GATE DA previous-year questions

2026 · session-8General AptitudeVerbal AptitudeMCQ · 1 mark

Verbosity : Brevity :: Insolence : __________ Choose the word that best fills the blank.

2026 · session-8General AptitudeQuantitative AptitudeMCQ · 1 mark

The product of the digits of a three-digit number is 70. The sum of the digits of this three-digit number is _____

2026 · session-8General AptitudeSpatial AptitudeMCQ · 1 mark

The four pieces of a puzzle are shown in the figure below. Which one of the figures labelled as P, Q, R, and S can be constructed by using each of the four pieces only once without overlaps?

2026 · session-8General AptitudeQuantitative AptitudeMCQ · 1 mark

Consider two distinct positive real numbers \(m, n\), with \(m > n\). Let \(x = n^{\log_{10}(m)}\) and \(y = m^{\log_{10}(n)}\). The relation between \(x\) and \(y\) is _______.

2026 · session-8General AptitudeVerbal AptitudeMCQ · 1 mark

'If his latest movie had been a commercial success, the actor would have made enough money to sponsor his next movie.' Based only on the above sentence, which one of the following statements is true?

2026 · session-8General AptitudeVerbal AptitudeMCQ · 2 marks

'My friend and I parted __ the door __ the cabin that I had rented __ the night.' Choose the option with the correct sequence of words to fill the blanks.

2026 · session-8General AptitudeAnalytical AptitudeMCQ · 2 marks

Five integers are picked from 0 to 20, with possible repetitions, such that their mean is 12, median is 18, and they have a single mode of 20. Ignoring permutations, the number of ways to pick these five integers is ___…

2026 · session-8General AptitudeAnalytical AptitudeMCQ · 2 marks

Rishi and Swathi are students of Class 5. Pavan and Tanvi are students of Class 4. Rishi and Pavan are boys. Swathi and Tanvi are girls. The four students played a total of three games of chess. The games were played one…

2026 · session-8General AptitudeQuantitative AptitudeMCQ · 2 marks

\(P, Q, R, S, X,\) and \(Y\) are distinct single-digit whole numbers taking values from 0 to 9. \(PQ\) is a two-digit number with \(Q\) in the units place and \(P\) in the tens place. Similarly, \(RS\) is a two-digit num…

2026 · session-8General AptitudeSpatial AptitudeMCQ · 2 marks

In the given figure, \(P\), \(Q\), and \(R\) are three points on a circle of radius 10 cm with \(O\) as its center, \(\overline{PQ} = \overline{RQ}\), and \(\angle PQR = 45^{\circ}\). The figure is representative. The ar…

2026 · session-8Machine LearningPrincipal Component AnalysisMCQ · 1 mark

For a classification problem, Principal Component Analysis (PCA) has been used to reduce the dimensionality of a feature space from 100 to 10. Which of the following options is true about the angle \(\theta\) between the…

2026 · session-8Machine LearningLeave-One-Out Cross-ValidationMCQ · 1 mark

Consider that you are training a classifier for a 10-class classification problem. Each input is represented as a 512-dimensional vector. There are 1000 samples, out of which the first 100 will be used for testing. Let L…

2026 · session-8AIUninformed SearchMCQ · 1 mark

Which of the following algorithms is NOT an example of uninformed search?

2026 · session-8AIPredicate LogicMCQ · 1 mark

Which of the following statements is NOT true? (The names of the predicates are intuitive.)

2026 · session-8Programming, Data Structures and AlgorithmsQuicksortMCQ · 1 mark

Consider that the quick sort algorithm is used to sort an array of \(n\) distinct randomly ordered elements. In every call, the pivot is chosen as the first element of the current subarray. Let \(T(n)\) denote the expect…

2026 · session-8Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 1 mark

Consider the given Python program. def append_to_lst(val, lst=[]): lst.append(val) return lst print(append_to_lst(1)) print(append_to_lst(2)) print(append_to_lst(3, [])) Which of the following is the correct output of…

2026 · session-8Database Management and WarehousingFunctional Dependencies and Normal FormMCQ · 1 mark

Let \(R(A,B,C,D,E)\) be a relational schema with functional dependency set \(F = \{A \rightarrow BC,\ CD \rightarrow E,\ E \rightarrow A\}\). Which of the following statements is correct?

2026 · session-8Database Management and WarehousingMultidimensional Data ModelsMCQ · 1 mark

Consider that the visualization of a 3-dimensional data cube is showing Sales Quantity for each combination of the attributes Product Type, Month and Country. From this, if we want to further visualize the Sales Quantity…

2026 · session-8Probability and StatisticsCounting and ProbabilityMCQ · 1 mark

Let \(M\) be a randomly chosen non-empty subset of \(S = \{1,2,3,\ldots,2026\}\). Which of the following is the probability that the product of all the elements of \(M\) is even?

2026 · session-8Probability and StatisticsCounting and CombinationsMCQ · 1 mark

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation \(n_1 + n_2 + n_3 + n_4 = 20\). Which of the following is the probability that all of \(n_1, n_2, n_3, n_4\) in t…

2026 · session-8Linear AlgebraMatricesMCQ · 1 mark

Let \(M = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) be a \(2 \times 2\) matrix, where \(\theta = \dfrac{2\pi}{5}\), and \(I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\). Wh…

2026 · session-8Linear AlgebraSubspacesMCQ · 1 mark

Consider a set \(S_1 = \{x = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \mid x^Tx \leq 16\}\). Let \(S_2\) be another set which is a subspace of \(\mathbb{R}^3\) with dimension two. Which of the following gives the area of \(S_1…

2026 · session-8Machine LearningSupervised and Unsupervised LearningMSQ · 1 mark

In the following table, the Task column lists a few tasks related to machine learning. The Algorithm column lists a few algorithms. Each entry "t" from the Task column is to be matched with an appropriate entry "a" from …

2026 · session-8AIPropositional LogicMSQ · 1 mark

Sentence \(X\) is said to entail Sentence \(Y\) if whenever \(X\) is \(TRUE\), \(Y\) also must hold \(TRUE\). Which of the following statements is/are correct if \(X\) entails \(Y\)?

