Search results for tifr2017 / GATE Overflow for GATE CSE

54 votes

7 answers

1

TIFR CSE 2017 | Part B | Question: 12
An undirected graph is complete if there is an edge between every pair of vertices. Given a complete undirected graph on $n$ vertices, in how many ways can you choose a direction for the edges so that there are no directed cycles? $n$ $\frac{n(n-1)}{2}$ $n!$ $2^n$ $2^m, \: \text{ where } m=\frac{n(n-1)}{2}$

An undirected graph is complete if there is an edge between every pair of vertices. Given a complete undirected graph on $n$ vertices, in how many ways can you choose a d...

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6.6k views

go_editor asked Dec 23, 2016

30 votes

5 answers

2

TIFR CSE 2017 | Part B | Question: 3
We have an implementation that supports the following operations on a stack (in the instructions below, $\mathsf{s}$ is the name of the stack). $\mathsf{isempty(s)}$ : returns $\mathsf{True}$ if $\mathsf{s}$ is empty, and $\mathsf{False}$ otherwise. $\mathsf{top(s)}$ : returns the top ... ()((())((((") is executed? $(((($ $))) \ (((($ $)))$ $(((()))$ $()()$

We have an implementation that supports the following operations on a stack (in the instructions below, $\mathsf{s}$ is the name of the stack).$\mathsf{isempty(s)}$ : ret...

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3.0k views

go_editor asked Dec 23, 2016

38 votes

4 answers

3

TIFR CSE 2017 | Part B | Question: 1
A vertex colouring with three colours of a graph $G=(V, E)$ is a mapping $c: V \rightarrow \{R, G, B\}$ so that adjacent vertices receive distinct colours. Consider the following undirected graph. How many vertex colouring with three colours does this graph have? $3^9$ $6^3$ $3 \times 2^8$ $27$ $24$

A vertex colouring with three colours of a graph $G=(V, E)$ is a mapping $c: V \rightarrow \{R, G, B\}$ so that adjacent vertices receive distinct colours. Consider the f...

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4.8k views

go_editor asked Dec 23, 2016

19 votes

4 answers

4

TIFR CSE 2017 | Part A | Question: 5
How many distinct ways are there to split $50$ identical coins among three people so that each person gets at least $5$ coins? $3^{35}$ $3^{50}-2^{50}$ $\binom{35}{2}$ $\binom{50}{15} \cdot 3^{35}$ $\binom{37}{2}$

How many distinct ways are there to split $50$ identical coins among three people so that each person gets at least $5$ coins?$3^{35}$$3^{50}-2^{50}$$\binom{35}{2}$$\bino...

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4.1k views

go_editor asked Dec 21, 2016

20 votes

5 answers

5

TIFR CSE 2017 | Part B | Question: 5
Consider the following psuedocode fragment, where $y$ is an integer that has been initialized. int i=1 int j=1 while (i<10): j=j*i i=i+1 if (i==y): break end if end while Consider the following statements: $(i==10)$ or $(i==y)$ If ... the end of the while loop? Choose from the following options. i only iii only ii and iii only i, ii, and iii None of the above

Consider the following psuedocode fragment, where $y$ is an integer that has been initialized.int i=1 int j=1 while (i<10): j=j*i i=i+1 if (i==y): break end if end whileC...

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3.1k views

go_editor asked Dec 23, 2016

12 votes

6 answers

6

TIFR CSE 2017 | Part B | Question: 9
Which of the following regular expressions correctly accepts the set of all $0/1$-strings with an even (possibly zero) number of $1$s? $(10^*10^*)^*$ $(0^*10^*1)^*$ $0^*1(10^*1)^*10^*$ $0^*1(0^*10^*10^*)^*10^*$ $(0^*10^*1)^*0^*$

Which of the following regular expressions correctly accepts the set of all $0/1$-strings with an even (possibly zero) number of $1$s?$(10^*10^*)^*$$(0^*10^*1)^*$$0^*1(10...

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2.3k views

go_editor asked Dec 23, 2016

20 votes

4 answers

7

TIFR CSE 2017 | Part B | Question: 8
For any natural number $n$, an ordering of all binary strings of length $n$ is a Gray code if it starts with $0^n$, and any successive strings in the ordering differ in exactly one bit (the first and last string must also differ by one ... two strings are separated by $k$ other strings in the ordering, then they must differ in exactly $k$ bits none of the above

For any natural number $n$, an ordering of all binary strings of length $n$ is a Gray code if it starts with $0^n$, and any successive strings in the ordering differ in e...

