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Answers by ankitgupta.1729
45
votes
61
GATE CSE 2021 Set 1 | Question: 30
Consider the following recurrence relation. $T\left ( n \right )=\left\{\begin{array} {lcl} T(n ∕ 2)+T(2n∕5)+7n & \text{if} \; n>0\\1 & \text{if}\; n=0 \end{array}\right.$ Which one of the following options is correct? $T(n)=\Theta (n^{5/2})$ $T(n)=\Theta (n\log n)$ $T(n)=\Theta (n)$ $T(n)=\Theta ((\log n)^{5/2})$
Consider the following recurrence relation.$$T\left ( n \right )=\left\{\begin{array} {lcl} T(n ∕ 2)+T(2n∕5)+7n & \text{if} \; n>0\\1 & \text{if}\; n=0 \end{array}\r...
23.8k
views
answered
Feb 19, 2021
Algorithms
gatecse-2021-set1
algorithms
recurrence-relation
time-complexity
2-marks
+
–
1
votes
62
TIFR CSE 2013 | Part A | Question: 7
For any complex number $z$, $arg$ $z$ defines its phase, chosen to be in the interval $0\leq arg z < 360^{∘}$. If $z_{1}, z_{2}$ and $z_{3}$ ... $\frac{1}{3}$ 1 3 $\frac{1}{2}$
For any complex number $z$, $arg$ $z$ defines its phase, chosen to be in the interval $0\leq arg z < 360^{∘}$. If $z_{1}, z_{2}$ and $z_{3}$ are three complex numbers w...
629
views
answered
Feb 9, 2021
Quantitative Aptitude
tifr2013
quantitative-aptitude
complex-number
non-gate
+
–
3
votes
63
TIFR CSE 2012 | Part A | Question: 11
Let $N$ be the sum of all numbers from $1$ to $1023$ except the five primes numbers: $2, 3, 11, 17, 31.$ Suppose all numbers are represented using two bytes (sixteen bits). What is the value of the least significant byte (the least significant eight bits) of $N$? $00000000$ $10101110$ $01000000$ $10000000$ $11000000$
Let $N$ be the sum of all numbers from $1$ to $1023$ except the five primes numbers: $2, 3, 11, 17, 31.$ Suppose all numbers are represented using two bytes (sixteen bits...
1.8k
views
answered
Feb 8, 2021
Digital Logic
tifr2012
digital-logic
number-representation
+
–
2
votes
64
TIFR CSE 2011 | Part A | Question: 5
Three distinct points $x, y, z$ lie on a unit circle of the complex plane and satisfy $x+y+z=0$. Then $x, y, z$ form the vertices of . An isosceles but not equilateral triangle. An equilateral triangle. A triangle of any shape. A triangle whose shape can't be determined. None of the above.
Three distinct points $x, y, z$ lie on a unit circle of the complex plane and satisfy $x+y+z=0$. Then $x, y, z$ form the vertices of .An isosceles but not equilateral tri...
711
views
answered
Feb 6, 2021
Quantitative Aptitude
tifr2011
quantitative-aptitude
geometry
complex-number
non-gate
+
–
1
votes
65
TIFR CSE 2014 | Part A | Question: 9
Solve min $x^{2}+y^{2}$ subject to $\begin {align*} x + y &\geq 10,\\ 2x + 3y &\geq 20,\\ x &\geq 4,\\ y &\geq 4. \end{align*}$ $32$ $50$ $52$ $100$ None of the above
Solve min $x^{2}+y^{2}$ subject to$$\begin {align*} x + y &\geq 10,\\2x + 3y &\geq 20,\\x &\geq 4,\\y &\geq 4.\end{align*}$$$32$$50$$52$$100$None of the above
1.8k
views
answered
Oct 2, 2020
Calculus
tifr2014
calculus
maxima-minima
+
–
0
votes
66
TIFR CSE 2012 | Part B | Question: 6
Let $n$ be a large integer. Which of the following statements is TRUE? $2^{\sqrt{2\log n}}< \frac{n}{\log n}< n^{1/3}$ $\frac{n}{\log n}< n^{1/3}< 2^{\sqrt{2\log n}}$ $2^\sqrt{{2\log n}}< n^{1/3}< \frac{n}{\log n}$ $n^{1/3}< 2^\sqrt{{2\log n}}<\frac{n}{\log n}$ $\frac{n}{\log n}< 2^\sqrt{{2\log n}}<n^{1/3}$
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^\sq...
4.3k
views
answered
Sep 24, 2020
Algorithms
tifr2012
algorithms
asymptotic-notation
+
–
1
votes
67
TIFR-2017-Maths-A: 30
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 $M_{2}\left ( \mathbb{R} \right )$ .
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 $x,y \in \mathbb{R}$ ar...
302
views
answered
Sep 6, 2020
TIFR
tifrmaths2017
true-false
+
–
1
votes
68
TIFR-2017-Maths-A: 29
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 $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
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 \l...
272
views
answered
Sep 6, 2020
TIFR
tifrmaths2017
true-false
+
–
0
votes
69
TIFR-2018-Maths-A: 7
True/False Question : In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions 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.
True/False Question :In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions on the closed interval $\lef...
