Recent questions tagged tifr2019

6 votes

6 answers

1

TIFR2019-A-1
Let $X$ be a set with $n$ elements. How many subsets of $X$ have odd cardinality? $n$ $2^n$ $2^{n/2}$ $2^{n-1}$ Can not be determined without knowing whether $n$ is odd or even

Let $X$ be a set with $n$ elements. How many subsets of $X$ have odd cardinality? $n$ $2^n$ $2^{n/2}$ $2^{n-1}$ Can not be determined without knowing whether $n$ is odd or even

asked Dec 18, 2018 in Set Theory & Algebra Arjun 1.5k views

4 votes

2 answers

2

TIFR2019-A-2
How many proper divisors (that is, divisors other than $1$ or $7200$) does $7200$ have ? $18$ $20$ $52$ $54$ $60$

How many proper divisors (that is, divisors other than $1$ or $7200$) does $7200$ have ? $18$ $20$ $52$ $54$ $60$

asked Dec 18, 2018 in Quantitative Aptitude Arjun 769 views

7 votes

2 answers

3

TIFR2019-A-3
$A$ is $n \times n$ square matrix for which the entries in every row sum to $1$. Consider the following statements: The column vector $[1,1,\ldots,1]^T$ is an eigen vector of $A.$ $ \text{det}(A-I) = 0.$ $\text{det}(A) = 0.$ Which of the above statements must be TRUE? Only $(i)$ Only $(ii)$ Only $(i)$ and $(ii)$ Only $(i)$ and $(iii)$ $(i),(ii) \text{ and }(iii)$

$A$ is $n \times n$ square matrix for which the entries in every row sum to $1$. Consider the following statements: The column vector $[1,1,\ldots,1]^T$ is an eigen vector of $A.$ $ \text{det}(A-I) = 0.$ $\text{det}(A) = 0.$ Which of the above statements must be TRUE? Only $(i)$ Only $(ii)$ Only $(i)$ and $(ii)$ Only $(i)$ and $(iii)$ $(i),(ii) \text{ and }(iii)$

asked Dec 18, 2018 in Linear Algebra Arjun 1.6k views

5 votes

1 answer

4

TIFR2019-A-4
What is the probability that a point $P=(\alpha,\beta)$ picked uniformly at random from the disk $x^2 +y^2 \leq 1$ satisfies $\alpha + \beta \leq 1$? $\frac{1}{\pi}$ $\frac{3}{4} + \frac{1}{4} \cdot \frac{1}{\pi}$ $\frac{3}{4}+ \frac{1}{4} \cdot \frac{2}{\pi}$ $1$ $\frac{2}{\pi}$

What is the probability that a point $P=(\alpha,\beta)$ picked uniformly at random from the disk $x^2 +y^2 \leq 1$ satisfies $\alpha + \beta \leq 1$? $\frac{1}{\pi}$ $\frac{3}{4} + \frac{1}{4} \cdot \frac{1}{\pi}$ $\frac{3}{4}+ \frac{1}{4} \cdot \frac{2}{\pi}$ $1$ $\frac{2}{\pi}$

asked Dec 18, 2018 in Probability Arjun 838 views

7 votes

4 answers

5

TIFR2019-A-5
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can only reply by saying Yes or no . Assume that Lata always tells the truth. What is the least ... within which she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$

Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can only reply by saying Yes or no . Assume that Lata always tells the truth. What is the least number of ... ask within which she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$

asked Dec 18, 2018 in Algorithms Arjun 1.9k views

1 vote

1 answer

6

TIFR2019-A-6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\mathbb{R}$ $→$ $\mathbb ... $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\mathbb{R}$ $→$ $ ... $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$

asked Dec 18, 2018 in Set Theory & Algebra Arjun 519 views

9 votes

2 answers

7

TIFR2019-A-7
What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$

What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$

asked Dec 18, 2018 in Quantitative Aptitude Arjun 628 views

0 votes

1 answer

8

TIFR2019-A-8
Consider the following toy model of traffic on a straight , single lane, highway. We think of cars as points, which move at the maximum speed $v$ that satisfies the following constraints: The speed is no more than the speed limit $v_{max}$ mandated ... Which of the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?