2026 · session-8Programming, Data Structures and AlgorithmsTreesMSQ · 1 mark

You are given the following Pre-order and In-order traversals of a Binary Tree \(T\) with nodes E, F, G, P, Q, R, S. Pre-order: P Q S E R F G In-order: S Q E P F R G Which of the following statements is/are true about …

2026 · session-8Database Management and WarehousingRelational Model and Integrity ConstraintsMSQ · 1 mark

Consider two relations \(r\) and \(s\) defined on the relational schemas \(R(A,B)\) and \(S(E,C)\), respectively. \(A\) is the primary key of \(R\) and \(E\) is a foreign key of \(S\) referencing \(A\) in \(R\). Which o…

2026 · session-8Calculus and OptimizationMaxima and MinimaMSQ · 1 mark

Let \(f(x) = x^3 - 3x^2 + 2\) be a function defined on \((-1, 3]\). Which of the following statements is/are correct?

2026 · session-8Probability and StatisticsNormal DistributionMSQ · 1 mark

Suppose a random variable \(Z\) follows \(Normal(\mu=0, \sigma^2=1)\) distribution with probability density function \(g(z)\) and cumulative distribution function \(G(z)\). Another random variable \(Y\) follows \(t_1\) d…

2026 · session-8Calculus and OptimizationOptimization Involving a Single VariableNAT · 1 mark

Consider that for a supervised learning task, the objective function being minimized is \(f_w(x) = wx\), where \(x \in \mathbb{R}\) is the input and \(w \in \mathbb{R}\) is the parameter. Stochastic Gradient Descent with…

2026 · session-8AIAdversarial SearchNAT · 1 mark

Consider the game tree for a two-player turn-taking minimax game as shown in the figure. The value of a terminal node represents the utility of the game state if the game ends there. The numbers written next to the edges…

2026 · session-8Programming, Data Structures and AlgorithmsBinary SearchNAT · 1 mark

Let \(A\) be a sorted array containing 1000 distinct integers. You perform a recursive binary search on \(A\) to find an element \(y\). Suppose each comparison checks whether the middle element computed during the curren…

2026 · session-8Database Management and WarehousingIndexingNAT · 1 mark

In a relational database, a B+ Tree Index is to be constructed for a relation on a key field. In a B+ Tree, a Node Pointer points to a sub-tree and a Data Record Pointer points to a block of database records. Let, Node …

2026 · session-8Probability and StatisticsCounting and PermutationsNAT · 1 mark

The number of bijections \(f(\cdot)\) from the set \(S = \{1, 2, 3, 4\}\) to itself such that \(f(f(n)) = n\), for all \(n \in S\), is ________. (Answer in integer)

2026 · session-8Probability and StatisticsExponential DistributionNAT · 1 mark

Let \(X\) be an exponentially distributed random variable with mean \(\lambda (> 0)\). If \(P(X > 5) = 0.35\), then the conditional probability \(P(X > 10 \mid X > 5)\) is ________. (Rounded off to two decimal places)

2026 · session-8Calculus and OptimizationLimitNAT · 1 mark

The value of \(\displaystyle\sum_{i=0}^{\infty}\sum_{j=1}^{\infty} 2^{-i}\,3^{-j}\) is ________. (Answer in integer)

2026 · session-8Machine LearningHierarchical ClusteringMCQ · 2 marks

Let four points in three-dimensional space be: \(P1: [2, 3, -1]\), \(P2: [3, 1, 1]\), \(P3: [5, -2, 3]\) and \(P4: [3, 3, 3]\). Hierarchical Agglomerative Clustering is used to cluster the above points. If Manhattan Di…

2026 · session-8Machine LearningRidge RegressionMCQ · 2 marks

Which of the following statements is true for Ridge Regression?

2026 · session-8AIPropositional LogicMCQ · 2 marks

Assume that a Creative (\(C\)) person will Succeed (\(S\)) if the person is also Disciplined (\(D\)), but will not succeed otherwise. Now, consider the following statements: (i) \(C \wedge S \Leftrightarrow D\) (ii) \(…

2026 · session-8Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 2 marks

A recursive function in Python is given. def mystery(n): if n <= 0: return 1 else: return mystery(n-1) + mystery(n-2) Now, consider the following function call: mystery(4) Assume that a typical runtime stack is used …

2026 · session-8Programming, Data Structures and AlgorithmsGraph TraversalsMCQ · 2 marks

Consider a directed graph \(G = (V, E)\), where \(V\) is the finite set of vertices and \(E\) is the set of directed edges between the vertices. \(G\) may contain cycles but there is no self-loop. Further, \(G\) may not …

2026 · session-8Database Management and WarehousingB+ Tree IndexingMCQ · 2 marks

Consider a B+ Tree where the maximum number of key values in each leaf node is 2 and the maximum number of pointers in each non-leaf node is 3. Let the content of the B+ Tree be as shown in the figure. Which of the foll…

2026 · session-8Database Management and WarehousingRelational AlgebraMCQ · 2 marks

Consider the given relations \(X\), \(Y\) and \(Z\). The relation \(X\) has three columns P, Q and R. The relation \(Y\) has three columns P, Q and S. The relation \(Z\) has two columns P and T. Relation X: | P | Q | R…

2026 · session-8Database Management and WarehousingConcept Hierarchies and Data CuboidsMCQ · 2 marks

Consider the concept hierarchies as shown in the figure. Which of the following options denotes the total number of possible data cuboids from these concept hierarchies?

