GATE DA 2027 | 25-30 Questions Daily - 29-10-2026

A fair die is rolled repeatedly. Given that the sum of the first two rolls is 8, what is the probability that the first roll was greater than 3?

A random variable 𝑋 has the probability mass function 𝑃(𝑋 = 𝑘) = 𝑐𝑘 for 𝑘 = 1,2,3,4, where 𝑐 is the normalization constant. Calculate 𝑃(𝑋 ≥ 3).

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

In an experiment, each trial succeeds independently with probability 𝑝. Let 𝑁 be the number of trials required to obtain the first success. Given that 𝑃(𝑁 > 2) = 9/25 , what is 𝑃(𝑁 = 1)?

For which value(s) of 𝑘 does the matrix 𝐴 have rank 1? 1 2 3 𝐴 = (2 4 6) 1 𝑘 𝑘 + 1

Let 𝐴 be a 3 × 3 matrix satisfying 𝐴2 − 5𝐴 + 6𝐼 = 0. If 𝐴 is invertible, which expression must equal 𝐴−1?

Let 𝑓(𝑥) = 𝑥3 − 3𝑥2 − 9𝑥 + 5 for 𝑥 ∈ [−2,4]. What is the absolute minimum value of 𝑓(𝑥) on this interval?

Evaluate 2 𝑥 ln(1 + 𝑥) − 𝑥 + 𝑥→0lim𝑥3 2.

Consider the objective function 𝐽(𝑤) = (𝑦 − 𝑋𝑤)𝑇(𝑦 − 𝑋𝑤) + 𝜆 𝑤𝑇𝑤, where 𝑋 has full column rank and 𝜆 > 0. Which expression gives the unique minimizer?

A binary logistic regression model predicts 1 𝑃(𝑌 = 1 ∣ 𝑥) = 𝑇 −(𝑤𝑥+𝑏) 1 + 𝑒 with classification threshold 0.5. If 𝑤 and 𝑏 are both multiplied by a positive constant 𝑐, what happens to the decision boundary?

A hard-margin linear SVM has the separating hyperplane 𝑤𝑇𝑥 + 𝑏 = 0. Suppose the minimum value of 𝑦𝑖 (𝑤𝑇𝑥𝑖 + 𝑏) over all training samples is 1, and ∥ 𝑤 ∥2= 4. What is the distance between the two supporting hyperplanes?

Two models are trained on the same dataset. • Model A has low training error but high test error. • Model B has higher training error but lower test error. Which conclusion is most justified?

A dynamic array starts with capacity 1. Whenever it becomes full, its capacity doubles and all existing elements are copied into the new array. Assume that inserting one element costs 1 and copying each existing element costs 1. After inserting 𝑛 elements into an initially empty array, what is the total cost?

Consider this directed weighted graph: • 𝑆 → 𝐴 with weight 4 • 𝑆 → 𝐵 with weight 1 • 𝐵 → 𝐴 with weight 2 • 𝐴 → 𝐶 with weight 1 • 𝐵 → 𝐶 with weight 5 Dijkstra’s algorithm starts at 𝑆. What are the final shortest distances (𝑑(𝐴), 𝑑(𝐵), 𝑑(𝐶))?

A hash table has 11 slots, indexed 0 to 10, and uses the hash function ℎ(𝑘) = 𝑘 mod 11. The keys 22, 1, 13, 11, 24 are inserted in this order using linear probing, with no deletions. Which statement is correct?

Let the strings be 𝑋 = ABCBDAB and 𝑌 = BDCABA. What is the length of their longest common subsequence (LCS)?

A university database has the following requirements: • Each Department must have at least one Professor. • Each Professor belongs to exactly one Department. • A Department may have multiple Professors. • Each Professor may supervise multiple Projects, but every Project must be supervised by exactly one Professor. Which option correctly represents the relationships?

Consider the relation Enroll(Student, Course, Marks): Student Course Marks A DBMS 80 A ML 60 B DBMS 70 B ML 90 C DBMS 50 C ML 70 What does the following query return?

Given the relations Student(Sid, Name), Enroll(Sid, Cid, Grade) and Course(Cid, Title), which relational algebra expression returns the names of students enrolled in a course titled ``DBMS’’? Assume natural joins use the common attribute names Sid and Cid.

Consider a relation 𝑅(𝐴, 𝐵, 𝐶, 𝐷) with the functional dependencies 𝐴 → 𝐵, 𝐵 → 𝐶, 𝐴𝐶 → 𝐷, 𝐷 → 𝐴. Which option gives all candidate keys of 𝑅?

A* search uses 𝑓(𝑛) = 𝑔(𝑛) + ℎ(𝑛). Consider the graph with edge costs: • 𝑆 → 𝐴: 2 • 𝑆 → 𝐵: 3 • 𝐴 → 𝐺: 6 • 𝐵 → 𝐺: 3 The heuristic values are ℎ(𝑆) = 4, ℎ(𝐴) = 5, ℎ(𝐵) = 2, ℎ(𝐺) = 0. If A* terminates when the goal is removed from the priority queue, which path does it return?

A MAX node has two MIN children, 𝐿 and 𝑅. Their leaf values, from left to right, are: • 𝐿: 8, 3, 6 • 𝑅: 5, 9, 2 What is the minimax value at the root?

Consider the knowledge base 𝑃 ⇒ 𝑄, 𝑄 ⇒ 𝑅, ¬𝑅. Which statement is necessarily entailed by the knowledge base?

Suppose a Bayesian network has the structure 𝐴 → 𝐶 ← 𝐵, where 𝐴 and 𝐵 are independent binary variables before observing 𝐶. Which statement is correct in general?

Random variables 𝑋, 𝑌, 𝑍 are such that 𝑋 and 𝑌 are independent given 𝑍. Also, 𝑃(𝑍 = 0) = 𝑃(𝑍 = 1) = 1/2 . It is given that 𝑃(𝑋 = 1 ∣ 𝑍 = 0) = 0.2, 𝑃(𝑋 = 1 ∣ 𝑍 = 1) = 0.8, 𝑃(𝑌 = 1 ∣ 𝑍 = 0) = 0.8, 𝑃(𝑌 = 1 ∣ 𝑍 = 1) = 0.2. What is 𝑃(𝑋 = 1, 𝑌 = 1)?