = 40, False Positive = 10, False Negative = 20, True Negative = 30. Calculate the F1-score. (Give your answer up to 3 decimal places.) Answer: ____________________

Consider a dataset containing four points on the real line: {1, 2, 8, 9}. K-means clustering is performed with 𝐾 = 2, and the initial centroids are 1 and 9. After convergence, what is the total within-cluster sum of squared errors (WCSS)?

Which statements about ensemble learning are correct?

What is the output?

What is printed by this code? Answer: ____________________

What is the output?

What is the output?

Consider a relation 𝑅(𝐴, 𝐵, 𝐶, 𝐷) with the functional dependencies 𝐹 = {𝐴𝐵 → 𝐶, 𝐶 → 𝐷, 𝐷 → 𝐴}. Which of the following is a candidate key?

Which statements about a B+ tree are correct?

Consider the following schedule: 𝑅1(𝑋), 𝑅2(𝑋), 𝑊1(𝑋), 𝑊2(𝑋). Which statement is correct?

Let 𝑅(𝐴, 𝐵) and 𝑆(𝐵, 𝐶) be two relations. Which relational algebra expression returns all pairs (𝐴, 𝐶) for which there exists at least one common value of 𝐵?

A CSP has three variables 𝑋, 𝑌, 𝑍, each with the domain {1,2,3}, and the constraints 𝑋 ≠ 𝑌, 𝑌 ≠ 𝑍 and 𝑋 ≠ 𝑍. How many complete assignments satisfy all constraints?

Consider the Bayesian network 𝐴 → 𝐵, 𝐴 → 𝐶. Which conditional independence statement is necessarily implied by this network?

A search algorithm expands the node with the smallest path cost 𝑔(𝑛). Suppose the following paths exist: 𝑆 → 𝐴: 2, 𝑆 → 𝐵: 5, 𝐴 → 𝐺: 8, 𝐵 → 𝐺: 2. What is the cost of the optimal path found by Uniform Cost Search?

Consider the domain of all integers and the predicates 𝑃(𝑥) and 𝑄(𝑥). Which statements correctly express the logical meaning of the formulas?

A can complete a piece of work in 12 days, while B can complete the same work in 18 days. They start working together. After 4 days, A leaves and B completes the remaining work alone. How many total days are required to complete the work?