2026 · session-8Probability and StatisticsIndependent Random VariablesMCQ · 2 marks

Let \(X\) and \(Y\) be two independent random variables. \(X\) follows \(Bernoulli(p=0.3)\) distribution and \(Y\) follows \(Normal(\mu=0, \sigma^{2}=100)\) distribution. Which of the following options is the variance o…

2026 · session-8Probability and StatisticsPoisson DistributionMCQ · 2 marks

Let \[ L = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n} n^{k}}{k!} \] Which of the following is the value of \(L\)?

2026 · session-8Linear AlgebraEigenvalues and EigenvectorsMCQ · 2 marks

Let \(\gamma_1, \gamma_2, \gamma_3\) be the eigenvalues of the matrix \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix} \] where \(t \in [-\pi, \pi]\) is in radians. Which one of …

2026 · session-8Machine LearningClassification ProblemsMSQ · 2 marks

Consider that 20 stories of Author X and 10 stories of Author Y were kept together without mentioning the names of the authors. A classifier was then asked to predict the author (X or Y) of each of these stories. Let, ou…

2026 · session-8AIPredicate LogicMSQ · 2 marks

Let \(P(x)\) be a predicate. Which of the following statements is/are NOT valid in first-order logic?

2026 · session-8Programming, Data Structures and AlgorithmsBubble Sort and Insertion SortMSQ · 2 marks

Consider the problem of sorting the given array in ascending order: \(P = [1, 2, 3, 5, 4]\) Consider two sorting algorithms Bubble Sort (BS) and Insertion Sort (IS). Let \(N_1\) be the total number of comparisons done…

2026 · session-8Programming, Data Structures and AlgorithmsProgramming in PythonMSQ · 2 marks

Consider the given Python program. def outer(): x = [] def inner(val): x.append(val) return x return inner f1 = outer() f2 = outer() print(f1(10)) # Line P print(f1(20)) # Line Q print(f2(30)) # Line R print(f1(4…

2026 · session-8Database Management and WarehousingSQLMSQ · 2 marks

Consider a table Employee(EmpID, TeamID), where the column EmpID (ID of an employee) is the primary key. The column TeamID denotes the team ID of the team of which the employee is a member. TeamID is a NOT NULL column. …

2026 · session-8Linear AlgebraProjection MatrixMSQ · 2 marks

Let \(M = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\). Which of the following options …

2026 · session-8Probability and StatisticsChi-Squared DistributionMSQ · 2 marks

Let \(X_1, X_2, \ldots, X_n\) be \(n\) independent random variables. Each of the random variables follows \(Normal(\mu=0, \sigma^2=1)\) distribution. Define \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\). Which of the foll…

2026 · session-8Probability and StatisticsCumulative Distribution FunctionMSQ · 2 marks

Let \(X\) be a discrete valued random variable with cumulative distribution function \(F(x)\). Which of the following statements is/are correct?

2026 · session-8Machine LearningRidge RegressionNAT · 2 marks

Consider that Linear Ridge Regression is being used to learn a prediction function \(y_{pred} = w^T x\), where \(w, x \in \mathbb{R}^2\) and Mean Absolute Error (MAE) is used to measure the prediction error. A weight of …

2026 · session-8Machine LearningMulti-Layer PerceptronNAT · 2 marks

Consider a fully-connected feed-forward multi-layer perceptron. It has 30 neurons in the input layer, followed by two hidden layers and an output layer. The first hidden layer has 4 neurons and the second hidden layer ha…

2026 · session-8Probability and StatisticsBayes TheoremNAT · 2 marks

A clinic specializes in testing for a disease \(D\). The result of the test can be either positive or negative. A study revealed that if a person suffers from the disease \(D\), the test result in that clinic comes out …

2026 · session-8Programming, Data Structures and AlgorithmsProgramming in PythonNAT · 2 marks

Consider the given Python program. def fun(L, i=0):     if i >= len(L)-1:         return 0     if L[i] > L[i+1]:         L[i+1], L[i] = L[i], L[i+1]         return 1+fun(L, i+1)     else:         return fun(L, i+1) dat…

2026 · session-8Database Management and WarehousingTuple CalculusNAT · 2 marks

Let there be two relations \(X\) and \(Y\) as shown. \(X\) has three columns \(P\), \(Q\) and \(R\). \(Y\) has two columns \(P\) and \(S\). Relation X: | P | Q | R | | --- | --- | --- | | P1 | Q1 | R1 | | P2 | Q2 | R2 …

2026 · session-8Database Management and WarehousingSQLNAT · 2 marks

Let Account be a relation as shown. Table: Account | AccNo | Balance | A1 | 5000 | A2 | 5000 | A3 | 10000 | A4 | 15000 | A5 | 18000 Consider the given SQL query. SELECT AccNo FROM Account AS A WHERE (SELECT COUN…

2026 · session-8Database Management and WarehousingER-ModelNAT · 2 marks

Consider an ER model with the entities E1(A11, A12, A13) and E2(A21, A22, A23), where A11, A12, A13 are the attributes of E1, and A21, A22, A23 are the attributes of E2. Let A22 be a multi-valued attribute. A11 and A21 a…

2026 · session-8Probability and StatisticsVariance and Standard DeviationNAT · 2 marks

For a given data set \(\{x_1, x_2, \ldots, x_n\}\), where \(n = 100\), it is known that \[ \frac{1}{2000}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i - x_j)^2 = 99 \] Let us denote \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\) The va…

2026 · session-8Probability and StatisticsConditional DistributionNAT · 2 marks

Let \(X\) be a random variable that follows \(Uniform(-1,1)\) distribution. The conditional distribution of the random variable \(Y\) given \(X = x\) is the \(Uniform(x^2-0.1, x^2+0.1)\) distribution. The value of \(cor…