Consider the following toy model of traffic on a straight , single lane, highway. We think of cars as points, which move at the maximum speed $v$ that satisfies the following constraints: The speed is no more than the speed limit $v_{max}$ mandated for the ... . Which of the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?

asked Dec 18, 2018 in Verbal Aptitude Arjun 495 views

2 votes

1 answer

9

TIFR2019-A-9
Let $A$ and $B$ be two containers. Container $A$ contains $50$ litres of liquid $X$ and container $B$ contains $100$ litres of liquid $Y$. Liquids $X$ and $Y$ are soluble in each other. We now take $30$ ml of liquid $X$ from container $A$ and put it into container $B$. The mixture in ... $V_{AY} > V_{BX}$ $V_{AY} = V_{BX}$ $V_{AY} + V_{BX}=30$ $V_{AY} + V_{BX}=20$

Let $A$ and $B$ be two containers. Container $A$ contains $50$ litres of liquid $X$ and container $B$ contains $100$ litres of liquid $Y$. Liquids $X$ and $Y$ are soluble in each other. We now take $30$ ml of liquid $X$ from container $A$ and put it into container $B$. The mixture in container $B$ is ... $V_{AY} > V_{BX}$ $V_{AY} = V_{BX}$ $V_{AY} + V_{BX}=30$ $V_{AY} + V_{BX}=20$

asked Dec 18, 2018 in Quantitative Aptitude Arjun 461 views

2 votes

1 answer

10

TIFR2019-A-10
Avni and Badal alternately choose numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ without replacement (starting with Avni). The first person to choose numbers of which any $3$ sum to $15$ wins the game (for example, Avni wins if she chooses ... a winning strategy Both of them have a winning strategy Neither of them has a winning strategy The Player that picks $9$ has a winning strategy

Avni and Badal alternately choose numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ without replacement (starting with Avni). The first person to choose numbers of which any $3$ sum to $15$ wins the game (for example, Avni wins if she chooses the ... has a winning strategy Both of them have a winning strategy Neither of them has a winning strategy The Player that picks $9$ has a winning strategy

asked Dec 18, 2018 in Analytical Aptitude Arjun 530 views

4 votes

5 answers

11

TIFR2019-A-11
Suppose there are $n$ guests at a party (and no hosts). As the night progresses, the guests meet each other and shake hands. The same pair of guests might shake hands multiple times. for some parties stretch late into the night , and it is hard to keep track.Still, they don't shake ... $2 \mid \text{Even} \mid - \mid \text{Odd} \mid$ $2 \mid \text{Odd} \mid - \mid \text{Even} \mid$

Suppose there are $n$ guests at a party (and no hosts). As the night progresses, the guests meet each other and shake hands. The same pair of guests might shake hands multiple times. for some parties stretch late into the night , and it is hard to keep track.Still, they don't shake hands with ... $2 \mid \text{Even} \mid - \mid \text{Odd} \mid$ $2 \mid \text{Odd} \mid - \mid \text{Even} \mid$

asked Dec 18, 2018 in Analytical Aptitude Arjun 739 views

2 votes

1 answer

12

TIFR2019-A-12
Let $f$ be a function with both input and output in the set $\{0,1,2, \dots ,9\}$, and let the function $g$ be defined as $g(x) = f(9-x)$. The function $f$ is non-decreasing, so that $f(x) \geq f(y)$ for $x \geq y$ ... $f$ and $g$ ? Only $\text{(i)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(iii)}$ None of them All of them

Let $f$ be a function with both input and output in the set $\{0,1,2, \dots ,9\}$, and let the function $g$ be defined as $g(x) = f(9-x)$. The function $f$ is non-decreasing, so that $f(x) \geq f(y)$ for $x \geq y$ ... functions $f$ and $g$ ? Only $\text{(i)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(iii)}$ None of them All of them

asked Dec 18, 2018 in Set Theory & Algebra Arjun 976 views

6 votes

2 answers

13

TIFR2019-A-13
Consider the integral $\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$ What is the value of this integral correct up to two decimal places? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$

