GATE DA 2027 | 25-30 Questions Daily - 08-11-2026

Let 𝑋1, 𝑋2, … , 𝑋5 be independent and identically distributed Uniform(0, 𝜃) random variables, and let 𝑀 = max(𝑋1, … , 𝑋5). Calculate 𝑃(𝑀 > 0.8𝜃 ∣ 𝑀 > 0.6𝜃). Give your answer up to 4 decimal places. Answer: ____________________

A random variable 𝑋 has the moment generating function 3 2 𝑀𝑋(𝑡) = ( 2 − 𝑡), 𝑡 < 2. Which of the following is the variance of 𝑋?

A sample of size 𝑛 = 25 is drawn from a normal population with known standard deviation 𝜎 = 10. We want to test 𝐻0: 𝜇 = 50 against 𝐻1: 𝜇 > 50. The observed sample mean is 𝑋‾ = 54. At significance level 𝛼 = 0.05, the critical value is 𝑧0.95 = 1.645. Calculate the p-value approximately, up to 3 decimal places. Answer: ____________________

Let 𝑋 ∼ 𝑈(−2,2) and define 𝑌 = 𝑋2. What is the probability density function of 𝑌?

Let 𝑇: ℝ4 → ℝ3 be defined by 𝑇(𝑥1, 𝑥2, 𝑥3, 𝑥4) = (𝑥1 + 𝑥2, 𝑥2 + 𝑥3, 𝑥3 + 𝑥4). Find the dimension of the null space of 𝑇. Answer: ____________________

Let 𝐴 be a 3 × 3 real matrix whose characteristic polynomial is 𝑝(𝜆) = (𝜆 − 2)2(𝜆 + 1). What is the determinant of 𝐴2 − 3𝐴 + 2𝐼?

Let 𝑢 = (1,2,2)𝑇 and 𝑣 = (2,1, −2)𝑇. Find the squared Euclidean distance between 𝑢 and its orthogonal projection onto the subspace spanned by 𝑣. That is, calculate ∥ 𝑢 − proj𝑣(𝑢) ∥2. Answer: ____________________

Consider the symmetric matrix 𝑎 1 0 𝐴 = (1 𝑎 1). 0 1 𝑎 For which values of 𝑎 is 𝐴 positive definite?

Consider the ridge regression objective 𝐽(𝑤) =∥ 𝑋𝑤 − 𝑦 ∥2+ 𝜆 ∥ 𝑤 ∥2. Given 𝑋𝑇𝑋 = ( 4/0 / 0/2 ) , 𝑋𝑇𝑦 = ( 8/6 ), and 𝜆 = 2, calculate the optimal weight vector 𝑤∗ and report the sum of its components. (Give your answer up to 3 decimal places.) Answer: ____________________

A dataset contains 10 samples: 6 positive and 4 negative. A binary split produces a left child with 4 positive and 0 negative samples, and a right child with 2 positive and 4 negative samples. Using base-2 logarithms, calculate the information gain. Which is approximately correct?

For a single training sample, logistic regression predicts 𝑝(𝑦 = 1 ∣ 𝑥) = 𝜎(𝑤𝑇𝑥), where 𝜎(𝑧) = 1/1+𝑒 −𝑧. Given 𝑤𝑇𝑥 = ln3 and 𝑦 = 1, calculate the negative log-likelihood loss 𝐿 = −[ 𝑦ln𝑝 + (1 − 𝑦)ln(1 − 𝑝) ]. Give your answer up to 3 decimal places. Answer: ____________________

A hard-margin linear SVM has the separating hyperplane 2𝑥1 − 2𝑥2 + 4 = 0. The classifier is expressed in the standard form 𝑤𝑇𝑥 + 𝑏 = 0. What is the width of the margin between the two supporting hyperplanes?

Consider the following Python code: What will be the output?

Consider the recurrence 𝑇(𝑛) = 2𝑇(𝑛/2) + 𝑛with 𝑇(2) = 𝛩(1). Find the asymptotic log𝑛 time complexity using a suitable recurrence-solving method.

Consider the postfix expression 5 2 3 ∗ + 8 4 / − Evaluate it using a stack.

Consider an undirected graph with vertices 𝑉 = {1,2,3,4,5,6} and edges 𝐸 = {(1,2), (1,3), (2,4), (2,5), (3,5), (4,6), (5,6)}. BFS starts from vertex 1. Whenever multiple adjacent unvisited vertices are available, they are visited in increasing numerical order. Find the BFS traversal sequence.

Consider relations 𝑅(𝐴, 𝐵) and 𝑆(𝐵, 𝐶), where 𝑅 contains 10,000 tuples and 𝑆 contains 2,000 tuples. Both relations are sorted on attribute 𝐵, and the join is an equijoin on 𝐵. Which algorithm can perform the join with the lowest asymptotic merge-processing cost?

Consider the table Employee: ID Salary 1 50000 2 NULL 3 70000 4 NULL Which statements are correct?

Consider a relation 𝑅(𝐴, 𝐵, 𝐶) with the functional dependencies 𝐴𝐵 → 𝐶 and 𝐶 → 𝐵. Which is the highest normal form satisfied by the relation?

A relation Orders(CustomerID, Amount) contains the following rows: CustomerID Amount 1 100 1 200 2 300 2 100 3 500 Consider the query: Find the number of rows returned. Answer: ____________________

A CSP has variables 𝑋, 𝑌, 𝑍, each with the domain {1,2,3}, and the constraints 𝑋 ≠ 𝑌, 𝑌 ≠ 𝑍 and 𝑋 ≠ 𝑍. Which statement is correct?

A MAX root has two MIN children, evaluated from left to right. The left MIN node has terminal leaves 8, 6, 10, and the right MIN node has terminal leaves 5, 7, 9. Using alpha-beta pruning, determine the number of terminal leaf nodes evaluated. Answer: ____________________

Consider the Bayesian network 𝐴 → 𝐶 ← 𝐵, where 𝐴 and 𝐵 are independent. Which statement is correct?

Consider the formula (𝑃 → 𝑄) ∧ (𝑄 → 𝑅). Which of the following are logically entailed?

A certain amount of money is divided among A, B and C in the ratio 2: 3: 5. If A receives ₹20 less than B, and C receives ₹40 more than B, what is the total amount?