2026 · session-8Probability and StatisticsBernoulli DistributionNAT · 2 marks

Let \(A_{5\times5}\) be a matrix such that each of its elements follows \(Bernoulli(p=0.50)\) distribution independently. The probability that the row-sum of the second row and the column-sum of the third column are bot…

2026 · session-8Linear AlgebraQuadratic Forms and Projection MatrixNAT · 2 marks

Let \(A = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\). The value of \(\max_{S} x^TAx\)…

2025 · session-5General AptitudeVerbal AptitudeMCQ · 1 mark

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

2025 · session-5General AptitudeVerbal AptitudeMCQ · 1 mark

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

2025 · session-5General AptitudeSpatial AptitudeMCQ · 1 mark

A 4 × 4 digital image has pixel intensities (U) as shown in the figure. The number of pixels with \( U \leq 4 \) is: * (1) 3 * (2) 8 * (3) 11 * (4) 9

2025 · session-5General AptitudeAnalytical AptitudeMCQ · 1 mark

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 \( R \)?

2025 · session-5General AptitudeQuantitative AptitudeMCQ · 1 mark

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 \(W\)? Select the most appropriate…

2025 · session-5General AptitudeVerbal AptitudeMCQ · 2 marks

Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.

2025 · session-5General AptitudeQuantitative AptitudeMCQ · 2 marks

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 ye…

2025 · session-5General AptitudeSpatial AptitudeMCQ · 2 marks

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…

2025 · session-5General AptitudeQuantitative AptitudeMCQ · 2 marks

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:

2025 · session-5General AptitudeQuantitative AptitudeMCQ · 2 marks

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 \(Rs. 1000(X - 0.2)\), what is the average amount (in Rs.) she has earned per shift in th…

2025 · session-5Probability and StatisticsConditional ExpectationMCQ · 1 mark

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:

2025 · session-5Linear AlgebraGaussian EliminationMCQ · 1 mark

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

2025 · session-5Linear AlgebraSystems of Linear EquationsMCQ · 1 mark

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 \( x \in \mathbb{R}^n \))?

2025 · session-5Calculus and OptimizationDifferentiabilityMCQ · 1 mark

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 \( f^{(10)}(0)? \) (Note: \( ! \) denotes factor…

2025 · session-5AIPropositional LogicMCQ · 1 mark

Let \( p \) and \( q \) be any two propositions. Consider the following propositional statements. S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \) where \( \land …

2025 · session-5Database Management and WarehousingNormal Form and Functional DependenciesMCQ · 1 mark

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?

2025 · session-5Database Management and WarehousingRelational AlgebraMCQ · 1 mark

Consider the following three relations: Car (model, year, serial, color) Make (maker, model) Own (owner, serial) A tuple in Car represents a specific car of a given model, made in a given year, with a serial number and a…

2025 · session-5Programming, Data Structures and AlgorithmsHash TablesMCQ · 1 mark

Consider a hash table of size 10 with indices \( \{0, 1, \dots, 9\} \), 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…

2025 · session-5Probability and StatisticsCumulative Distribution FunctionMCQ · 1 mark

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 & if x \leq t \frac{x - t}{4 - t} & if t \leq x \l…

2025 · session-5Probability and StatisticsNormal DistributionMCQ · 1 mark

Let \( X = aZ + b \), where \( Z \) is a standard normal random variable, and \( a, b \) are two unknown constants. It is given that \[ E[X] = 1, \quad E[(X - E[X]) | Z] = -2, \quad E[(X - E[X])^2] = 4, \] where \( E[X] …

2025 · session-5Probability and StatisticsExponential DistributionMCQ · 1 mark

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…

2025 · session-5Machine LearningClassification ProblemsMCQ · 1 mark

Consider designing a linear classifier \[ y = sign(f(x; w, b)), \quad f(x; w, b) = w^T x + b \] on a dataset \( D = \{(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)\}, x_i \in \mathbb{R}^d, y_i \in \{+1, -1\}, i = 1, 2, \dots…

2025 · session-5Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 1 mark

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

2025 · session-5Calculus and OptimizationDifferentiabilityMSQ · 1 mark

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 \(…

2025 · session-5Linear AlgebraOrthonormal Vectors and Linear IndependenceMSQ · 1 mark

Which of the following statements is/are correct?

2025 · session-5AIBayesian Networks and InferenceMSQ · 1 mark

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

2025 · session-5Programming, Data Structures and AlgorithmsBinary SearchMSQ · 1 mark

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

2025 · session-5Linear AlgebraMatrices and EigenvaluesMSQ · 1 mark

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?

2025 · session-5Programming, Data Structures and AlgorithmsInsertion SortMSQ · 1 mark

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 \( x \).

2025 · session-5Machine LearningHierarchical Clustering: LinkageMSQ · 1 mark

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_…

2025 · session-5Probability and StatisticsBayes TheoremNAT · 1 mark

There are three boxes containing white balls and black balls. Box-1 contains 2 black and 1 white ball. Box-2 contains 1 black and 2 white balls. Box-3 contains 3 black and 3 white balls. In a random experiment, one of th…

2025 · session-5Calculus and OptimizationLimitNAT · 1 mark

Evaluate the following limit: \[ \lim_{t \to \infty} \sqrt{t^2 + t - t} \]

2025 · session-5Database Management and WarehousingSQLNAT · 1 mark

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');

2025 · session-5Machine LearningSimple Linear RegressionNAT · 1 mark

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)

2025 · session-5Machine LearningNaive Bayes ClassifierNAT · 1 mark

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 d…

2025 · session-5Probability and StatisticsChi-Squared DistributionMCQ · 2 marks

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

2025 · session-5Linear AlgebraMatrices and EigenvaluesMCQ · 2 marks

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

2025 · session-5Linear AlgebraRank and Quadratic FormsMCQ · 2 marks

Let \( \{ x_1, x_2, \dots, x_n \} \) 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 \( A_{ij} = x_i^T x_j \), wher…