Consider the integral $\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$ What is the value of this integral correct up to two decimal places? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$

asked Dec 18, 2018 in Calculus Arjun 1.1k views

5 votes

1 answer

14

TIFR2019-A-14
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is the probability that two red pens are drawn in succession ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$

A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is the probability that two red pens are drawn in succession ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$

asked Dec 18, 2018 in Probability Arjun 927 views

5 votes

5 answers

15

TIFR2019-A-15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\displaystyle \lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 & 0 & 0\\ ... $\text{The limit exists, but it is none of the above}$

Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\displaystyle \lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$ $\ ... $\text{The limit exists, but it is none of the above}$

asked Dec 18, 2018 in Calculus Arjun 1.2k views

2 votes

1 answer

16

TIFR2019-B-1
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ? $0.1$ $0.2$ $0.4$ $0.5$ All the above

Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ? $0.1$ $0.2$ $0.4$ $0.5$ All the above

asked Dec 18, 2018 in Digital Logic Arjun 1k views

9 votes

5 answers

17

TIFR2019-B-2
How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $8$ $16$ $32$ $64$ None of the above

How many distinct minimum weight spanning trees does the following undirected, weighted graph have ? $8$ $16$ $32$ $64$ None of the above

asked Dec 18, 2018 in Algorithms Arjun 1.8k views

3 votes

1 answer

18

TIFR2019-B-3
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE? $d$ divides $n$ Both $d$ and $n$ are even Both $d$ and $n$ are odd At least one of $d$ and $n$ is odd At least one of $d$ and $n$ is even

A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE? $d$ divides $n$ Both $d$ and $n$ are even Both $d$ and $n$ are odd At least one of $d$ and $n$ is odd At least one of $d$ and $n$ is even

asked Dec 18, 2018 in Graph Theory Arjun 803 views

2 votes

2 answers

19

TIFR2019-B-4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\varphi$ be a propositional formula on a set of variables $B$ , such that $\varphi$ $\Rightarrow$ $\psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ on ... Craig interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$

Let $\varphi$ be a propositional formula on a set of variables $A$ and $\varphi$ be a propositional formula on a set of variables $B$ , such that $\varphi$ $\Rightarrow$ $\psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ ... is a Craig interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$

asked Dec 18, 2018 in Mathematical Logic Arjun 652 views

7 votes

2 answers

20

TIFR2019-B-5
Stirling's approximation for $n!$ states for some constants $c_1,c_2$ $c_1 n^{n+\frac{1}{2}}e^{-n} \leq n! \leq c_2 n^{n+\frac{1}{2}}e^{-n}.$ What are the tightest asymptotic bounds that can be placed on $n!$ $?$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$

Stirling's approximation for $n!$ states for some constants $c_1,c_2$ $c_1 n^{n+\frac{1}{2}}e^{-n} \leq n! \leq c_2 n^{n+\frac{1}{2}}e^{-n}.$ What are the tightest asymptotic bounds that can be placed on $n!$ $?$ $n! = \Omega(n^n) \text{ and } n! = \mathcal{O}(n^{n+\frac{1}{2}})$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$

asked Dec 18, 2018 in Algorithms Arjun 1.3k views

12 votes

3 answers

21

TIFR2019-B-6
Given the following pseudocode for function $\text{printx()}$ below, how many times is $x$ printed if we execute $\text{printx(5)}?$ void printx(int n) { if(n==0){ printf(“x”); } for(int i=0;i<=n-1;++i){ printx(n-1); } } $625$ $256$ $120$ $24$ $5$

Given the following pseudocode for function $\text{printx()}$ below, how many times is $x$ printed if we execute $\text{printx(5)}?$ void printx(int n) { if(n==0){ printf(“x”); } for(int i=0;i<=n-1;++i){ printx(n-1); } } $625$ $256$ $120$ $24$ $5$

asked Dec 18, 2018 in Programming Arjun 1.2k views

2 votes

1 answer

22

TIFR2019-B-7
A formula is said to be a $3$-CF-formula if it is a conjunction (i.e., an AND) of clauses, and each clause has at most $3$ literals. Analogously, a formula is said to be a $3$-DF-formula if it is a disjunction (i.e., an OR) of clauses of at most $3$ literals each. ... $\text{3-DF-SAT}$ is NP-complete Neither $\text{3-CF-SAT}$ nor $\text{3-DF-SAT}$ are in P

