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Recent questions tagged tifr2019
8
votes
6
answers
1
TIFR CSE 2019 | Part A | Question: 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 w...
Arjun
3.5k
views
Arjun
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
set-theory
+
–
5
votes
2
answers
2
TIFR CSE 2019 | Part A | Question: 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$
Arjun
1.5k
views
Arjun
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
modular-arithmetic
quantitative-aptitude
+
–
9
votes
2
answers
3
TIFR CSE 2019 | Part A | Question: 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 ... 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...
Arjun
3.2k
views
Arjun
asked
Dec 18, 2018
Linear Algebra
tifr2019
engineering-mathematics
linear-algebra
eigen-value
+
–
7
votes
1
answer
4
TIFR CSE 2019 | Part A | Question: 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}$$\frac...
Arjun
1.8k
views
Arjun
asked
Dec 18, 2018
Probability
tifr2019
engineering-mathematics
discrete-mathematics
probability
+
–
10
votes
6
answers
5
TIFR CSE 2019 | Part A | Question: 5
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can only reply by saying Yes or no . Assume that Lata always tells the truth. What is ... she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can onl...
Arjun
4.4k
views
Arjun
asked
Dec 18, 2018
Algorithms
tifr2019
algorithm-design
binary-search
+
–
2
votes
1
answer
6
TIFR CSE 2019 | Part A | Question: 6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\ ... . Which of the functions $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(...
Arjun
1.1k
views
Arjun
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
set-theory&algebra
functions
convex-sets-functions
non-gate
+
–
12
votes
2
answers
7
TIFR CSE 2019 | Part A | Question: 7
What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$
What are the last two digits of $1! + 2! + \dots +100!$?$00$$13$$30$$33$$73$
Arjun
1.4k
views
Arjun
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
modular-arithmetic
+
–
1
votes
1
answer
8
TIFR CSE 2019 | Part A | Question: 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$ ... 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 follo...
Arjun
1.2k
views
Arjun
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
speed-time-distance
non-gate
+
–
3
votes
1
answer
9
TIFR CSE 2019 | Part A | Question: 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$. ... $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 solub...
Arjun
955
views
Arjun
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
alligation-mixture
+
–
3
votes
1
answer
10
TIFR CSE 2019 | Part A | Question: 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 ... 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 ...
Arjun
1.2k
views
Arjun
asked
Dec 18, 2018
Analytical Aptitude
tifr2019
general-aptitude
analytical-aptitude
logical-reasoning
+
–
7
votes
5
answers
11
TIFR CSE 2019 | Part A | Question: 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, ... $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 mul...
Arjun
2.1k
views
Arjun
asked
Dec 18, 2018
Analytical Aptitude
tifr2019
general-aptitude
analytical-aptitude
logical-reasoning
+
–
3
votes
1
answer
12
TIFR CSE 2019 | Part A | Question: 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$. Consider the following statements ... 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-decreas...
Arjun
1.7k
views
Arjun
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
functions
+
–
7
votes
2
answers
13
TIFR CSE 2019 | Part A | Question: 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$$1.00$
Arjun
2.8k
views
Arjun
asked
Dec 18, 2018
Calculus
tifr2019
engineering-mathematics
calculus
definite-integral
+
–
8
votes
1
answer
14
TIFR CSE 2019 | Part A | Question: 14
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is ... that two red pens are drawn in succession ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that ...
Arjun
2.4k
views
Arjun
asked
Dec 18, 2018
Probability
tifr2019
engineering-mathematics
probability
conditional-probability
+
–
6
votes
5
answers
15
TIFR CSE 2019 | Part A | Question: 15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\displaystyle \lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 ... $\text{The limit exists, but it is none of the above}$
Consider the matrix$$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}$$What is $\displaystyle ...
