List of top Questions asked in GATE Data Science and Artificial Intelligence

Consider designing a linear binary classifier $ f(x) = \text{sign}(w^T x + b), x \in \mathbb{R}^2 $ on the following training data:

Class-2: $ \left\{ \left( \begin{array}{c} 0 \\ 0 \end{array} \right) \right\} $

Hard-margin support vector machine (SVM) formulation is solved to obtain $ w $ and $ b $. Which of the following options is/are correct?

In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of $ P $, $ Q $, and $ R $?

Let $ X $ be a continuous random variable whose cumulative distribution function (CDF) $ F_X(x) $, for some $ t $, is given as follows:
\[ F_X(x) = \begin{cases} 0 & \text{if } x \leq t \\ \frac{x - t}{4 - t} & \text{if } t \leq x \leq 4 \\ 1 & \text{if } x \geq 4 \end{cases} \]
If the median of $ X $ is 3, then what is the value of $ t $?

Consider a hash table of size 10 with indices $ \{0, 1, \dots, 9\} $, with the hash function \[ h(x) = 3x \, (\text{mod} \, 10), \] where linear probing is used to handle collisions. The hash table is initially empty and then the following sequence of keys is inserted into the hash table: 1, 4, 5, 6, 14, 15. The indices where the keys 14 and 15 are stored are, respectively:

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $ r $ cm as shown in the figure. The side of the dodecagon is $ d $ cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side $ r $ cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:

Consider the following Python code snippet.

def f(a, b):
    if (a == 0):
        return b
    if (a % 2 == 1):
        return 2 * f((a - 1) / 2, b)
    return b + f(a - 1, b)

print(f(15, 10))

The value printed by the code snippet is 160 (Answer in integer).

Consider the following pseudocode.

Create empty stack S
Set x = 0, flag = 0, sum = 0
Push x onto S
while (S is not empty){
    if (flag equals 0){
        Set x = x + 1
        Push x onto S
    }
    if (x equals 8):
        Set flag = 1
    if (flag equals 1){
        x = Pop(S)
        if (x is odd):
            Pop(S)
        Set sum = sum + x
    }
}
Output sum

The value of $ sum $ output by a program executing the above pseudocode is:

Given the equation: $$ 3^{x^2} = 27 \times 9^x $$ Find the value of: $$ \frac{2^{x^2}}{(2^x)^2} $$

Consider a directed graph $ G = (V,E) $, where $ V = \{0,1,2,\dots,100\} $ and
\[ E = \{(i,j) : 0 < j - i \leq 2, \text{ for all } i,j \in V \}. \] Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is:

Consider the following tables, Loan and Borrower, of a bank.

Query: \[ \pi_{\text{branch\_name}, \text{customer\_name}} (\text{Loan} \bowtie \text{Borrower}) \div \pi_{\text{branch\_name}}(\text{Loan}) \] where $ \bowtie $ denotes natural join. The number of tuples returned by the above relational algebra query is 1 (Answer in integer).

Let $ D = \{ x^{(1)}, x^{(2)}, \dots, x^{(n)} \} $ be a dataset of $ n $ observations where each $ x^{(i)} \in \mathbb{R}^{100} $. It is given that

\[ \sum_{i=1}^{n} x^{(i)} = 0. \] The covariance matrix computed from $ D $ has eigenvalues $ \lambda_i = 100^2 - i $, for $ 1 \leq i \leq 100 $. Let $ u \in \mathbb{R}^{100} $ be the direction of maximum variance with $ u^T u = 1 $. The value of

\[ \frac{1}{n} \sum_{i=1}^{n} \left( u^T x^{(i)} \right)^2 = \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Let $ G $ be a simple, unweighted, and undirected graph. A subset of the vertices and edges of $ G $ are shown below.

It is given that $ a - b - c - d $ is a shortest path between $ a $ and $ d $; $ e - f - g - h $ is a shortest path between $ e $ and $ h $; $ a - f - c - h $ is a shortest path between $ a $ and $ h $. Which of the following is/are NOT the edges of $ G $?

