Courage : Bravery :: Yearning : ______ ?. Select the most appropriate option to complete the analogy.

We _______ tennis in the lawn when it suddenly started to rain. Select the most appropriate option to complete the above sentence.

A 4 × 4 digital image has pixel intensities (𝑈) as shown in the figure. The number of pixels with 𝑈≤4 is:

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?

A rectangle has a length L and a width W, where L > W. If the width, W, is increased by 10, which one of the following statements is correct for all values of L and W?

Column-I has statements made by Shanthala and Column-II has responses given by Kanishk. Identify the option that has the correct match between Column-I and Column-II.

Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let $a_1$ and $a_2$ (in years) denote the ages of the brothers and $a_1 < a_2$. Which one of the following statements is correct about their age on the day when they attain the same weight?

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

If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^x)^2}$ is?

The number of patients per shift (𝑋) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is ₹ 1000(𝑋−0.2), what is the average amount (in ₹) she has earned per shift in the past 100 shifts?

Suppose X and Y are random variables. The conditional expectation of X given Y is denoted by $E[X|Y]$. Then $E[E[X|Y]]$ equals

The number of additions and multiplications involved in performing Gaussian elimination on any $n \times n$ upper triangular matrix is of the order

The sum of the elements in each row of $A \in \mathbb{R}^{n \times n}$ is 1. If $B = A^3 - 2A^2 + A$, which one of the following statements is correct ( for $x \in \mathbb{R}^n$)?

Let $f(x) = \frac{e^x - e^{-x}}{2}$, $x \in \mathbb{R}$. Let $f^{(k)}(a)$ denote the $k$th derivative of $f$ evaluated at $a$. What is the value of $f^{(10)}(0)$?

Let p and q be any two propositions. Consider the following propositional statements: S1: $p \to q$, S2: $\lnot p \land q$, S3: $\lnot p \lor q$, S4: $\lnot p \lor \lnot q$, where $\land$ denotes conjunction (AND operation), $\lor$ denotes disjunction (OR operation), and $\lnot$ denotes negation (NOT operation). Which one of the following options is correct? (Note: $\equiv$ denotes logical equivalence)

If a relational decomposition is not dependency-preserving, which one of the following relational operators will be executed more frequently in order to maintain the dependencies?

Consider a hash table of size 10 with indices {0, 1, ..., 9}, with the hash function
$h(x) = 3x (\mod 10$),
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

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

Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a, b$ are two unknown constants. It is given that
$E[X] = 1$, $E[(X-E[X])Z] = -2$, $E[(X -E[X])^2] = 4$,
where $E[X]$ denotes the expectation of random variable $X$. The values of $a, b$ are

It is given that $P(X \geq 2) = 0.25$ for an exponentially distributed random variable $X$ with $E[X] = \frac{1}{\lambda}$, where $E[X]$ denotes the expectation of $X$. What is the value of $\lambda$?

Consider designing a linear classifier $y = \mathrm{sign}(f(x; w, b))$, where $f(x; w, b) = w^T x + b$, on a dataset $D = \{(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)\}$, with $x_i \in \mathbb{R}^d$ and $y_i \in \{+1, -1\}$ , $i = 1, 2, \ldots, N$. Recall that the sign function outputs $+1$ if the argument is positive, and $-1$ if the argument is non-positive. The parameters $w$ and $b$ are updated as per the following training algorithm:
$w_{\text{new}} = w_{\text{old}} + y_n x_n$, $b_{\text{new}} = b_{\text{old}} + y_n$
whenever $\mathrm{sign}(f(x_n; w_{\text{old}}, b_{\text{old}})) \neq y_n$. In other words, whenever the classifier wrongly predicts a sample $(x_n, y_n)$ from the dataset, $w_{\text{old}}$ gets updated to $w_{\text{new}}$, and likewise $b_{\text{old}}$ gets updated to $b_{\text{new}}$. Consider the case $(x_n, +1)$ with $f(x_n; w_{\text{old}}, b_{\text{old}}) < 0$. Then

Consider the following Python declarations of two lists. A = [1,2,3], B = [4,5,6]. Which one of the following statements results in A = [1, 2, 3, 4, 5, 6]?