Arjun
2.8k
views
Arjun
asked
Dec 18, 2018
Calculus
tifr2019
engineering-mathematics
calculus
limits
matrix
+
–
4
votes
2
answers
16
TIFR CSE 2019 | Part B | Question: 1
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ? $0.1$ $0.2$ $0.4$ $0.5$ All the above
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ?$0.1$$0.2$$0.4$$0.5$All the above
Arjun
4.2k
views
Arjun
asked
Dec 18, 2018
Digital Logic
tifr2019
digital-logic
number-representation
+
–
11
votes
5
answers
17
TIFR CSE 2019 | Part B | Question: 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$$64$None of the above
Arjun
4.6k
views
Arjun
asked
Dec 18, 2018
Algorithms
tifr2019
algorithms
minimum-spanning-tree
+
–
5
votes
1
answer
18
TIFR CSE 2019 | Part B | Question: 3
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE? $d$ divides $n$ Both $d$ and $n$ are even Both $d$ and $n$ are odd At least one of $d$ and $n$ is odd At least one of $d$ and $n$ is even
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE?$d$ divides $n$Both $d$ and $n...
Arjun
1.6k
views
Arjun
asked
Dec 18, 2018
Graph Theory
tifr2019
graph-theory
degree-of-graph
+
–
7
votes
2
answers
19
TIFR CSE 2019 | Part B | Question: 4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ ... interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \p...
Arjun
1.4k
views
Arjun
asked
Dec 18, 2018
Mathematical Logic
tifr2019
mathematical-logic
propositional-logic
+
–
10
votes
2
answers
20
TIFR CSE 2019 | Part B | Question: 5
Stirling's approximation for $n!$ states for some constants $c_1,c_2$ $c_1 n^{n+\frac{1}{2}}e^{-n} \leq n! \leq c_2 n^{n+\frac{1}{2}}e^{-n}.$ What are the tightest asymptotic bounds that can be placed on $n!$ $?$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$
Stirling’s approximation for $n!$ states for some constants $c_1,c_2$$$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 asym...
Arjun
3.5k
views
Arjun
asked
Dec 18, 2018
Algorithms
tifr2019
algorithms
asymptotic-notation
+
–
12
votes
3
answers
21
TIFR CSE 2019 | Part B | Question: 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...
Arjun
2.5k
views
Arjun
asked
Dec 18, 2018
Programming in C
tifr2019
programming
programming-in-c
+
–
3
votes
1
answer
22
TIFR CSE 2019 | Part B | Question: 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$ ... $\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 ...
Arjun
1.0k
views
Arjun
asked
Dec 18, 2018
Theory of Computation
tifr2019
theory-of-computation
p-np-npc-nph
+
–
5
votes
2
answers
23
TIFR CSE 2019 | Part B | Question: 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 p...
Arjun
2.0k
views
Arjun
asked
Dec 18, 2018
Programming in C
tifr2019
programming
parameter-passing
+
–
12
votes
2
answers
24
TIFR CSE 2019 | Part B | Question: 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 ...
Arjun
3.5k
views
Arjun
asked
Dec 18, 2018
Programming in C
tifr2019
programming
loop-invariants
+
–
9
votes
2
answers
25
TIFR CSE 2019 | Part B | Question: 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$ ... $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}...
Arjun
2.0k
views
Arjun
asked
Dec 18, 2018
Theory of Computation
tifr2019
theory-of-computation
identify-class-language
+
–
11
votes
3
answers
26
TIFR CSE 2019 | Part B | Question: 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 ... $(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 l...
Arjun
2.1k
views
Arjun
asked
Dec 18, 2018
Theory of Computation
tifr2019
theory-of-computation
regular-expression
+
–
6
votes
1
answer
27
TIFR CSE 2019 | Part B | Question: 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$ ... is acyclic $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 arborescen...
Arjun
2.5k
views
Arjun
asked
Dec 18, 2018
Graph Theory
tifr2019
graph-connectivity
graph-theory
difficult
+
–
20
votes
7
answers
28
TIFR CSE 2019 | Part B | Question: 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 r...
Arjun
5.0k
views
Arjun
asked
Dec 18, 2018
Combinatory
tifr2019
combinatory
counting
+
–
3
votes
2
answers
29
TIFR CSE 2019 | Part B | Question: 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^...
Arjun
1.3k
views
Arjun
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
modular-arithmetic
+
–
7
votes
2
answers
30
TIFR CSE 2019 | Part B | Question: 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 ? $\displaystyle \sum_{k=1}^{n-1} k!(n-k)!$ ... $n!\bigg[\displaystyle \sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ 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-loo...
Arjun
2.2k
views
Arjun
asked
Dec 18, 2018
Graph Theory
tifr2019
graph-connectivity
graph-theory
+
–
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