A formula is said to be a $3$-CF-formula if it is a conjunction (i.e., an AND) of clauses, and each clause has at most $3$ literals. Analogously, a formula is said to be a $3$-DF-formula if it is a disjunction (i.e., an OR) of clauses of at most $3$ literals each. Define the ... $\text{3-DF-SAT}$ is NP-complete Neither $\text{3-CF-SAT}$ nor $\text{3-DF-SAT}$ are in P

asked Dec 18, 2018 in Algorithms Arjun 472 views

5 votes

2 answers

23

TIFR2019-B-8
Consider the following program fragment: var a,b : integer; procedure G(c,d: integer); begin c:=c-d; d:=c+d; c:=d-c end; a:=2; b:=3; G(a,b); If both parameters to $G$ are passed by reference, what are the values of $a$ and $b$ at the end of the above program fragment ? $a=0$ and $b=2$ $a=3$ and $b=2$ $a=2$ and $b=3$ $a=1$ and $b=5$ None of the above

Consider the following program fragment: var a,b : integer; procedure G(c,d: integer); begin c:=c-d; d:=c+d; c:=d-c end; a:=2; b:=3; G(a,b); If both parameters to $G$ are passed by reference, what are the values of $a$ and $b$ at the end of the above program fragment ? $a=0$ and $b=2$ $a=3$ and $b=2$ $a=2$ and $b=3$ $a=1$ and $b=5$ None of the above

asked Dec 18, 2018 in Programming Arjun 895 views

9 votes

2 answers

24

TIFR2019-B-9
Consider the following program fragment: var x, y: integer; x := 1; y := 0; while y < x do begin x := 2*x; y := y+1 end; For the above fragment , which of the following is a loop invariant ? $x=y+1$ $x=(y+1)^2$ $x=(y+1)2^y$ $x=2^y$ None of the above, since the loop does not terminate

Consider the following program fragment: var x, y: integer; x := 1; y := 0; while y < x do begin x := 2*x; y := y+1 end; For the above fragment , which of the following is a loop invariant ? $x=y+1$ $x=(y+1)^2$ $x=(y+1)2^y$ $x=2^y$ None of the above, since the loop does not terminate

asked Dec 18, 2018 in Programming Arjun 1.3k views

9 votes

2 answers

25

TIFR2019-B-10
Let the language $D$ be defined in the binary alphabet $\{0,1\}$ as follows: $D:= \{ w \in \{0,1\}^* \mid \text{ substrings 01 and 10 occur an equal number of times in w} \}$ For example , $101 \in D$ while $1010 \notin D$. Which of the ... ? $D$ is regular $D$ is context-free but not regular $D$ is decidable but not context-free $D$ is decidable but not in NP $D$ is undecidable

Let the language $D$ be defined in the binary alphabet $\{0,1\}$ as follows: $D:= \{ w \in \{0,1\}^* \mid \text{ substrings 01 and 10 occur an equal number of times in w} \}$ For example , $101 \in D$ while $1010 \notin D$. Which of the following must ... $D$ is regular $D$ is context-free but not regular $D$ is decidable but not context-free $D$ is decidable but not in NP $D$ is undecidable

asked Dec 18, 2018 in Theory of Computation Arjun 918 views

10 votes

3 answers

26

TIFR2019-B-11
Consider the following non-deterministic automaton,where $s_1$ is the start state and $s_4$ is the final (accepting) state. The alphabet is $\{a,b\}$. A transition with label $\epsilon$ can be taken without consuming any symbol from the input. Which of the following regular expressions correspond to the language ... $(a+b)^*ba^*$ $(a+b)^*ba(aa)^*$ $(a+b)^*$ $(a+b)^*baa^*$

