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Arjun
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Questions by Arjun
8
votes
6
answers
1141
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...
3.4k
views
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
set-theory
+
–
5
votes
2
answers
1142
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$
1.5k
views
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
modular-arithmetic
quantitative-aptitude
+
–
9
votes
2
answers
1143
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...
3.1k
views
asked
Dec 18, 2018
Linear Algebra
tifr2019
engineering-mathematics
linear-algebra
eigen-value
+
–
7
votes
1
answer
1144
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}$$\frac...
1.7k
views
asked
Dec 18, 2018
Probability
tifr2019
engineering-mathematics
discrete-mathematics
probability
+
–
10
votes
6
answers
1145
TIFR CSE 2019 | Part A | Question: 5
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can only reply by saying Yes or no . Assume that Lata always tells the truth. What is ... she can always find out the number Lata has thought of? $10$ $32$ $100$ $999$ $\text{None of the above}$
Asha and Lata play a game in which Lata first thinks of a natural number between $1$ and $1000$. Asha must find out that number by asking Lata questions, but Lata can onl...
4.3k
views
asked
Dec 18, 2018
Algorithms
tifr2019
algorithm-design
binary-search
+
–
2
votes
1
answer
1146
TIFR CSE 2019 | Part A | Question: 6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$. Let $f:$\ ... . Which of the functions $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(...
1.0k
views
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
set-theory&algebra
functions
convex-sets-functions
non-gate
+
–
12
votes
2
answers
1147
TIFR CSE 2019 | Part A | Question: 7
What are the last two digits of $1! + 2! + \dots +100!$? $00$ $13$ $30$ $33$ $73$
What are the last two digits of $1! + 2! + \dots +100!$?$00$$13$$30$$33$$73$
1.3k
views
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
modular-arithmetic
+
–
1
votes
1
answer
1148
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...
1.1k
views
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
speed-time-distance
non-gate
+
–
3
votes
1
answer
1149
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...
918
views
asked
Dec 18, 2018
Quantitative Aptitude
tifr2019
quantitative-aptitude
alligation-mixture
+
–
3
votes
1
answer
1150
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 ...
1.2k
views
asked
Dec 18, 2018
Analytical Aptitude
tifr2019
general-aptitude
analytical-aptitude
logical-reasoning
+
–
7
votes
5
answers
1151
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...
2.0k
views
asked
Dec 18, 2018
Analytical Aptitude
tifr2019
general-aptitude
analytical-aptitude
logical-reasoning
+
–
3
votes
1
answer
1152
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...
1.6k
views
asked
Dec 18, 2018
Set Theory & Algebra
tifr2019
engineering-mathematics
discrete-mathematics
set-theory&algebra
functions
+
–
7
votes
2
answers
1153
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$$1.00$
2.7k
views
asked
Dec 18, 2018
Calculus
tifr2019
engineering-mathematics
calculus
definite-integral
+
–
8
votes
1
answer
1154
TIFR CSE 2019 | Part A | Question: 14
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that each pen is drawn with equal probability from the remaining pens in the drawer. What is ... that two red pens are drawn in succession ? $7/12$ $1/6$ $1/12$ $1/81$ $\text{None of the above}$
A drawer contains $9$ pens, of which $3$ are red, $3$ are blue, and $3$ are green. The nine pens are drawn from the drawer one at at time (without replacement) such that ...
2.3k
views
asked
Dec 18, 2018
Probability
tifr2019
engineering-mathematics
probability
conditional-probability
+
–
6
votes
5
answers
1155
TIFR CSE 2019 | Part A | Question: 15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\displaystyle \lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 ... $\text{The limit exists, but it is none of the above}$
Consider the matrix$$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}$$What is $\displaystyle ...
2.7k
views
asked
Dec 18, 2018
Calculus
tifr2019
engineering-mathematics
calculus
limits
matrix
+
–
4
votes
2
answers
1156
TIFR CSE 2019 | Part B | Question: 1
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ? $0.1$ $0.2$ $0.4$ $0.5$ All the above
Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ?$0.1$$0.2$$0.4$$0.5$All the above
4.1k
views
asked
Dec 18, 2018
Digital Logic
tifr2019
digital-logic
number-representation
+
–
11
votes
5
answers
1157
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$$64$None of the above
4.4k
views
asked
Dec 18, 2018
Algorithms
tifr2019
algorithms
minimum-spanning-tree
+
–
5
votes
1
answer
1158
TIFR CSE 2019 | Part B | Question: 3
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE? $d$ divides $n$ Both $d$ and $n$ are even Both $d$ and $n$ are odd At least one of $d$ and $n$ is odd At least one of $d$ and $n$ is even
A graph is $d$ – regular if every vertex has degree $d$. For a $d$ – regular graph on $n$ vertices, which of the following must be TRUE?$d$ divides $n$Both $d$ and $n...
1.6k
views
asked
Dec 18, 2018
Graph Theory
tifr2019
graph-theory
degree-of-graph
+
–
6
votes
2
answers
1159
TIFR CSE 2019 | Part B | Question: 4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ ... interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \p...
1.3k
views
asked
Dec 18, 2018
Mathematical Logic
tifr2019
mathematical-logic
propositional-logic
+
–
10
votes
2
answers
1160
TIFR CSE 2019 | Part B | Question: 5
Stirling's approximation for $n!$ states for some constants $c_1,c_2$ $c_1 n^{n+\frac{1}{2}}e^{-n} \leq n! \leq c_2 n^{n+\frac{1}{2}}e^{-n}.$ What are the tightest asymptotic bounds that can be placed on $n!$ $?$ ... $n! =\Theta((\frac{n}{e})^{n+\frac{1}{2}})$ $n! =\Theta(n^{n+\frac{1}{2}}2^{-n})$
Stirling’s approximation for $n!$ states for some constants $c_1,c_2$$$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...
3.4k
views
asked
Dec 18, 2018
Algorithms
tifr2019
algorithms
asymptotic-notation
+
–
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