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41
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 $\_\_\_\_\_\_\_\_\_$
Arjun
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Feb 16
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Arjun
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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 $\_\_\_\_\_\_\_\_\_$
Arjun
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Feb 16
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Arjun
921
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gate-ds-ai-2024
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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
Arjun
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Feb 16
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Arjun
773
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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 $\_\_\_\_\_\_\_$.
Arjun
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Feb 16
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Arjun
963
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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).
Arjun
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Feb 16
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Arjun
1.0k
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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).
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Feb 16
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Arjun
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47
lost my gate 2023 scorecard,, Help
Hello GO , I'm GO reader from last 2 years and I love this forum so I have given gate 2024 also but the thing is I lost my gate 2023 scorecard it wasn't good rank and I didn't know about last date to download I downloaded ... though it was bad rank but i qualified after studying what should i do , I sent mail to iitK today evening I'm worried
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48
how i give free mock test on previous year
Shruti bhurse
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49
TIFR Mathematics 2024 | Part B | Question: 1
If $\text{G}$ is a group of order $361$, then $\text{G}$ has a normal subgroup $\text{H}$ such that $H \cong G / H$.
admin
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Jan 19
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admin
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tifrmaths2024
true-false
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50
TIFR Mathematics 2024 | Part B | Question: 2
There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.
admin
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Jan 19
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admin
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51
TIFR Mathematics 2024 | Part B | Question: 3
The function $d: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ given by $d(x, y)=\left|e^{x}-e^{y}\right|$ defines a metric on $\mathbb{R}$, and $(\mathbb{R}, d)$ is a complete metric space.
admin
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Jan 19
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admin
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TIFR Mathematics 2024 | Part B | Question: 4
Let $n$ be a positive integer, and $A$ an $n \times n$ matrix over $\mathbb{R}$ such that $A^{3}=\mathrm{Id}$. Then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{R})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{R})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.
admin
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Jan 19
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admin
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TIFR Mathematics 2024 | Part B | Question: 5
If $A \in \mathrm{M}_{n}(\mathbb{Q})$ is such that the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$, then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{C})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{C})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.
admin
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Jan 19
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admin
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TIFR Mathematics 2024 | Part B | Question: 6
The complement of any countable union of lines in $\mathbb{R}^{3}$ is path connected.
admin
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Jan 19
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admin
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tifrmaths2024
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55
TIFR Mathematics 2024 | Part B | Question: 7
The subsets $\left\{(x, y) \in \mathbb{R}^{2} \mid\left(y^{2}-x\right)\left(y^{2}-x-1\right)=0\right\}$ and $\left\{(x, y) \in \mathbb{R}^{2} \mid y^{2}-x^{2}=1\right\}$ of $\mathbb{R}^{2}$ (with the induced metric) are homeomorphic.
admin
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Jan 19
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admin
51
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tifrmaths2024
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56
TIFR Mathematics 2024 | Part B | Question: 8
$\mathbb{Q} \cap[0,1]$ is a compact subset of $\mathbb{Q}$.
admin
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Jan 19
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admin
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tifrmaths2024
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57
TIFR Mathematics 2024 | Part B | Question: 9
Suppose $f: X \rightarrow Y$ is a function between metric spaces, such that whenever a sequence $\left\{x_{n}\right\}$ converges to $x$ in $X$, the sequence $\left\{f\left(x_{n}\right)\right\}$ converges in $Y$ (but it is not given that the limit of $\left\{f\left(x_{n}\right)\right\}$ is $\left.f(x)\right)$. Then $f$ is continuous.
admin
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Jan 19
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admin
53
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tifrmaths2024
true-false
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58
TIFR Mathematics 2024 | Part B | Question: 10
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and assume that $\left|f^{\prime}(x)\right| \geq 1$ for all $x \in \mathbb{R}$. Then for each compact set $C \subset \mathbb{R}$, the set $f^{-1}(C)$ is compact.
admin
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Jan 19
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admin
54
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tifrmaths2024
true-false
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59
TIFR Mathematics 2024 | Part B | Question: 11
There exists a function $f:[0,1] \rightarrow \mathbb{R}$, which is not Riemann integrable and satisfies \[ \sum_{i=1}^{n}\left|f\left(t_{i}\right)-f\left(t_{i-1}\right)\right|^{2}<1 \]
admin
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Jan 19
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admin
48
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60
TIFR Mathematics 2024 | Part B | Question: 12
Let $E \subset[0,1]$ be the subset consisting of numbers that have a decimal expansion which does not contain the digit 8 . Then $E$ is dense in $[0,1]$.
admin
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Jan 19
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admin
55
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