Consider two functions $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (1, \infty)$. Both functions are differentiable at a point $c$. Which of the following functions is/are ALWAYS differentiable at $c$? The symbol $\cdot$ denotes product and the symbol $\circ$ denotes composition of functions.

Which of the following statements is/are correct?

Which of the following statements is/are correct in a Bayesian network?

For which of the following inputs does binary search take time $O(\log n)$ in the worst case?

Let $A = I_n - xx^T$, where $I_n$ is the $n \times n$ identity matrix and $x \in \mathbb{R}^n$, $x^T x = 1$. Which of the following options is/are correct?

Suppose that insertion sort is applied to the array [1, 3, 5, 7, 9, 11, x, 15, 13] and it takes exactly two swaps to sort the array. Select all possible values of $x$.

Let $C_1$ and $C_2$ be two sets of objects. Let $D(x, y)$ be a measure of dissimilarity between two objects $x$ and $y$. Consider the following definitions of dissimilarity between $C_1$ and $C_2$.
DIS-1: $D(C_1, C_2) = \max_{x \in C_1, y \in C_2} D(x, y)$,
DIS-2: $D(C_1, C_2) = \min_{x \in C_1, y \in C_2} D(x, y)$.
Which of the following statements is/are correct?

There are three boxes containing white balls and black balls.
Box-1 contains 2 black and 1 white balls.
Box-2 contains 1 black and 2 white balls.
Box-3 contains 3 black and 3 white balls.
In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is $\frac{1}{2}$, Box-2 is $\frac{1}{6}$, and Box-3 is $\frac{1}{3}$. A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ______. (Round off to two decimal places)

$\lim_{t\to+\infty} \left(\sqrt{t^2 + t} - t\right) =$ ______. (Round off to one decimal place)

On a relation named Loan of a bank: The following SQL query is executed.
SELECT L1.loan_number
FROM Loan L1
WHERE L1.amount > (SELECT MAX(L2.amount)
FROM Loan L2
WHERE L2.branch_name = 'SR Nagar').
The number of rows returned by the query is ______. (Answer in integer)

Given data {(-1, 1),(2, -5),(3, 5)} of the form $(x, y)$, we fit a model $y = wx$ using linear least-squares regression. The optimal value of $w$ is ______. (Round off to three decimal places)

The naive Bayes classifier is used to solve a two-class classification problem with class labels $y_1, y_2$. Suppose the prior probabilities are $P(y_1) = \frac{1}{3}$ and $P(y_2) = \frac{2}{3}$. Assuming a discrete feature space with $P(x|y_1) = \frac{3}{4}$ and $P(x|y_2) = \frac{1}{4}$ for a specific feature vector $x$. The probability of misclassifying $x$ is ______. (Round off to two decimal places)

Let $Y = Z^2$, where $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

Let $A \in \mathbb{R}^{n \times n}$ be such that $A^3 = A$. Which one of the following statements is ALWAYS correct?

Let $\{x_1, x_2, ..., x_n\}$ be a set of linearly independent vectors in $\mathbb{R}^n$. Let the $(i, j)$-th element of matrix $A \in \mathbb{R}^{n \times n}$ be given by $A_{ij} = x_i^T x_j$, $1 \leq i, j \leq n$. Which one of the following statements is correct?

Consider the cumulative distribution function (CDF) of a random variable $X$: $F_X(x) = \begin{cases} 0 & x < -1 \\ \frac{1}{4}(x+1)^2 & -1 \leq x \leq 1 \\ 1 & x \geq 1 \end{cases}$. The value of $P(X^2 \leq 0.25)$ is

A random variable $X$ is said to be distributed as $\text{Bernoulli}(\theta)$, denoted by $X \sim \text{Bernoulli}(\theta)$,
if $P(X = 1) = \theta$ and $P(X = 0) = 1 - \theta$
for $0 < \theta < 1$. Let $Y = \sum_{i=1}^{300} X_i$, where $X_i \sim \text{Bernoulli}(\theta)$ for $i = 1, 2, \ldots, 300$ are 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. Recall that $\phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt$.

For $x \in \mathbb{R}$, the floor function is denoted by $f(x) = \lfloor x \rfloor$ and defined as follows: $\lfloor x \rfloor = k$, where $k \leq x < k + 1$, where $k$ is an integer. Let $Y = \lfloor X \rfloor$, 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

Consider the neural network shown in the figure with
inputs $u, v$;
weights $a, b, c, d, e, f$;
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?

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 alpha-beta pruning algorithm?

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

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, $P(A) =$

Consider a fact table in an OLAP application: $\mathbf{Facts}(D_1, D_2, \text{val})$, where $D_1$ and $D_2$ are its dimension attributes and $\text{val}$ is a dependent attribute. Suppose attribute $D_1$ takes 3 values and $D_2$ takes 2 values, and all combinations of these values are present in the table $\mathbf{Facts}$. How many tuples are there in the result of the following query?
$\texttt{SELECT } D_1, D_2, \texttt{SUM(val) FROM Facts GROUP BY CUBE}(D_1, D_2)$

Consider the following Python code snippet. When the below program is executed, at the end, which of the following sets contains "this"?

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

Consider the function $f(x) = \frac{x^3}{3} + \frac{7x^2}{2} + 10x + \frac{133}{2}$, $x \in [-8, 0]$. Which of the following statements is/are 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 $A = x_1 x_1^T + ... + x_5 x_5^T$. Which of the following statements is/are 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?

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

Consider designing a linear binary classifier $f(x) = \mathrm{sign}(w^T x + b)$, $x \in \mathbb{R}^2$, on the following training data: $\mathbf{Class-1:}$ $\{ \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \end{bmatrix} \}$; $\mathbf{Class-2:}$ $\{ \begin{bmatrix} 0 \\ 0 \end{bmatrix} \}$. The hard-margin support vector machine (SVM) formulation is solved to obtain $w$ and $b$. Which of the following options 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 a two-class problem in $\mathbb{R}^d$ with class labels red and green. Let $\mu_{red}$ and $\mu_{green}$ be the means of the two classes. Given test sample $x \in \mathbb{R}^d$, a classifier calculates the squared Euclidean distance denoted by $\|\cdot\|^2$ between $x$ and the means of the two classes and assigns the class label that the sample $x$ is closest to. That is, the classifier computes $f(x) = \|\mu_{red} - x\|^2 - \|\mu_{green} - x\|^2$ and assigns the label red to $x$ if $f(x) < 0$, and green otherwise. Which of the following statements is/are correct?

Consider the following two relations, named Customer and Person, in a database. Which of the following statements is/are correct?

Consider a database relation $R$ with attributes $ABCDEFG$, and having the following functional dependencies: $A \to BCEF$, $E \to DG$, $BC \to A$. Which of the following statements is/are correct?

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: $a \to b \to c \to d$ is a shortest path between $a$ and $d$; $e \to f \to g \to h$ is a shortest path between $e$ and $h$; $a \to f \to c \to h$ is a shortest path between $a$ and $h$. Which of the following is/are NOT the edges of $G$?

Let $f : \mathbb{R} \to \mathbb{R}$ be such that $|f(x) - f(y)| \leq (x - y)^2$ for all $x, y \in \mathbb{R}$. Then $f(1) - f(0) =$ (Answer in integer).

Let $D = \{x^{(1)}, \dots, x^{(n)}\}$ 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 $\lambda_i = 100^{2-i}$, $1 \leq i \leq 100$. Let $u \in \mathbb{R}^{100}$ be the direction of maximum variance with ${u^T}u = 1$. The value of $\frac{1}{n} \sum_{i=1}^{n} (u^T x^{(i)})^2$ = ________ (Answer in integer).

A bag contains 5 white balls and 10 black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black balls drawn in the experiment. The expectation of $S_{100}$ denoted by $E[S_{100}] =$ ______. (Round off to one decimal place)

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

Consider the following Python code snippet. The value printed by the code snippet is ______.

Consider the following pseudocode. The value of sum output by a program executing the above pseudocode is ______.

Consider a directed graph $G = (V, E)$, where $V = \{0, 1, 2, ..., 100\}$ 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 ______.