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3.2k views

go_editor asked Dec 23, 2016

9 votes

4 answers

8

TIFR CSE 2017 | Part A | Question: 14
Consider the following game with two players, Aditi and Bharat. There are $n$ tokens in a bag. The two players know $n$, and take turns removing tokens from the bag. In each turn, a player can either remove one token or two tokens. The player ... a winning strategy. For both $n=7$ and $n=8$, Bharat has a winning strategy. Bharat never has a winning strategy.

Consider the following game with two players, Aditi and Bharat. There are $n$ tokens in a bag. The two players know $n$, and take turns removing tokens from the bag. In e...

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2.9k views

go_editor asked Dec 23, 2016

6 votes

2 answers

9

TIFR CSE 2017 | Part A | Question: 2
For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ ... $a, \: b$? Choose from the following options. ii only i and ii iii only iv only iv and v

For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, ...

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1.6k views

go_editor asked Dec 21, 2016

19 votes

2 answers

10

TIFR CSE 2017 | Part B | Question: 14
Consider the following grammar $G$ with terminals $\{[, ]\}$, start symbol $S$, and non-terminals $\{A, B, C\}$: $S \rightarrow AC \mid SS \mid AB$ $C \rightarrow SB$ $A \rightarrow [$ $B \rightarrow ]$ A language $L$ is called ... $L(G)$ can be recognized by a deterministic push down automaton $L(G)$ is prefix-closed $L(G)$ is recursive

Consider the following grammar $G$ with terminals $\{[, ]\}$, start symbol $S$, and non-terminals $\{A, B, C\}$:$$S \rightarrow AC \mid SS \mid AB$$$$C \rightarrow SB$$$...

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2.9k views

go_editor asked Dec 23, 2016

32 votes

3 answers

11

TIFR CSE 2017 | Part B | Question: 11
Given that $B(x)$ means "$x$ is a bat", $F(x)$ means "$x$ is a fly", and $E(x, y)$ means "$x$ eats $y$", what is the best English translation of $ \forall x(F(x) \rightarrow \forall y (E(y, x) \rightarrow B(y)))?$ all flies eat bats every fly is eaten by some bat bats eat only flies every bat eats flies only bats eat flies

Given that$B(x)$ means "$x$ is a bat",$F(x)$ means "$x$ is a fly", and$E(x, y)$ means "$x$ eats $y$",what is the best English translation of $$ \forall x(F(x) \rightarrow...

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3.6k views

go_editor asked Dec 23, 2016

22 votes

3 answers

12

TIFR CSE 2017 | Part A | Question: 4
Which of the following functions asymptotically grows the fastest as $n$ goes to infinity? $(\log \: \log \: n)!$ $(\log \: \log \: n)^ {\log \: n}$ $(\log \: \log \: n)^{\log \: \log \: \log \: n}$ $(\log \: n)^{\log \: \log \: n}$ $2^{\sqrt{\log \: \log \: n}}$

Which of the following functions asymptotically grows the fastest as $n$ goes to infinity?$(\log \: \log \: n)!$$(\log \: \log \: n)^ {\log \: n}$$(\log \: \log \: n)^{\l...

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4.0k views

go_editor asked Dec 21, 2016

15 votes

4 answers

13

TIFR CSE 2017 | Part A | Question: 15
Let $T(a, b)$ be the function with two arguments (both nonnegative integral powers of 2) defined by the following recurrence: $ T(a, b) = T \left( \frac{a}{2}, b \right) +T\left( a, \frac{b}{2} \right)\quad \quad \quad \text{if } a, b \geq 2$ ... $\begin{pmatrix} r+s \\ r \end{pmatrix}$ $2^{r-s}$ if $r \geq s$, otherwise $2^{s-r}$

Let $T(a, b)$ be the function with two arguments (both nonnegative integral powers of 2) defined by the following recurrence:$ T(a, b) = T \left( \frac{a}{2}, b \right) +...

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2.4k views

go_editor asked Dec 23, 2016

26 votes

4 answers

14

TIFR CSE 2017 | Part A | Question: 11
Let $f \: \circ \: g$ denote function composition such that $(f \circ g)(x) = f(g(x))$. Let $f: A \rightarrow B$ such that for all $g \: : \: B \rightarrow A$ and $h \: : \: B \rightarrow A$ ... ) $f$ is one-to-one (injective) $f$ is both one-to-one and onto (bijective) the range of $f$ is finite the domain of $f$ is finite

Let $f \: \circ \: g$ denote function composition such that $(f \circ g)(x) = f(g(x))$. Let $f: A \rightarrow B$ such that for all $g \: : \: B \rightarrow A$ and $h \: :...

