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

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 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:

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 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?

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?

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?

Consider the neural network shown in the figure with \[ \text{inputs: } u = 2, \, v = 3 \] \[ \text{weights: } a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 \] \[ \text{output: } y \] R denotes the ReLU function, $ R(x) = \max(0, x) $.

Given $ u = 2, v = 3, a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 $, which one of the following is correct?

For $ x \in \mathbb{R} $, the floor function is denoted by $ f(x) = \lfloor x \rfloor $ and defined as follows \[ \lfloor x \rfloor = k, \quad k \leq x<k + 1, \] where $ k $ is an integer. Let $ Y = |X| $, where $ X $ is an exponentially distributed random variable with mean $ \frac{1}{\ln 10} $, where $ \ln $ denotes natural logarithm. For any positive integer $ \ell $, one can write the probability of the event $ Y = \ell $ as follows: \[ P(Y = \ell) = q^\ell (1 - q) \] The value of $ q $ is:

A random variable $ X $ is said to be distributed as $ \text{Bernoulli}(\theta) $, denoted by $ X \sim \text{Bernoulli}(\theta) $, if \[ P(X = 1) = \theta, \quad P(X = 0) = 1 - \theta \] for $ 0<\theta<1 $. Let $ Y = \sum_{i=1}^{300} X_i $, where $ X_i \sim \text{Bernoulli}(\theta) $, $ i = 1, 2, \dots, 300 $ be independent and identically distributed random variables with $ \theta = 0.25 $. The value of $ P(60 \leq Y \leq 90) $, after approximation through the Central Limit Theorem, is given by

Consider the cumulative distribution function (CDF) of a random variable $ X $:

\[ F_X(x) = \begin{cases} 0 & \text{if } x \leq -1 \\ \frac{1}{4}(x + 1)^2 & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } x \geq 1 \end{cases} \]

The value of $ P(X^2 \leq 0.25) $ is: