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ISI 2019 | PCB CS | Question: 1
Let $A$ be a sorted array containing $n$ distinct integers, such that, for all $1 \leq i<j \leq n$, we have $A[i]<A[j]$. Note that the integers stored in the array $A$ ... time of the algorithm should be asymptotically better than $O(n)$. Prove the correctness of your algorithm and state its asymptotic time complexity.
Let $A$ be a sorted array containing $n$ distinct integers, such that, for all $1 \leq i<j \leq n$, we have $A[i]<A[j]$. Note that the integers stored in the array $A$ ar...
vaibhav_mani
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GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 1
Let $\text{b}$ be the branching factor of a search tree. If the optimal goal is reached after $\text{d}$ actions from the initial state, in the worst case, how many times will the initial state be expanded for iterative deepening depth-first ... $\text{IDDFS}$ $\text{-b}^{d}, \mathrm{IDA}^{*}\text{-b}^{d}$.
Let $\text{b}$ be the branching factor of a search tree. If the optimal goal is reached after $\text{d}$ actions from the initial state, in the worst case, how many times...
Riya_23
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GATE DS&AI 2024 | Question: 55
Two fair coins are tossed independently. $X$ is a random variable that takes a value of $1$ if both tosses are heads and $0$ otherwise. $Y$ is a random variable that takes a value of $1$ if at least one of the tosses is heads and $0$ otherwise. The value of the covariance of $X$ and $Y$ is $\_\_\_\_\_\_\_$ (rounded off to three decimal places).
Two fair coins are tossed independently. $X$ is a random variable that takes a value of $1$ if both tosses are heads and $0$ otherwise. $Y$ is a random variable that take...
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GATE DS&AI 2024 | Question: 54
Given the following Bayesian Network consisting of four Bernoulli random variables and the associated conditional probability tables: \begin{array}{|c|c|} \hline & P(\cdot) \\ \hline U=0 & 0.5 \\ \hline U=1 & 0.5 \\ \hline \end{array} \begin{array}{|c|c|c|} \ ... The value of $P(U=1, V=1, W=1, Z=1)= \_\_\_\_\_\_\_$ (rounded off to three decimal places).
Given the following Bayesian Network consisting of four Bernoulli random variables and the associated conditional probability tables:\begin{array}{|c|c|}\hline & P(\cdot)...
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GATE DS&AI 2024 | Question: 53
Given the two-dimensional dataset consisting of $5$ data points from two classes (circles and squares) and assume that the Euclidean distance is used to measure the distance between two points. The minimum odd value of $k$ in $k$-nearest neighbor algorithm for which the diamond $(\diamond)$ shaped data point is assigned the label square is $\_\_\_\_\_\_\_$.
Given the two-dimensional dataset consisting of $5$ data points from two classes (circles and squares) and assume that the Euclidean distance is used to measure the dista...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 52
Details of ten international cricket games between two teams "Green" and "Blue" are given in Table $\mathrm{C}$. This table consists of matches played on different pitches, across formats along with their winners. The attribute Pitch can take one of two values: spin-friendly ( ... $S$ $O$ Green $8$ $F$ $T$ Blue $9$ $F$ $O$ Blue $10$ $S$ $O$ Green
Details of ten international cricket games between two teams "Green" and "Blue" are given in Table $\mathrm{C}$. This table consists of matches played on different pitche...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 37
Select all choices that are subspaces of $\mathbb{R}^{3}$. Note: $\mathbb{R}$ ...
Select all choices that are subspaces of $\mathbb{R}^{3}$.Note: $\mathbb{R}$ denotes the set of real numbers.$\left\{\mathbf{x}=\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_...
Sachin Mittal 1
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GATE DS&AI 2024 | Question: 51
Let $\text{u}=\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]$, and let $\sigma_{1}, \sigma_{2}, \sigma_{3}, \sigma_{4}, \sigma_{5}$ be the singular values of the matrix $\text{M}=\text{u} \text{u}^{\text{T}}$ (where $\text{u}^{\text{T}}$ is the transpose of $\text{u}$ ). The value of $\sum_{i=1}^{5} \sigma_{i}$ is $\_\_\_\_\_\_\_\_\_$
Let $\text{u}=\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]$, and let $\sigma_{1}, \sigma_{2}, \sigma_{3}, \sigma_{4}, \sigma_{5}$ be the singular values ...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 49
Consider a joint probability density function of two random variables $X$ and $Y$ \[ f_{X, Y}(x, y)=\left\{\begin{array}{rll}2 x y, & 0<x<2, & 0<y<x \\ 0, & \text { otherwise } & \end{array}\right. \] Then, $E[Y \mid X=1.5]$ is $\_\_\_\_\_\_\_\_\_$
Consider a joint probability density function of two random variables $X$ and $Y$\[f_{X, Y}(x, y)=\left\{\begin{array}{rll}2 x y, & 0<x<2, & 0<y<x \\ 0, & \text { otherwi...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 46
Let $X$ be a random variable uniformly distributed in the interval $[1,3]$ and $Y$ be a random variable uniformly distributed in the interval $[2, 4]$. If $X$ and $Y$ are independent of each other, the probability $P(X \geq Y)$ is $\_\_\_\_\_\_\_\_$ (rounded off to three decimal places).
Let $X$ be a random variable uniformly distributed in the interval $[1,3]$ and $Y$ be a random variable uniformly distributed in the interval $[2, 4]$. If $X$ and $Y$ are...
makhdoom ghaya
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makhdoom ghaya
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GATE DS&AI 2024 | Question: 45
An OTT company is maintaining a large disk-based relational database of different movies with the following schema: \[ \begin{array}{l} \text { Movie (ID, CustomerRating) } \\ \text { Genre (ID, Name) } \\ \text { Movie_Genre ... attributes. Hash index on Movie.CustomerRating and $\mathrm{B}^{+}$tree on the remaining attributes. Hash index on all the attributes.
An OTT company is maintaining a large disk-based relational database of different movies with the following schema:\[\begin{array}{l}\text { Movie (ID, ...
makhdoom ghaya
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makhdoom ghaya
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GATE DS&AI 2024 | Question: 44
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: $\forall$ ball $\neg$ game(ball, rugby) $\Rightarrow$ shape(ball, round) ... used in rugby"? $s 1 \wedge s 3$ $s 1 \wedge s 2$ $s 2 \wedge s 3$ $s 3 \wedge s 4$
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.Cons...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 43
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? $\text{(i)}$ is linearly separable. $\text{(ii)}$ is linearly separable. $\text{(iii)}$ is linearly separable. $\text{(iv)}$ is linearly separable.
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 ...
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GATE DS&AI 2024 | Question: 42
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? $L \leq I+1$ $H+1 \leq N \leq 2^{H+1}-1$ $H \leq I \leq 2^{H}-1$ $H \leq L \leq 2^{H-1}$
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...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 40
Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ where $\mathbb{R}$ is the set of all real numbers. \[ f(x)=\frac{x^{4}}{4}-\frac{2 x^{3}}{3}-\frac{3 x^{2}}{2}+1 \] Which of the following statements is/are TRUE? $x=0$ is a local maximum of $f$ $x=3$ is a local minimum of $f$ $x=-1$ is a local maximum of $f$ $x=0$ is a local minimum of $f$
Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ where $\mathbb{R}$ is the set of all real numbers.\[f(x)=\frac{x^{4}}{4}-\frac{2 x^{3}}{3}-\fr...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 39
Let $\mathbb{R}$ be the set of real numbers, $U$ be a subspace of $\mathbb{R}^{3}$ and $\text{M} \in \mathbb{R}^{3 \times 3}$ be the matrix corresponding to the projection on to the subspace $U$. Which of the following statements is/are TRUE? If $U$ is a ... of $\mathbb{R}^{3}$, then the null space of $\text{M}$ is a $1$-dimensional subspace. $M^{2}=M$ $M^{3}=M$
Let $\mathbb{R}$ be the set of real numbers, $U$ be a subspace of $\mathbb{R}^{3}$ and $\text{M} \in \mathbb{R}^{3 \times 3}$ be the matrix corresponding t...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 38
Which of the following statements is/are TRUE? Note: $\mathbb{R}$ denotes the set of real numbers. There exist $\text{M} \in \mathbb{R}^{3 \times 3}, \text{p} \in \mathbb{R}^{3}$, and $\text{q} \in \mathbb{R}^{3}$ ... $\text{Mx}=\text{p}$ has a unique solution and $\text{M x}=\text{q}$ has no solutions.
Which of the following statements is/are TRUE?Note: $\mathbb{R}$ denotes the set of real numbers.There exist $\text{M} \in \mathbb{R}^{3 \times 3}, \tex...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 34
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 the ascending order of ... than BFS. Both BFS and DFS expand equal number of states. Both BFS and DFS do not reach the goal state number $6$.
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 $n+2$. ...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 33
Consider the two neural networks (NNs) shown in Figures $1$ and $2$, with $R e L U$ activation $(\text{ReLU}(z)=\max \{0, z\}, \forall z \in \text{R})$. The connections and their corresponding weights are shown in the Figures. The biases at every neuron are set to $0$. ... real numbers. $p=36, q=24, r=24$ $p=24, q=24, r=36$ $p=18, q=36, r=24$ $p=36, q=36, r=36$
Consider the two neural networks (NNs) shown in Figures $1$ and $2$, with $R e L U$ activation $(\text{ReLU}(z)=\max \{0, z\}, \forall z \in \text{R})$....
makhdoom ghaya
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GATE DS&AI 2024 | Question: 32
Consider the table below, where the $(i, j)^{t h}$ element of the table is the distance between points $x_{i}$ and $x_{j}$. Single linkage clustering is performed on data points, $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$. \begin{array} ... & 3 & 5 & 1 & 0 \\ \hline \end{array} Which ONE of the following is the correct representation of the clusters produced?
Consider the table below, where the $(i, j)^{t h}$ element of the table is the distance between points $x_{i}$ and $x_{j}$. Single linkage clustering is performed on data...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 29
Consider the function computes $(X)$ whose pseudocode is given below: computes $(X)$ $S[1] \leftarrow 1$ for $i \leftarrow 2$ to length $(X)$ $S[i] \leftarrow 1$ if $X[i-1] \leq X[i]$ $S[i] \leftarrow S[i]+S[i-1]$ end if end for return $S$ Which ONE of the following values is ... for $X=[6,3,5,4,10]$ ? $[1,1,2,3,4]$ $[1,1,2,3,3]$ $[1,1,2,1,2]$ $[1,1,2,1,5]$
Consider the function computes $(X)$ whose pseudocode is given below:computes $(X)$$S \leftarrow 1$for $i \leftarrow 2$ to length $(X)$$S[i] \leftarrow 1$...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 27
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers. \[ f(x)=\left\{\begin{array}{cl} -x, & \text { if } x<-2 \\ a x^{2}+b x+c, & \text { if } x \in[-2,2] \\ x, & \text { if } x>2 \end ... differentiable? $a=\frac{1}{4}, b=0, c=1$ $a=\frac{1}{2}, b=0, c=0$ $a=0, b=0, c=0$ $a=1, b=1, c=-4$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers.\[f(x)=\left\{\begin{array}{cl}-x, & \text { i...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 25
Consider the $3 \times 3$ matrix $\boldsymbol{M}=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6\end{array}\right]$. The determinant of $\left(\boldsymbol{M}^{2}+12 \boldsymbol{M}\right)$ is $\_\_\_\_\_\_\_\_\_$.
Consider the $3 \times 3$ matrix $\boldsymbol{M}=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6\end{array}\right]$.The determinant of $\left(\boldsymbol{M}^{...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 24
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 $\_\_\_\_\_\_\_\_$.
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 $\_\_\_\_\_\_\_\_$.
makhdoom ghaya
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GATE DS&AI 2024 | Question: 23
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function $f(x)=\frac{1}{1+e^{-x}}$. The value of the derivative of $f$ at $x$ where $f(x)=0.4$ is $\_\_\_\_\_\_\_$. (rounded off to two decimal places). Note: $\mathbb{R}$ denotes the set of real numbers.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function $f(x)=\frac{1}{1+e^{-x}}$.The value of the derivative of $f$ at $x$ where $f(x)=0.4$ is $\_\_\_\_\_\_\_$. (roun...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 3
Consider the matrix $\boldsymbol{M}=\left[\begin{array}{cc}2 & -1 \\ 3 & 1\end{array}\right]$. Which ONE of the following statements is TRUE? The eigenvalues of $\boldsymbol{M}$ are non-negative and real. The eigenvalues of ... zero. One eigenvalue of $\boldsymbol{M}$ is non-negative and real, and another eigenvalue of $\boldsymbol{M}$ is negative and real.
Consider the matrix $\boldsymbol{M}=\left[\begin{array}{cc}2 & -1 \\ 3 & 1\end{array}\right]$.Which ONE of the following statements is TRUE?The eigenvalues of...
Thomas_18
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GATE DS&AI 2024 | Question: 19
Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies? $(\neg x \wedge y) \Rightarrow(y \Rightarrow x)$ $(x \wedge \neg y) \Rightarrow(\neg x \Rightarrow y)$ $(\neg x \wedge y) \Rightarrow(\neg x \Rightarrow y)$ $(x \wedge \neg y) \Rightarrow(y \Rightarrow x)$
Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies?$(\neg x \wedge y) \Rightarrow(y \Rightarrow x)$$...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 18
Consider the following tree traversals on a full binary tree: Preorder Inorder Postorder Which of the following traversal options is/are sufficient to uniquely reconstruct the full binary tree? $\text{(i) and (ii)}$ $\text{(ii) and (iii)}$ $\text{(i) and (iii)}$ $\text{(ii) only}$
Consider the following tree traversals on a full binary tree:PreorderInorderPostorderWhich of the following traversal options is/are sufficient to uniqu...
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GATE DS&AI 2024 | Question: 17
Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹$46000$, ₹ $170000$, ₹ $96000$, and ₹ $21000$, respectively. The $z$-score normalized income value of ₹ $106000$ is closest to which ONE of the following options? $0.217$ $0.476$ $0.623$ $2.304$
Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹$46000$, ₹ $170000$, ₹ $96000$, and ₹ ...
makhdoom ghaya
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GATE DS&AI 2024 | Question: 16
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 checks that every name occurring in Team appears in ... $\left.)\right) \backslash \Pi_{\text {name }}($ Team $)=\phi$
Consider a database that includes the following relations:Defender(name, rating, side, goals)Forward(name, rating, assists, goals)Team(name, club, price)Wh...
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