2025 · session-5Probability and StatisticsCumulative Distribution FunctionMCQ · 2 marks

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

2025 · session-5Probability and StatisticsBernoulli DistributionMCQ · 2 marks

A random variable \( X \) is said to be distributed as \( Bernoulli(\theta) \), denoted by \( X \sim Bernoulli(\theta) \), if \[ P(X = 1) = \theta, \quad P(X = 0) = 1 - \theta \] for \( 0 < \theta < 1 \). Let \( Y = \sum…

2025 · session-5Probability and StatisticsExponential DistributionMCQ · 2 marks

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 \( Y = |X| \), wher…

2025 · session-5Machine LearningMulti-Layer PerceptronMCQ · 2 marks

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

2025 · session-5AIAdversarial SearchMCQ · 2 marks

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 …

2025 · session-5AIInformed SearchMCQ · 2 marks

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…

2025 · session-5Probability and StatisticsBinomial DistributionMCQ · 2 marks

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, \( P(A) = \)

2025 · session-5Database Management and WarehousingData Warehouse ModellingMCQ · 2 marks

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 combinat…

2025 · session-5Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 2 marks

Consider the following Python code snippet. \[ A = \{"this", "that"\}, \quad B = \{"that", "other"\}, \quad C = \{"other", "this"\} \] \[ while "other" in C: \] \[ \quad if "this" in A: \] \[ \quad \quad A, B, C = A - B,…

2025 · session-5Machine LearningReLU Activation in Neural NetworksMSQ · 2 marks

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

2025 · session-5Calculus and OptimizationMaxima and MinimaMSQ · 2 marks

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?

2025 · session-5Linear AlgebraProjection MatrixMSQ · 2 marks

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…

2025 · session-5Calculus and OptimizationMaxima and MinimaMSQ · 2 marks

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 cor…

2025 · session-5Linear AlgebraOrthogonal MatrixMSQ · 2 marks

An \( n \times n \) matrix \( A \) with real entries satisfies the property: \[ \|Ax\|^2 = \|x\|^2, \quad for all x \in \mathbb{R}^n, \] where \( \| \cdot \| \) denotes the Euclidean norm. Which of the following statemen…

2025 · session-5Machine LearningSupport Vector MachineMSQ · 2 marks

Consider designing a linear binary classifier \(f(x)=\operatorname{sign}(w^Tx+b)\), where \(x\in\mathbb{R}^2\), on the following training data: \[ \text{Class 1: }\left\{\begin{bmatrix}2\\0\end{bmatrix},\begin{bmatrix}0…

2025 · session-5Probability and StatisticsBernoulli DistributionMSQ · 2 marks

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} \s…

2025 · session-5Machine LearningClassification ProblemsMSQ · 2 marks

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 calc…

2025 · session-5Database Management and WarehousingSQL Integrity ConstraintsMSQ · 2 marks

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

2025 · session-5Database Management and WarehousingFunctional Dependencies and Normal FormMSQ · 2 marks

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 follow…

2025 · session-5Programming, Data Structures and AlgorithmsGraph TraversalsMSQ · 2 marks

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 …

2025 · session-5Calculus and OptimizationDifferentiabilityNAT · 2 marks

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 \[ f(1) - f(0) = \ \hspace{2cm} \quad (Answer in integer) \]

2025 · session-5Linear AlgebraEigenvalues and Covariance MatrixNAT · 2 marks

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 f…

2025 · session-5Probability and StatisticsBinomial DistributionNAT · 2 marks

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 …

2025 · session-5Database Management and WarehousingRelational AlgebraNAT · 2 marks

Consider the following tables of a bank. | Loan number | Branch name | Amount | | --- | --- | --- | | L11 | Banjara Hills | 90000 | | L14 | Kondapur | 50000 | | L15 | SR Nagar | 40000 | | L22 | SR Nagar | 25000 | | L23 …

2025 · session-5Programming, Data Structures and AlgorithmsProgramming in PythonNAT · 2 marks

Consider the following Python code snippet. The value printed by the code snippet is 160 (answer in integer).

2025 · session-5Programming, Data Structures and AlgorithmsStacksNAT · 2 marks

Consider the following pseudocode. The value of \( sum \) output by a program executing the above pseudocode is:

2025 · session-5Programming, Data Structures and AlgorithmsGraph TraversalsNAT · 2 marks

Consider a directed graph \( G = (V,E) \), where \( V = \{0,1,2,\dots,100\} \) and \[ E = \{(i,j) : 0 < j - i \leq 2, for all i,j \in V \}. \] Suppose the adjacency list of each vertex is in decreasing order of vertex nu…

2024 · session-1General AptitudeVerbal AptitudeMCQ · 1 mark

If '→' denotes increasing order of intensity, then the meaning of the words [sick → infirm → moribund] is analogous to [silly → - → daft]. Which one of the given options is appropriate to fill the blank?

2024 · session-1General AptitudeSpatial AptitudeMCQ · 1 mark

The 15 parts of the given figure are to be painted such that no two adjacent parts with shared boundaries (excluding corners) have the same color. The minimum number of colors required is:

2024 · session-1General AptitudeQuantitative AptitudeMCQ · 1 mark

How many 4-digit positive integers divisible by 3 can be formed using only the digits {1, 3, 4, 6, 7}, such that no digit appears more than once in a number?

2024 · session-1General AptitudeQuantitative AptitudeMCQ · 1 mark

The sum of the following infinite series is: \[2+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{8}+\frac{1}{9}+\frac{1}{16}+\frac{1}{27}+\cdots\]

2024 · session-1General AptitudeQuantitative AptitudeMCQ · 1 mark

In an election, the share of valid votes received by the four candidates A, B, C, and D is represented by the pie chart shown. The total number of votes cast in the election were 1,15,000, out of which 5,000 were invalid…

2024 · session-1General AptitudeVerbal AptitudeMCQ · 2 marks

Thousands of years ago, some people began dairy farming. This coincided with a number of mutations in a particular gene that resulted in these people developing the ability to digest dairy milk. Based on the given passag…

2024 · session-1General AptitudeQuantitative AptitudeMCQ · 2 marks

The probability of a boy or a girl being born is \(\frac{1}{2}\). For a family having only three children, what is the probability of having two girls and one boy?

2024 · session-1General AptitudeQuantitative AptitudeMCQ · 2 marks

Person 1 and Person 2 invest in three mutual funds A, B, and C. The amounts they invest in each of these mutual funds are given in the table. Mutual fund | A | B | C --- | --- | --- | --- Person 1 | Rs 10,000 | Rs 20,000…

2024 · session-1General AptitudeSpatial AptitudeMCQ · 2 marks

Three different views of a dice are shown in the figure below. The piece of paper that can be folded to make this dice is: 1. 2. 3. 4.

2024 · session-1General AptitudeSpatial AptitudeMCQ · 2 marks

Visualize two identical right circular cones such that one is inverted over the other and they share a common circular base. If a cutting plane passes through the vertices of the assembled cones, what shape does the oute…

2024 · session-1Probability and StatisticsPoisson and Normal DistributionsMCQ · 1 mark

• (i) The mean and variance of a Poisson random variable are equal. • (ii) For a standard normal random variable, the mean is zero and the variance is one. Which ONE of the following options is correct?

2024 · session-1Probability and StatisticsIndependent EventsMCQ · 1 mark

Three fair coins are tossed independently. T is the event that two or more tosses result in heads. S is the event that two or more tosses result in tails. What is the probability of the event T ∩ S?

2024 · session-1Linear AlgebraEigenvalues and EigenvectorsMCQ · 1 mark

Consider the matrix \[ M=\begin{bmatrix}2&-1\\3&1\end{bmatrix}. \] Which ONE of the following statements is TRUE?

2024 · session-1Programming, Data Structures and AlgorithmsGraph Traversals and Shortest PathMCQ · 1 mark

Consider performing depth-first search (DFS) on an undirected and unweighted graph G starting at vertex s. For any vertex u in G, d[u] is the length of the shortest path from s to u. Let (u, v) be an edge in G such that …

2024 · session-1Calculus and OptimizationMaxima and MinimaMCQ · 1 mark

For any twice differentiable function f : R → R, if at some x* ∈ R, f'(x*) = 0 and f''(x*) > 0, then the function f necessarily has a _______ at x*.

2024 · session-1Programming, Data Structures and AlgorithmsStacks, Queues and Hash TablesMCQ · 1 mark

Match the items in Column 1 with the items in Column 2 in the following table: Column 1 | Column 2 --- | --- (p) First In First Out | (i) Stacks (q) Lookup Operation | (ii) Queues (r) Last In First Out | (iii) Hash Table…

2024 · session-1Machine LearningSupport Vector MachineMCQ · 1 mark

Consider the dataset with six datapoints: {(x1, y1), (x2, y2), ..., (x6, y6)}, where x1 = [0], x2 = [1], x3 = [−1], x4 = [2], x5 = [−2], and the labels are given by y1 = y2 = 5, y3 = y4 = 5, and y5 = y6 = −1. A hard marg…

2024 · session-1Machine LearningPrincipal Component AnalysisMCQ · 1 mark

Match the items in Column 1 with the items in Column 2 in the following table: Column 1 | Column 2 --- | --- (p) Principal Component Analysis | (i) Discriminative Model (q) Naive Bayes Classification | (ii) Dimensionalit…

2024 · session-1Machine LearningK-Means ClusteringMCQ · 1 mark

Euclidean-distance based \(k\)-means clustering was run on a dataset of 100 points with \(k=3\). If the points \(\begin{bmatrix}1\\1\end{bmatrix}\) and \(\begin{bmatrix}-1\\1\end{bmatrix}\) are both part of cluster 3, wh…

2024 · session-1Machine LearningNaive Bayes ClassifierMCQ · 1 mark

Given a dataset with \(K\) binary-valued attributes, where \(K>2\), for a two-class classification task, how many parameters must be estimated to learn a naive Bayes classifier?

2024 · session-1Programming, Data Structures and AlgorithmsHash TablesMCQ · 1 mark

Consider uniform hashing in an open-address hash table with load factor \(\alpha<1\), where \(n\) elements are stored in \(m\) slots. The expected number of probes in an unsuccessful search is at most \(\frac{1}{1-\alpha…

2024 · session-1Machine LearningLinear Discriminant AnalysisMCQ · 1 mark

For a binary classification dataset, let \(S_B\in\mathbb{R}^{d\times d}\) and \(S_W\in\mathbb{R}^{d\times d}\) be the between-class and within-class scatter matrices, respectively. The Fisher linear discriminant is \[J(…

2024 · session-1AIInformed SearchMCQ · 1 mark

Let \(h_1\) and \(h_2\) be two admissible heuristics used in A* search. Which ONE of the following expressions is always an admissible heuristic?

2024 · session-1AIConditional Independence RepresentationMCQ · 1 mark

Consider five random variables U, V, W, X, Y whose joint distribution satisfies: P(U, V, W, X, Y) = P(U)P(V)P(W|U,V)P(X|W)P(Y|W) Which ONE of the following statements is FALSE?

2024 · session-1AIAdversarial SearchMCQ · 1 mark

Consider the following statement: In adversarial search, α-β pruning can be applied to game trees of any depth where α is the (m) value choice we have formed so far at any choice point along the path for the MAX player a…

2024 · session-1Database Management and WarehousingRelational AlgebraMCQ · 1 mark

Consider a database that includes the following relations: Defender(name, rating, side, goals), Forward(name, rating, assists, goals), Team(name, club, price). Which ONE of the following relational algebra expressions ch…

2024 · session-1Database Management and WarehousingData Transformation: NormalizationMCQ · 1 mark

Let the minimum, maximum, mean, and standard deviation values for the attribute income of data scientists be Rs 246000, Rs 170000, Rs 96000, and Rs 21000, respectively. The Z-score normalized income value of Rs 106000 is…

2024 · session-1Programming, Data Structures and AlgorithmsTreesMSQ · 1 mark

Consider the following tree traversals on a full binary tree: • (i) Preorder • (ii) Inorder • (iii) Postorder Which of the following traversal options is/are sufficient to uniquely reconstruct the full binary tree?

2024 · session-1AIPropositional LogicMSQ · 1 mark

Let x and y be two propositions. Which of the following statements is a tautology/are tautologies?

2024 · session-1Programming, Data Structures and AlgorithmsQuicksortNAT · 1 mark

Consider sorting the following array of integers in ascending order using an in-place Quicksort algorithm that uses the last element as the pivot. {60, 70, 80, 90, 100} The minimum number of swaps performed during this Q…

2024 · session-1Database Management and WarehousingSQLNAT · 1 mark

Consider the following two tables named Raider and Team in a relational database maintained by a Kabaddi league. The attribute ID in table Team references the primary key of the Raider table, ID. Raider table: ID | Name …

2024 · session-1Programming, Data Structures and AlgorithmsQueuesNAT · 1 mark

The fundamental operations in a double-ended queue D are: • insertFirst(e) = Insert a new element e at the beginning of D. • insertLast(e) = Insert a new element e at the end of D. • removeFirst() = Remove and return the…

2024 · session-1Calculus and OptimizationDifferentiabilityNAT · 1 mark

Let \(f:\mathbb{R}\to\mathbb{R}\) be the function \(f(x)=\frac{1}{1+e^{-x}}\). The value of \(f'(x)\) at a point where \(f(x)=0.4\) is ______ (rounded off to two decimal places).

2024 · session-1Probability and StatisticsMeanNAT · 1 mark

The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ____

2024 · session-1Linear AlgebraDeterminantNAT · 1 mark

Consider the matrix \[ M=\begin{bmatrix}1&2&3\\3&1&3\\4&3&6\end{bmatrix}. \] The determinant of \(M^2+12M\) is ______.

2024 · session-1Probability and StatisticsConditional ExpectationMCQ · 2 marks

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?

2024 · session-1Calculus and OptimizationContinuity and DifferentiabilityMCQ · 2 marks

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=\begin{cases} -x, & x<-2,\\ ax^2+bx+c, & -2\le x\le2,\\ x, & x>2. \end{cases} \] Which ONE of the following choices gives the values of \(a\), \(b\), and \(c\) th…

2024 · session-1Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 2 marks

Consider the following Python code: ` def count(child_dict, i): if i not in child_dict.keys(): return 1 ans = 1 for j in child_dict[i]: ans += count(child_dict, j) return ans child_dict = dict() child_dict[0] = [1,2] chi…

2024 · session-1Programming, Data Structures and AlgorithmsBasic Sorting AlgorithmsMCQ · 2 marks

Consider the function computeS(X) whose pseudocode is given below: ` computeS(X) S[1] = 1 for i = 2 to length(X) S[i] = -1 if X[i-1] <= X[i] S[i] = S[i-1] + S[i-1] end if end for return S ` Which ONE of the following val…

2024 · session-1Programming, Data Structures and AlgorithmsBinary SearchMCQ · 2 marks

Let F(n) denote the maximum number of comparisons made while searching for an entry in a sorted array of size n using binary search. Which ONE of the following options is TRUE?

2024 · session-1Programming, Data Structures and AlgorithmsProgramming in PythonMCQ · 2 marks

Consider the following Python function: ` def fun(D, s1, s2): if s1 < s2: D[s1], D[s2] = D[s2], D[s1] fun(D, s1+1, s2-1) ` What does this Python function fun() do? Select the ONE appropriate option below.

2024 · session-1Machine LearningHierarchical Clustering: Single LinkageMCQ · 2 marks

Consider the table below, where the \((i,j)\)-th element is the distance between points \(x_i\) and \(x_j\). Single-linkage clustering is performed on \(x_1,x_2,x_3,x_4,x_5\): | | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) …

2024 · session-1Machine LearningFeed-Forward Neural NetworkMCQ · 2 marks

Consider the two neural networks (NNs) shown in Figures 1 and 2, with ReLU activation, where ReLU(z) = max{0,z}, ∀z∈ R. The connections and their corresponding weights are shown in the figures. The biases at every neuron…

2024 · session-1AIUninformed SearchMCQ · 2 marks

Consider a state space where the start state is number 1. The successor function for the state numbered n returns two states numbered n + 1 and n + 2. Assume that the states in the unexpanded state list are expanded in t…

2024 · session-1Programming, Data Structures and AlgorithmsSelection Sort, Bubble Sort and Insertion SortMCQ · 2 marks

Consider the following sorting algorithms: • (i) Bubble sort • (ii) Insertion sort • (iii) Selection sort Which ONE among the following choices of sorting algorithms sorts the numbers in the array [4, 3, 2, 1, 5] in incr…

2024 · session-1Database Management and WarehousingNormal Form and Functional DependenciesMSQ · 2 marks

Given the relational schema R = (U, V, W, X, Y, Z) and the set of functional dependencies: {U → V, U → W, W → Y, W → Z, X → Y, X → Z} Which of the following functional dependencies can be derived from the above set?

