GATE DA 2027 | 25-30 Questions Daily - 28-10-2026

Let 𝑋 and 𝑌 be independent random variables with 𝐸[𝑋] = 2, Var(𝑋) = 3, 𝐸[𝑌] = 1 and Var(𝑌) = 4. Define 𝑍 = 𝑋 − 2𝑌. What is Var(𝑍)?

A continuous random variable 𝑋 has density 𝑓(𝑥) = 2𝑥 for 0 < 𝑥 < 1 and 0 otherwise. What is 𝑃 (𝑋 > 1/2 | 𝑋 > 1/4 )?

Let $X_1, \ldots, X_n$ be an independent and identically distributed sample from a distribution with mean $\mu$ and finite variance $\sigma^2$. Which estimator of $\mu$ is unbiased and has variance $\sigma^2/n$?

Suppose 𝑋 and 𝑌 have finite second moments, Var(𝑋) = 4, Var(𝑌) = 9 and Corr(𝑋, 𝑌) = − 1/2 . What is Var(𝑋 + 𝑌)?

A random variable 𝑋 has 𝐸[𝑋] = 0 and 𝐸[𝑋2] = 1. Which is the strongest conclusion that must hold?

Let 𝑋 follow a Binomial(𝑛, 𝑝) distribution, where 𝑛 is known and 𝑛 ≥ 2. An experiment reports that 𝑃(𝑋 = 0) = 𝑃(𝑋 = 1). Which value of 𝑝 is implied, assuming 0 < 𝑝 < 1?

Let 𝑋 have a Poisson distribution with unknown parameter 𝜆 > 0. It is known that 𝑃(𝑋 = 2) = 𝑃(𝑋 = 3). What is 𝜆?

Let 𝑋 have an exponential distribution with rate 𝜆 > 0. For 𝑡 > 0 and 𝑠 > 0, which expression equals 𝑃(𝑋 > 𝑠 + 𝑡 ∣ 𝑋 > 𝑠)?

A fair six-sided die is rolled repeatedly until the first 6 appears. Let 𝑁 be the number of rolls. What is 𝑃(𝑁 is odd)?

Let 𝐴 be a 3 × 3 real matrix satisfying 𝐴2 = 𝐴. Which set of values can occur as tr(𝐴), the trace of 𝐴?

Define 𝑓(𝑥) = 𝑥4 − 4𝑥2 + 1. How many distinct real solutions does 𝑓′(𝑥) = 0 have?

Let 𝐴 be a real symmetric matrix whose eigenvalues are −2, 1 and 4. Define 𝐵 = 𝐴2 − 2𝐴. How many positive, negative and zero eigenvalues does 𝐵 have, counting multiplicity? (The answer is written as: number positive, number negative, number zero.)

Let 𝑓(𝑥) = 𝑥3 − 3𝑥. On the interval [−2,2], what is the absolute maximum value of 𝑓?

A binary classifier outputs probability 𝑝 = 0.7 for class 1. Predicting class 1 incurs cost 2 when the true class is 0, while predicting class 0 incurs cost 5 when the true class is 1. Correct predictions have zero cost. Which prediction minimizes the expected cost?

In a linear regression model with design matrix 𝑋 and target vector 𝑦, suppose a nonzero vector 𝑣 satisfies 𝑋𝑣 = 0. If 𝑤 is a least-squares minimizer of ∥ 𝑋𝑤 − 𝑦 ∥2, which statement must hold for every real 𝑡?

A decision tree node contains 8 positive and 8 negative examples. A split creates two children: the first contains 6 positive and 2 negative examples; the second contains 2 positive and 6 negative examples. Using Gini impurity 𝐺 = 1 − ∑ 𝑘 2/ 𝑝𝑘, what is the weighted Gini impurity after the split?

For binary logistic regression, the predicted probability is 𝜎(𝑧), where 𝜎(𝑧) = 1/1+𝑒 −𝑧 and 𝑧 = 𝑤𝑇𝑥. For one training example with label 𝑦 = 1, the negative log-likelihood loss is −log𝜎(𝑧). What is its derivative with respect to 𝑧?