Consider the following non-deterministic automaton,where $s_1$ is the start state and $s_4$ is the final (accepting) state. The alphabet is $\{a,b\}$. A transition with label $\epsilon$ can be taken without consuming any symbol from the input. Which of the following regular expressions correspond to the language accepted by ... $(a+b)^*ba^*$ $(a+b)^*ba(aa)^*$ $(a+b)^*$ $(a+b)^*baa^*$

asked Dec 18, 2018 in Theory of Computation Arjun 1k views

5 votes

1 answer

27

TIFR2019-B-12
Let $G=(V,E)$ be a directed graph with $n(\geq 2)$ vertices, including a special vertex $r$. Each edge $e \in E$ has a strictly positive edge weight $w(e)$. An arborescence in $G$ rooted at $r$ is a subgraph $H$ of $G$ in which every ... $w^*$ is less than the weight of the minimum weight directed Hamiltonian cycle in $G$, when $G$ has a directed Hamiltonian cycle

Let $G=(V,E)$ be a directed graph with $n(\geq 2)$ vertices, including a special vertex $r$. Each edge $e \in E$ has a strictly positive edge weight $w(e)$. An arborescence in $G$ rooted at $r$ is a subgraph $H$ of $G$ ... $H^*$ is acyclic $w^*$ is less than the weight of the minimum weight directed Hamiltonian cycle in $G$, when $G$ has a directed Hamiltonian cycle

asked Dec 18, 2018 in Graph Theory Arjun 1.1k views

16 votes

7 answers

28

TIFR2019-B-13
A row of $10$ houses has to be painted using the colours red, blue, and green so that each house is a single colour, and any house that is immediately to the right of a red or a blue house must be green. How many ways are there to paint the houses? $199$ $683$ $1365$ $3^{10}-2^{10}$ $3^{10}$

A row of $10$ houses has to be painted using the colours red, blue, and green so that each house is a single colour, and any house that is immediately to the right of a red or a blue house must be green. How many ways are there to paint the houses? $199$ $683$ $1365$ $3^{10}-2^{10}$ $3^{10}$

asked Dec 18, 2018 in Combinatory Arjun 2k views

2 votes

2 answers

29

TIFR2019-B-14
Let $m$ and $n$ be two positive integers. Which of the following is NOT always true? If $m$ and $n$ are co-prime, there exist integers $a$ and $b$ such that $am + bn=1$ $m^{n-1} \equiv 1 (\text{ mod } n)$ ... $m+1$ is a factor of $m^{n(n+1)}-1$ If $2^n -1$ is prime, then $n$ is prime

Let $m$ and $n$ be two positive integers. Which of the following is NOT always true? If $m$ and $n$ are co-prime, there exist integers $a$ and $b$ such that $am + bn=1$ $m^{n-1} \equiv 1 (\text{ mod } n)$ ... $m+1$ is a factor of $m^{n(n+1)}-1$ If $2^n -1$ is prime, then $n$ is prime

asked Dec 18, 2018 in Quantitative Aptitude Arjun 579 views

4 votes

2 answers

30

TIFR2019-B-15
Consider directed graphs on $n$ labelled vertices $\{1,2, \dots ,n\}$, where each vertex has exactly one edge coming in and exactly one edge going out. We allow self-loops. How many graphs have exactly two cycles ? $\sum_{k=1}^{n-1} k!(n-k)!$ $\frac{n!}{2}\bigg[\sum_{k=1}^{n-1} \frac{1}{k(n-k)}\bigg]$ $n!\bigg[\sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ None of the above

Consider directed graphs on $n$ labelled vertices $\{1,2, \dots ,n\}$, where each vertex has exactly one edge coming in and exactly one edge going out. We allow self-loops. How many graphs have exactly two cycles ? $\sum_{k=1}^{n-1} k!(n-k)!$ $\frac{n!}{2}\bigg[\sum_{k=1}^{n-1} \frac{1}{k(n-k)}\bigg]$ $n!\bigg[\sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ None of the above

asked Dec 18, 2018 in Graph Theory Arjun 992 views

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