2024 · session-1Linear AlgebraSubspacesMSQ · 2 marks

Select all choices that are subspaces of \(\mathbb{R}^3\). Note: \(\mathbb{R}\) denotes the set of real numbers.

2024 · session-1Linear AlgebraMatrices and Systems of Linear EquationsMSQ · 2 marks

Which of the following statements is/are TRUE? Note: R denotes the set of real numbers.

2024 · session-1Linear AlgebraProjection MatrixMSQ · 2 marks

Let \(\mathbb{R}\) be the set of real numbers, let \(U\) be a subspace of \(\mathbb{R}^3\), and let \(M\in\mathbb{R}^{3\times3}\) be the matrix for the projection onto \(U\). Which statements are TRUE?

2024 · session-1Calculus and OptimizationMaxima and MinimaMSQ · 2 marks

Consider the function \[ f(x)=\frac{x^4}{4}-\frac{2x^3}{3}-\frac{3x^2}{2}+1. \] Which of the following statements is/are TRUE?

2024 · session-1Programming, Data Structures and AlgorithmsGraph Algorithms: TraversalsMSQ · 2 marks

Consider the directed acyclic graph (DAG) below: Which of the following is/are valid vertex orderings that can be obtained from a topological sort of the DAG?

2024 · session-1Programming, Data Structures and AlgorithmsTreesMSQ · 2 marks

Let H, I, L, and N represent height, number of internal nodes, number of leaf nodes, and the total number of nodes respectively in a rooted binary tree. Which of the following statements is/are always TRUE?

2024 · session-1Machine LearningClassification ProblemsMSQ · 2 marks

Consider the following figures representing datasets consisting of two-dimensional features with two classes denoted by circles and squares. Which of the following is/are TRUE?

2024 · session-1AIPredicate LogicMSQ · 2 marks

Let game(ball, rugby) be true if the ball is used in rugby and false otherwise. Let shape(ball, round) be true if the ball is round and false otherwise. Consider the following logical sentences: s1 : ∀ball ⇒ game(ball, r…

2024 · session-1Database Management and WarehousingIndexing and SQLMSQ · 2 marks

An OTT company is maintaining a large disk-based relational database of different movies with the following schema: • Movie (ID, CustomerRating) • Genre (ID, Name) • Movie_Genre (MovieID, GenreID) The given SQL query is:…

2024 · session-1Probability and StatisticsUniform DistributionNAT · 2 marks

Let \(X\) be uniformly distributed on \([1,3]\), and let \(Y\) be uniformly distributed on \([2,4]\). If \(X\) and \(Y\) are independent, then \(P(X\ge Y)\) is ______ (rounded off to three decimal places).

2024 · session-1Probability and StatisticsExponential DistributionNAT · 2 marks

Let \(X\) be exponentially distributed with parameter \(\lambda>0\), with density \[ f_X(x)=\begin{cases}\lambda e^{-\lambda x},&x\ge0,\\0,&\text{otherwise}. \end{cases} \] If \(5E[X]=\operatorname{Var}(X)\), then \(\l…

2024 · session-1Probability and StatisticsConditional ProbabilityNAT · 2 marks

Consider two events \(T\) and \(S\), and let \(\overline{T}\) denote the complement of \(T\). It is given that \[ P(\overline{T})=0.6,\qquad P(S\mid T)=0.3,\qquad P(S\mid\overline{T})=0.6. \] Then \(P(T\mid S)\) is ___…

2024 · session-1Calculus and OptimizationLimitNAT · 2 marks

Evaluate the limit \[ \lim_{x\to0}\frac{\ln\!\left((x^2+1)\cos x\right)}{x^2}=\boxed{\phantom{0}}. \]

2024 · session-1Linear AlgebraSingular Value DecompositionNAT · 2 marks

Let \[ \mathbf{u}=\begin{bmatrix}1\\2\\3\\4\\5\end{bmatrix}, \] and let \(\sigma_1,\sigma_2,\sigma_3,\sigma_4,\sigma_5\) be the singular values of \(M=\mathbf{u}\mathbf{u}^{T}\). The value of \(\sum_{i=1}^{5}\sigma_i\)…

2024 · session-1Machine LearningDecision TreesNAT · 2 marks

Details of ten international cricket games between teams Green and Blue are given in Table C. The pitch is spin-friendly (\(S\)) or pace-friendly (\(F\)); the format is one-day (\(O\)) or test (\(T\)). | Match | Pitch |…

2024 · session-1Machine LearningK-Nearest NeighbourNAT · 2 marks

Given the two-dimensional dataset of five points from two classes (circles and squares), assume Euclidean distance. The minimum odd value of \(k\) in the \(k\)-nearest-neighbour algorithm for which the diamond-shaped poi…

2024 · session-1AIConditional Independence and Exact InferenceNAT · 2 marks

Given the Bayesian network of four Bernoulli random variables and its conditional probability tables, the value of \(P(U=1,V=1,W=1,Z=1)\) is ______ (rounded off to three decimal places).

2024 · session-1Probability and StatisticsJoint ProbabilityNAT · 2 marks

Two fair coins are tossed independently. Let \(X=1\) if both tosses are heads and \(X=0\) otherwise. Let \(Y=1\) if at least one toss is heads and \(Y=0\) otherwise. The covariance of \(X\) and \(Y\) is ______ (rounded …

2024 · session-1Probability and StatisticsConditional ExpectationNAT · 2 marks

The joint density of \(X\) and \(Y\) is \[ f_{X,Y}(x,y)=\begin{cases}2xy,&0<x<2,\ 0<y<x,\\0,&\text{otherwise}. \end{cases} \] Then \(E[Y\mid X=1.5]\) is ______.