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

A fair coin is tossed 4 times. Given that at least one head occurs, what is the probability of obtaining exactly 2 heads?

The number of requests received by a server in one minute follows a Poisson distribution with parameter 𝜆 = 2. Given that at least one request has arrived, what is the probability that exactly two requests arrived? −2

Let 𝑋1, 𝑋2, … , 𝑋𝑛 be independent observations from the probability density function 1 , 0 ≤ 𝑥 ≤ 𝜃 𝑓(𝑥; 𝜃) = { 𝜃 0, otherwise where 𝜃 > 0. What is the maximum likelihood estimator of 𝜃?

A 3 × 3 matrix 𝐴 has eigenvalues 1, 2 and 4. Define 𝐵 = 𝐴2 − 5𝐴 + 4𝐼. What is det(𝐵)?

Find the global minimum value of 𝑓(𝑥) = 𝑥4 − 4𝑥3 + 4𝑥2 + 1 over 𝑥 ∈ ℝ.

Evaluate sin𝑥 − 𝑥cos𝑥 𝑥→0lim𝑥3 .

Consider the directed graph with edges 𝐴 → 𝐶, 𝐵 → 𝐶, 𝐵 → 𝐷, 𝐶 → 𝐸, 𝐷 → 𝐸, 𝐸 → 𝐹. Kahn’s algorithm is used with a FIFO queue. Initially, the zero-indegree vertices are inserted in alphabetical order. What is the resulting topological ordering?

A knapsack has capacity 𝑊 = 7. There are four items: Item Weight Value 1 1 1 2 3 4 3 4 5 4 5 7 Each item can be selected at most once. What is the maximum achievable value?

A stack supports three operations: • PUSH(x): insert an element. • POP(): remove the top element. • MULTIPOP(k): remove at most 𝑘 elements. Suppose an initially empty stack performs a sequence of 𝑛 operations, where every element is pushed at most once. What is the tight upper bound on the total cost of all operations, assuming each individual push or pop costs 𝑂(1)?

Consider this code: Now consider: Assuming dp is initially filled with −1, what are the time complexities of 𝑓(𝑛) and 𝑔(𝑛) respectively?

A university has the following requirements: • Every student must belong to exactly one department. • A department may have zero or more students. • Every department has exactly one head. • A professor can head at most one department. Which is the correct interpretation?

Consider the relation Employee(Eid, Dept, Salary): Eid Dept Salary 1 A 40 2 A 60 3 B 50 4 B 70 5 C 80 6 C 20 Execute the following query: Which departments are returned?

Consider a relation 𝑅(𝐴, 𝐵, 𝐶, 𝐷) with the functional dependencies 𝐴 → 𝐵, 𝐵 → 𝐶, 𝐶 → 𝐴, 𝐶 → 𝐷. Which statement is correct?

Consider the schedule 𝑆: 𝑟1(𝑋), 𝑟2(𝑌), 𝑤1(𝑌), 𝑟3(𝑋), 𝑤2(𝑋), 𝑤3(𝑌) Which statement is correct?

Consider the dataset 1 01 𝑋 = (1 1) , 𝑦 = (2). 1 22 Using ordinary least squares, what is the estimated coefficient vector 𝛽̂ = (𝑋𝑇𝑋)−1𝑋𝑇𝑦? 1

For a single observation with 𝑦 = 1, logistic regression predicts 𝑝 = 𝜎(𝑤𝑇𝑥) = 0.8. What is the negative log-likelihood loss for this observation?

Consider the polynomial kernel 𝐾(𝑥, 𝑧) = (𝑥𝑇𝑧 + 1)2. For 𝑥 = (1,2)𝑇 and 𝑧 = (2,1)𝑇, what is 𝐾(𝑥, 𝑧)?

A graph has the following edge costs: 𝑆 → 𝐴: 2, 𝑆 → 𝐵: 4, 𝐴 → 𝐶: 2, 𝐵 → 𝐶: 1, 𝐶 → 𝐺: 3. The heuristic values are ℎ(𝑆) = 5, ℎ(𝐴) = 4, ℎ(𝐵) = 3, ℎ(𝐶) = 2, ℎ(𝐺) = 0. Using 𝑓(𝑛) = 𝑔(𝑛) + ℎ(𝑛), which path does A* return, assuming it terminates when the goal is removed from the priority queue?

A MAX root has two MIN children, evaluated from left to right. • Left subtree leaves: 7, 9, 5 • Right subtree leaves: 6, 4, 8 How many leaf nodes are pruned by alpha-beta pruning?

Consider the Bayesian network 𝐴 → 𝐵 → 𝐶. Which statement is correct?

If 2𝑥+1 + 2𝑥−1 = 40, then the value of 𝑥 is:

Two random variables 𝑋 and 𝑌 are independent and uniformly distributed over {1,2,3,4,5,6}. Define the events 𝐴 = {𝑋 + 𝑌 is even} and 𝐵 = {𝑋𝑌 is divisible by 3}. Find 𝑃(𝐴 ∣ 𝐵).

Let 𝐴 be an 𝑛 × 𝑛 matrix satisfying 𝐴2 = 𝐴. Suppose rank(𝐴) = 𝑟. What is the rank of 𝐼 − 𝐴?

A binary classification problem has classes 𝐶1 and 𝐶2 with equal prior probabilities, 𝑃(𝐶1) = 𝑃(𝐶2) = 1/2 . Two binary features 𝑋1 and 𝑋2 are conditionally independent given the class, with 𝑃(𝑋1 = 1 ∣ 𝐶1) = 0.8, 𝑃(𝑋2 = 1 ∣ 𝐶1) = 0.3, 𝑃(𝑋1 = 1 ∣ 𝐶2) = 0.4, 𝑃(𝑋2 = 1 ∣ 𝐶2) = 0.7. An observation has 𝑋1 = 1 and 𝑋2 = 1. What is the posterior probability 𝑃(𝐶1 ∣ 𝑋1 = 1, 𝑋2 = 1)?

Consider the Bayesian network 𝐴 → 𝐶 ← 𝐵 and 𝐶 → 𝐷, where all variables are binary. The joint distribution factorizes as 𝑃(𝐴, 𝐵, 𝐶, 𝐷) = 𝑃(𝐴) 𝑃(𝐵) 𝑃(𝐶 ∣ 𝐴, 𝐵) 𝑃(𝐷 ∣ 𝐶). You want to compute 𝑃(𝐴 ∣ 𝐷) using variable elimination. Which expression correctly represents the unnormalized posterior factor over 𝐴 after eliminating 𝐵 and 𝐶?