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4.0k views

go_editor asked Dec 22, 2016

8 votes

1 answer

15

TIFR CSE 2017 | Part B | Question: 6
Consider the First Order Logic (FOL) with equality and suitable function and relation symbols. Which of the following is FALSE? Partial orders cannot be axiomatized in FOL FOL has a complete proof system Natural numbers cannot be axiomatized in FOL Real numbers cannot be axiomatized in FOL Relational numbers cannot be axiomatized in FOL

Consider the First Order Logic (FOL) with equality and suitable function and relation symbols. Which of the following is FALSE?Partial orders cannot be axiomatized in FOL...

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1.2k views

go_editor asked Dec 23, 2016

16 votes

2 answers

16

TIFR CSE 2017 | Part B | Question: 7
An array of $n$ distinct elements is said to be un-sorted if for every index $i$ such that $ 2 \leq i \leq n-1$, either $A[i] > \text{max} \{A [i-1], A[i+1]\}$, or $A[i] < \text{min} \{A[i-1], A[i+1]\}$. What is the time-complexity of the fastest ... $O(n)$ but not $O(\sqrt{n})$ $O(\sqrt{n})$ but not $O(\log n)$ $O(\log n)$ but not $O(1)$ $O(1)$

An array of $n$ distinct elements is said to be un-sorted if for every index $i$ such that $ 2 \leq i \leq n-1$, either $A[i] \text{max} \{A [i-1], A[i+1]\}$, or $A[i] <...

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2.7k views

go_editor asked Dec 23, 2016

17 votes

2 answers

17

TIFR CSE 2017 | Part B | Question: 13
For an undirected graph $G=(V, E)$, the line graph $G'=(V', E')$ is obtained by replacing each edge in $E$ by a vertex, and adding an edge between two vertices in $V'$ if the corresponding edges in $G$ are ... vertex in the line graph is at most the maximum degree in the original graph each vertex in the line graph has degree one or two

For an undirected graph $G=(V, E)$, the line graph $G'=(V', E')$ is obtained by replacing each edge in $E$ by a vertex, and adding an edge between two vertices in $V'$ if...

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2.9k views

go_editor asked Dec 23, 2016

23 votes

2 answers

18

TIFR CSE 2017 | Part B | Question: 10
A vertex colouring of a graph $G=(V, E)$ with $k$ coulours is a mapping $c: V \rightarrow \{1, \dots , k\}$ such that $c(u) \neq c(v)$ for every $(u, v) \in E$. Consider the following statements: If every vertex in $G$ has ... the above statements is/are TRUE? Choose from the following options: only i only i and ii only i and iii only ii and iii i, ii, and iii

A vertex colouring of a graph $G=(V, E)$ with $k$ coulours is a mapping $c: V \rightarrow \{1, \dots , k\}$ such that $c(u) \neq c(v)$ for every $(u, v) \in E$. Consider ...

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2.8k views

go_editor asked Dec 23, 2016

14 votes

2 answers

19

TIFR CSE 2017 | Part A | Question: 10
For a set $A$ define $P(A)$ to be the set of all subsets of $A$. For example, if $A = \{1, 2\}$ then $P(A) = \{ \emptyset, \{1, 2\}, \{1\}, \{ 2 \} \}$. Let $A \rightarrow P(A)$ be a function and $A$ is not ... be onto (surjective) $f$ is both one-to-one and onto (bijective) there is no such $f$ possible if such a function $f$ exists, then $A$ is infinite

For a set $A$ define $P(A)$ to be the set of all subsets of $A$. For example, if $A = \{1, 2\}$ then $P(A) = \{ \emptyset, \{1, 2\}, \{1\}, \{ 2 \} \}$. Let $A \rightarro...

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2.2k views

go_editor asked Dec 22, 2016

15 votes

2 answers

20

TIFR CSE 2017 | Part A | Question: 9
Consider the $majority$ function on three bits, $\textbf{maj}: \{0, 1\}^3 \rightarrow \{0, 1\}$ where $\textbf{maj}(x_1, x_2, x_3)=1$ if and only if $x_1+x_2+x_3 \geq 2$. Let $p(\alpha)$ be the probability that the output is $1$ when each input is set to ... $3 \alpha$ $\alpha^2$ $6\alpha(1-\alpha)$ $3\alpha^2 (1-\alpha)$ $6\alpha(1-\alpha)+\alpha^2$

Consider the $majority$ function on three bits, $\textbf{maj}: \{0, 1\}^3 \rightarrow \{0, 1\}$ where $\textbf{maj}(x_1, x_2, x_3)=1$ if and only if $x_1+x_2+x_3 \geq 2$....

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1.4k views

go_editor asked Dec 21, 2016

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