List of practice Questions

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 \[ P(x | y_1) = \frac{3}{4} \quad \text{and} \quad P(x | y_2) = \frac{1}{4} \] for a specific feature vector \( x \). The probability of misclassifying \( x \) is: (Round off to two decimal places)

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

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

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 \( E[S_{100}] \) is \( \ \ \ \ \ \) (Round off to one decimal place).

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

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

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:

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 \textit{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:}

Describe any three features of the multiparty system.

A vertically held bar magnet is dropped along the axis of a copper ring having a cut as shown in the diagram. The acceleration of the falling magnet is:

Why was the Indian subcontinent significant to trade networks before European intervention? Explain.

Consider the discrete-time system below with input \( x[n] \) and output \( y[n] \). In the figure, \( h_1[n] \) and \( h_2[n] \) denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also, \( \delta[n] \) is the unit impulse, and \( b>0 \).
Assuming \( h_2[n] \neq \delta[n] \), the overall system (denoted by the dashed box) is _________.
\includegraphics[width=0.5\linewidth]{17.png}

Kavya _________ go to work yesterday as she _________ feeling well. Select the most appropriate option to complete the above sentence.

Good : Evil :: Genuine : _________ Select the most appropriate option to complete the analogy.

The relationship between two variables \( x \) and \( y \) is given by \( x + py + q = 0 \) and is shown in the figure. Find the values of \( p \) and \( q \). Note: The figure shown is representative.

Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in Column-I. Identify the option that has the most appropriate match between Column-I and Column-II. Note: The figures shown are representative.

What is the value of \( \left( \frac{3^{81}}{27^4} \right)^{\frac{1}{3}} \)?

Identify the option that has the most appropriate sequence such that a coherent paragraph is formed:
P. It is because deer, like most of the animals that tigers normally prey on, run much faster! It simply means, another day of empty stomach for the big cats.
Q. Tigers spend most of their life searching for food.
R. If they trace the scent of deer, tigers follow the trail, chase the deer for a mile or two in the dark, and yet may not catch them.
S. For several nights, they relentlessly prowl through the forest, hunting for a trail that may lead to their prey.

In the given figure, EF and HJ are coded as 30 and 80, respectively. Which one among the given options is most appropriate for the entries marked (i) and (ii)?

Scores obtained by two students P and Q in seven courses are given in the table below. Based on the information given in the table, which one of the following statements is INCORRECT?

Spheres of unit diameter are centered at \( (l, m, n) \), where \( l, m, \) and \( n \) take every possible integer value. The distance between two spheres is computed from the center of one sphere to the center of another sphere. For a given sphere, \( x \) is the distance to its nearest sphere and \( y \) is the distance to its next nearest sphere. The value of \( \frac{y}{x} \) is:

In triangle \( PQR \), the lengths of \( PT \) and \( TR \) are in the ratio \( 3:2 \). ST is parallel to QR. Two semicircles are drawn with \( PS \) and \( PQ \) as diameters, as shown in the figure. Which one of the following statements is true about the shaded area \( PQS \)? (Note: The figure shown is representative.)

Consider the set \( S \) of points \( (x, y) \in \mathbb{R}^2 \) which minimize the real-valued function \( f(x, y) = (x + y - 1)^2 + (x + y)^2 \). Which of the following statements is true about the set \( S \)?

Let \( \mathbf{A} = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix} \), and \( \mathbf{b} = \begin{bmatrix} \frac{1}{3} \\ -\frac{1}{3} \\ 0 \end{bmatrix} \). Then, the system of linear equations \( \mathbf{A} \mathbf{x} = \mathbf{b} \) has

Let \( P = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and let \( I \) be the identity matrix. Then \( P^2 \) is equal to