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Answers by ankitgupta.1729

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1
Let $n$ be a large integer. Which of the following statements is TRUE? $2^{\sqrt{2\log n}}< \frac{n}{\log n}< n^{1/3}$ $\frac{n}{\log n}< n^{1/3}< 2^{\sqrt{2\log n}}$ $2^\sqrt{{2\log n}}< n^{1/3}< \frac{n}{\log n}$ $n^{1/3}< 2^\sqrt{{2\log n}}<\frac{n}{\log n}$ $\frac{n}{\log n}< 2^\sqrt{{2\log n}}<n^{1/3}$
answered 6 days ago in Algorithms 1.6k views
1 vote
2
True/False Question : The matrices $\begin{pmatrix} x &0 \\ 0 & y \end{pmatrix} and \begin{pmatrix} x &1 \\ 0 & y \end{pmatrix}, x\neq y,$ for any $x,y \in \mathbb{R}$ are conjugate in $M_{2}\left ( \mathbb{R} \right )$ .
answered Sep 6 in TIFR 23 views
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3
True/False Question : Let $y\left ( t \right )$ be a real valued function defined on the real line such that ${y}'=y \left ( 1-y \right )$, with $y\left ( 0\right ) \in \left [ 0,1 \right ]$. Then $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
answered Sep 6 in TIFR 17 views
0 votes
4
True/False Question : In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued function on the closed interval $\left [ 0,1 \right ]$, the set $S=\left \{ sin\left ( x \right ) , cos\left ( x \right ),tan\left ( x \right )\right \}$ is linearly independent.
answered Sep 6 in TIFR 14 views
0 votes
5
Let $S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$ Then which of the following describes $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
answered Aug 31 in TIFR 14 views
0 votes
6
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
answered Aug 31 in TIFR 27 views
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7
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
answered Aug 31 in TIFR 16 views
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8
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy: $\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right )dx=\frac{1}{4}.$ Then the cardinality of this set is: $0$. $1$. $2$. more than $2$.
answered Aug 31 in TIFR 15 views
0 votes
9
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
answered Aug 30 in TIFR 13 views
0 votes
10
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$? $T$ is invertible as well as diagonalizable. $T$ is invertible, but not necessearily diagonalizable. $T$ is diagonalizable, but not necessary invertible. None of the other three statements.
answered Aug 29 in TIFR 18 views
0 votes
11
Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m$. Then the sequence diverges to.$+\infty$ has more than one limit point. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$.
answered Aug 29 in TIFR 13 views
0 votes
12
Consider the sequences $\left \{ a_{n}\right \}_{n=1}^{\infty }$ and $\left \{ b_{n}\right \}_{n=1}^{\infty }$ defined by $a_{n}=\left ( 2^{n}+3^{n} \right )^{1/n}$ and $b_{n}=\frac{n}{\sum_{i=1}^{n}\frac{1}{a_{i}}}$. What is the limit of $\left \{ b_{n}\right \}_{n=1}^{\infty }$? $2$. $3$. $5$. The limit does not exist.
answered Aug 29 in TIFR 21 views
0 votes
13
What is the greatest integer less than or equal to $\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$ $1332$ $1352$ $1372$ $1392$
answered Aug 29 in TIFR 14 views
0 votes
14
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $K-xy$ $K+xy$ none of the above
answered Aug 27 in Calculus 157 views
0 votes
15
Let $n> 1$ be an odd integer. The number of zeros at the end of the number $99^{n}+1$ is. $1$ $2$ $3$ $4$ None of the above.
answered Aug 27 in Numerical Ability 429 views
0 votes
16
How many ways are there to assign colours from range $\left\{1,2,\ldots,r\right\}$ to vertices of the following graph so that adjacent vertices receive distinct colours? $r^{4}$ $r^{4} - 4r^{3}$ $r^{4}-5r^{3}+8r^{2}-4r$ $r^{4}-4r^{3}+9r^{2}-3r$ $r^{4}-5r^{3}+10r^{2}-15r$
answered Aug 27 in Graph Theory 1.6k views
0 votes
17
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix} \sin\frac{\pi}{18}&-\sin \frac{4\pi}{9} \\ \sin \frac{4\pi}{9}&\sin \frac {\pi}{18} \end{pmatrix}$. Then the smallest number $n \in \mathbb{N}$ such that $A^{n}=1$ is. $3$ $9$ $18$ $27$
answered Aug 25 in Linear Algebra 214 views
0 votes
18
For a real number $t >0$, let $\sqrt{t}$ denote the positive square root of $t$. For a real number $x > 0$, let $F(x)= \int_{x^{2}}^{4x^{2}} \sin \sqrt{t} dt$. If $F'$ is the derivative of $F$, then $F'(\frac{\pi}{2}) = 0$ $F'(\frac{\pi}{2}) = \pi$ $F'(\frac{\pi}{2}) = - \pi$ $F'(\frac{\pi}{2}) = 2\pi$
answered Aug 24 in Calculus 138 views
0 votes
19
If $z=\dfrac{\sqrt{3}-i}{2}$ and $\large(z^{95}+ i^{67})^{97}= z^{n}$, then the smallest value of $n$ is? $1$ $10$ $11$ $12$ None of the above.