Consider a database relation $ R $ with attributes $ A, B, C, D, E, F, G $, and having the following functional dependencies: \[ A \rightarrow BCEF \quad E \rightarrow DG \quad BC \rightarrow A \] Which of the following statements is/are correct?

Consider the following two relations, named Customer and Person, in a database:

\[ \text{Person} \left( \begin{array}{l} \text{aadhaar CHAR(12) PRIMARY KEY,} \\ \text{name VARCHAR (32);} \end{array} \right) \]

\[ \text{Customer} \left( \begin{array}{l} \text{name VARCHAR (32),} \\ \text{email VARCHAR(32) PRIMARY KEY,} \\ \text{phone CHAR(10),} \\ \text{aadhaar CHAR(12),} \\ \text{FOREIGN KEY (aadhaar) REFERENCES Person (aadhaar);} \end{array} \right) \]

Which of the following statements is/are correct?

Consider a coin-toss experiment where the probability of head showing up is $ p $. In the $ i $-th coin toss, let $ X_i = 1 $ if head appears, and $ X_i = 0 $ if tail appears. Consider \[ \hat{p} = \frac{1}{n} \sum_{i=1}^n X_i, \] where $ n $ is the total number of independent coin tosses. Which of the following statements is/are correct?

An $ n \times n $ matrix $ A $ with real entries satisfies the property: \[ \|Ax\|^2 = \|x\|^2, \quad \text{for all } x \in \mathbb{R}^n, \] where $ \| \cdot \| $ denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice-differentiable function, and suppose its second derivative satisfies $ f''(x)>0 $ for all $ x \in \mathbb{R} $. Which of the following statements is/are ALWAYS correct?

Let $ x_1, x_2, x_3, x_4, x_5 $ be a system of orthonormal vectors in $ \mathbb{R}^{10} $. Consider the matrix \[ A = x_1 x_1^T + x_2 x_2^T + x_3 x_3^T + x_4 x_4^T + x_5 x_5^T. \] Which of the following statements is/are correct?

Consider the function $ f(x) = \frac{x^3}{3} + \frac{7}{2}x^2 + 10x + \frac{133}{2} $, $ x \in [-8, 0] $. Which of the following statements is/are correct?

Which of the following statements is/are correct about the rectified linear unit (ReLU) activation function defined as ReLU(x) = max(x, 0), where $ x \in \mathbb{R} $?

Consider the following Python code snippet. \[ A = \{\text{"this"}, \text{"that"}\}, \quad B = \{\text{"that"}, \text{"other"}\}, \quad C = \{\text{"other"}, \text{"this"}\} \] \[ \text{while "other" in C:} \] \[ \quad \text{if "this" in A:} \] \[ \quad \quad A, B, C = A - B, B - C, C - A \] \[ \quad \text{if "that" in B:} \] \[ \quad \quad A, B, C = C | A, A | B, B | C \] When the above program is executed, at the end, which of the following sets contains "this"?

Consider a fact table in an OLAP application: Facts(D1, D2, val), where D1 and D2 are its dimension attributes and val is a dependent attribute. Suppose attribute D1 takes 3 values and D2 takes 2 values, and all combinations of these values are present in the table Facts. How many tuples are there in the result of the following query? \[ \text{SELECT D1, D2, sum(val)} \] \[ \text{FROM Facts} \] \[ \text{GROUP BY CUBE (D1, D2);} \]

A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6. An event $ A $ represents the set of all outcomes where at least one of the dice shows a 1. Then, $ P(A) = $

The state graph shows the action cost along the edges and the heuristic function $ h $ associated with each state. Suppose the A algorithm is applied on this state graph using a priority queue to store the frontier. In what sequence are the nodes expanded?

Consider game trees Tree-1 and Tree-2 as shown. The first level is a MAX agent and the second level is a MIN agent. The value in the square node is the output of the utility function.

For what ranges of $ x $ and $ y $, the right child of node B and the right child of node E will be pruned by the alpha-beta pruning algorithm?