334
views
answered
Sep 6, 2020
TIFR
tifrmaths2018
true-false
+
–
1
votes
70
TIFR-2019-Maths-A: 10
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
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...
281
views
answered
Aug 31, 2020
TIFR
tifrmaths2019
+
–
1
votes
71
TIFR-2019-Maths-B: 2
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
True/False Question :If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
331
views
answered
Aug 31, 2020
TIFR
tifrmaths2019
true-false
+
–
0
votes
72
TIFR-2018-Maths-B: 9
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
What are the last $3$ digits of $2^{2017}$?$072$$472$$512$$912.$
291
views
answered
Aug 31, 2020
TIFR
tifrmaths2018
+
–
0
votes
73
TIFR-2020-Maths-A: 2
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy: $\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right )dx=\frac{1}{4}.$ Then the cardinality of this set is: $0$. $1$. $2$. more than $2$.
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy:$$\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right...
492
views
answered
Aug 31, 2020
TIFR
tifrmaths2020
+
–
0
votes
74
TIFR-2020-Maths-B: 10
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
True/False Question :Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then$$det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$$
245
views
answered
Aug 29, 2020
TIFR
tifrmaths2020
true-false
+
–
0
votes
75
TIFR-2020-Maths-A: 17
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.
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 ...
357
views
answered
Aug 29, 2020
TIFR
tifrmaths2020
+
–
1
votes
76
TIFR-2020-Maths-A: 7
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 $\{a_{m}\} \infty_{m=1}$ diverges ... limit point converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$ converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$
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...
267
views
answered
Aug 29, 2020
TIFR
tifrmaths2020
+
–
0
votes
77
TIFR-2020-Maths-A: 1
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
Consider the sequences $\left \{ a_{n}\right \}_{n=1}^{\infty }$ and $\left \{ b_{n}\right \}_{n=1}^{\infty }$ defined by $a_...
350
views
answered
Aug 29, 2020
TIFR
tifrmaths2020
+
–
0
votes
78
TIFR-2020-Maths-A: 9
What is the greatest integer less than or equal to $\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$ $1332$ $1352$ $1372$ $1392$
What is the greatest integer less than or equal to$$\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$$$1332$$1352$$1372$$1392$
325
views
answered
Aug 29, 2020
TIFR
tifrmaths2020
+
–
1
votes
79
ISI2015-MMA-30
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $K-xy$ $K+xy$ none of the above
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions:$$\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ a...
673
views
answered
Aug 27, 2020
Calculus
isi2015-mma
calculus
functions
non-gate
+
–
1
votes
80
TIFR CSE 2011 | Part A | Question: 20
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
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
1.2k
views
answered
Aug 27, 2020
Quantitative Aptitude
tifr2011
quantitative-aptitude
modular-arithmetic
+
–
2
votes
81
TIFR CSE 2018 | Part A | Question: 9
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$
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?...
4.6k
views
answered
Aug 27, 2020
Graph Theory
tifr2018
graph-theory
graph-coloring
+
–
0
votes
82
TIFR-2015-Maths-A-6
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$
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 num...
692
views
answered
Aug 25, 2020
Linear Algebra
tifrmaths2015
matrix
linear-algebra
+
–
0
votes
83
TIFR-2015-Maths-A-13
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$
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'$ ...
481
views
answered
Aug 24, 2020
Calculus
tifrmaths2015
calculus
+
–
2
votes
84
TIFR CSE 2011 | Part A | Question: 13
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
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
1.4k
views
answered
Aug 18, 2020
Quantitative Aptitude
tifr2011
quantitative-aptitude
complex-number
+
–
7
votes
85
TIFR CSE 2011 | Part A | Question: 13
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
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
1.4k
views
answered
Aug 18, 2020
Quantitative Aptitude
tifr2011
quantitative-aptitude
complex-number
+
–
0
votes
86
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...
2.9k
views
answered
Apr 16, 2020
Analytical Aptitude
tifr2017
analytical-aptitude
logical-reasoning
+
–
0
votes
87
ISI2018-PCB-A2
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 ... game, that is, whatever moves his opponent makes, he can always make his moves in a certain way ensuring his win? Justify your answer.
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 o...
742
views
answered
Apr 16, 2020
Analytical Aptitude
isi2018-pcb-a
general-aptitude
analytical-aptitude
logical-reasoning
descriptive
+
–
23
votes
88
GATE CSE 2020 | Question: 51
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 _________
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 ...
13.5k
views
answered
Mar 9, 2020
Theory of Computation
gatecse-2020
numerical-answers
theory-of-computation
regular-language
2-marks
+
–
1
votes
89
TIFR CSE 2020 | Part A | Question: 2
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
Let $M$ be a real $n\times n$ matrix such that for$ every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x 0.$ ThenSuch an $M$ cannot existSuch $Ms$ exist and th...
1.4k
views
answered
Mar 8, 2020
Linear Algebra
tifr2020
engineering-mathematics
linear-algebra
rank-of-matrix
eigen-value
+
–
1
votes
90
TIFR CSE 2020 | Part B | Question: 4
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 $y$ ... outputs the maximum of $x$ and $y?$ $1$ $2$ $3$ $4$ No circuit composed only of clamp gates can compute the max function
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 numbe...
800
views
answered
Mar 7, 2020
Calculus
tifr2020
calculus
maxima-minima
+
–
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