answered Aug 18 in Numerical Ability 482 views
0 votes
20
If $z=\dfrac{\sqrt{3}-i}{2}$ and $\large(z^{95}+ i^{67})^{97}= z^{n}$, then the smallest value of $n$ is? $1$ $10$ $11$ $12$ None of the above.
answered Aug 18 in Numerical Ability 482 views
0 votes
21
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 that removes the last token from the ... has a winning strategy. Edit : Option (D) is : For both $n=7$ and $n=8$, Bharat has a winning strategy. (Source)
answered Apr 16 in Numerical Ability 942 views
0 votes
22
Let there be a pile of $2018$ chips in the center of a table. Suppose there are two players who could alternately remove one, two or three chips from the pile. At least one chip must be removed, but no more than three chips can be removed in a single move. ... this game, that is, whatever moves his opponent makes, he can always make his moves in a certain way ensuring his win? Justify your answer.
answered Apr 16 in Numerical Ability 177 views
5 votes
23
Consider the following language. $L = \{{ x\in \{a,b\}^*\mid}$number of $a$’s in $x$ divisible by $2$ but not divisible by $3\}$ The minimum number of states in DFA that accepts $L$ is _________
answered Mar 9 in Theory of Computation 2k views
0 votes
24
Let $M$ be a real $n\times n$ matrix such that for$ every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above
answered Mar 8 in Linear Algebra 120 views
0 votes
25
A $\textit{clamp}$ gate is an analog gate parametrized by two real numbers $a$ and $b$, and denoted as $\text{clamp}_{a,b}$. It takes as input two non-negative real numbers $x$ and $y$ ... $y$ outputs the maximum of $x$ and $y?$ $1$ $2$ $3$ $4$ No circuit composed only of clamp gates can compute the max function
answered Mar 7 in Others 71 views
0 votes
26
Let $G$ be an undirected graph. An Eulerian cycle of $G$ is a cycle that traverses each edge of $G$ exactly once. A Hamiltonian cycle of $G$ is a cycle that traverses each vertex of $G$ exactly once. Which of the following must be true? Checking if ... , then it has a Hamiltonian cycle A complete graph always has both an Eulerian cycle and a Hamiltonian cycle All of the other statements are true
answered Mar 1 in Others 89 views
1 vote
27
Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly one global minimum inside $(0,1)$. What can you say about the behaviour of the first derivative $f'$ and and second derivative $f''$ ... $f'$ is zero at at least two points, $f''$ is zero at at least two points
answered Mar 1 in Calculus 122 views
1 vote
28
Suppose we toss $m=3$ labelled balls into $n=3$ numbered bins. Let $A$ be the event that the first bin is empty while $B$ be the event that the second bin is empty. $P(A)$ and $P(B)$ denote their respective probabilities. Which of the following is true? $P(A)>P(B)$ $P(A) = \dfrac{1}{27}$ $P(A)>P(A\mid B)$ $P(A)<P(A\mid B)$ None of the above
answered Mar 1 in Probability 120 views
1 vote
29
Suppose $X_{1a}, X_{1b},X_{2a},X_{2b},\dots , X_{5a},X_{5b}$ are ten Boolean variables each of which can take the value TRUE or FLASE. Recall the Boolean XOR $X\oplus Y:=(X\wedge \neg Y)\vee (\neg X \wedge Y)$ ... $H$ have? $20$ $30$ $32$ $160$ $1024$
answered Feb 29 in Others 115 views
1 vote
30
A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so large that each winner is equally likely to be born on any of the seven days of the week independent of the other winners. ... $\dfrac{48}{2401} \\$ $\dfrac{105}{2401} \\$ $\dfrac{175}{2401} \\$ $\dfrac{294}{2401}$
answered Feb 26 in Probability 196 views
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