What is printed by this Python code?

What is printed?

What is the output?

A hash table uses separate chaining. Keys are inserted in this order: 12, 7, 22, 17, 5. The table has 5 buckets and the hash function is ℎ(𝑘) = 𝑘 mod 5. How many keys are stored in bucket 2 after all insertions?

A graph has the following edge costs: • 𝑆 → 𝐴 = 2, 𝑆 → 𝐵 = 4 • 𝐴 → 𝐺 = 7, 𝐴 → 𝐵 = 1 • 𝐵 → 𝐺 = 3 Heuristic values: ℎ(𝑆) = 4, ℎ(𝐴) = 4, ℎ(𝐵) = 2, ℎ(𝐺) = 0. A* search uses 𝑓(𝑛) = 𝑔(𝑛) + ℎ(𝑛). Which statement is correct?

The root is a MAX node. Its two children, 𝐿 and 𝑅, are both MIN nodes. The leaves, in left- to-right order, are: • 𝐿: 8, 5, 9 • 𝑅: 6, 4, 7 Alpha-beta pruning is applied in left-to-right order. Which result is correct?

Variables 𝑋, 𝑌, 𝑍 each have the domain {1,2,3,4}. The constraints are: 𝑋 < 𝑌, 𝑌 < 𝑍, 𝑋 + 𝑍 = 5 Which option correctly identifies all ordered triples (𝑋, 𝑌, 𝑍) that satisfy every constraint?

A Bayesian network has the structure 𝐴 → 𝐶 ← 𝐵, and 𝐴 and 𝐵 are initially independent. Which statement is correct?

Let 𝑋 be a nonnegative integer-valued random variable with 𝐸[𝑋] = 𝑚, where 𝑚 is a positive integer. Which statement describes the largest possible value of 𝑃(𝑋 = 0)?

Let 𝐴 be a real symmetric 2 × 2 matrix with eigenvalues 1 and 3. Define 𝑔(𝑥) = 𝑥𝑇 𝐴𝑥 for 𝑥 ∈ ℝ2 subject to 𝑥𝑇𝑥 = 1. What is the minimum possible value of 𝑔(𝑥), and why?

A binary classifier is evaluated on a fixed dataset. Its ROC curve is unchanged after applying a strictly increasing function 𝑔 to every predicted score, replacing each score 𝑠 by 𝑔(𝑠). Which conclusion must hold, assuming ties are preserved exactly?

A directed acyclic graph is processed by Kahn’s topological-sort algorithm. Initially the zero-indegree queue contains only vertex 𝐴. Processing 𝐴 makes both 𝐵 and 𝐶 have indegree zero, and the algorithm enqueues 𝐵 before 𝐶. Later, processing 𝐵 makes 𝐷 have indegree zero, while 𝐶 is still waiting in the queue. Which vertex is processed next after 𝐵?

Consider a relation 𝑅(𝐴, 𝐵, 𝐶, 𝐷, 𝐸) with the functional dependencies 𝐹 = {𝐴 → 𝐵𝐶, 𝐶𝐷 → 𝐸, 𝐵 → 𝐷, 𝐸 → 𝐴} Which option correctly identifies all candidate keys of the relation?

Consider the schedule 𝑆: 𝑟1(𝑋), 𝑟2(𝑌), 𝑤1(𝑌), 𝑟2(𝑋), 𝑤2(𝑋) Here 𝑟𝑖(𝑍) denotes a read and 𝑤𝑖(𝑍) denotes a write of data item 𝑍 by transaction 𝑇𝑖. Which statement is correct?

A relation 𝑅(𝐴, 𝐵, 𝐶) has the functional dependencies 𝐹 = {𝐴 → 𝐵, 𝐵 → 𝐶} It is decomposed into 𝑅1(𝐴, 𝐵) and 𝑅2(𝐴, 𝐶). Which conclusion is correct?

A knowledge base contains the rules 𝑃 ⇒ 𝑄, 𝑄 ⇒ 𝑅 and 𝑅 ⇒ ¬𝑃, together with the fact that 𝑃 is true. In classical propositional logic, what can be concluded about the knowledge base?