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4 answers
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
0 answers
2
A continuous random variable $x$ is distributed over the interval $[0,2]$ with probability density function $f(x) =ax^2 +bx$, where $a$ and $b$ are constants. If the mean of the distribution is $\frac{1}{2}$. Find the values of the constants $a$ and $b$. $a=2, b=- \frac{13}{6}$ $a= – \frac{15}{8}, b=3$ $a= – \frac{29}{6}, b=2$ $a=3, b= – \frac{7}{2}$
commented Sep 13 in Probability 180 views
5 answers
3
$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals $1$ $-1$ $\infty$ $-\infty$
commented Sep 11 in Calculus 3.6k views
3 answers
4
A list of $n$ elements is commonly written as a sequence of $n$ elements enclosed in a pair of square brackets. For example. $[10, 20, 30]$ is a list of three elements and $[]$ is a nil list. Five functions are defined below: $car (l)$ returns the first element of its argument list $l$; $cdr (l)$ ... $f ([32, 16, 8], [9, 11, 12])$ (b) $g ([5, 1, 8, 9])$
commented Sep 10 in DS 1k views
1 answer
5
Let the characteristic equation of matrix $M$ be $\lambda ^{2} - \lambda - 1 = 0$. Then. $M^{-1}$ does not exist. $M^{-1}$ exists but cannot be determined from the data. $M^{-1} = M + I$ $M^{-1} = M - I$ $M^{-1}$ exists and can be determined from the data but the choices (c) and (d) are incorrect.
commented Sep 7 in Linear Algebra 870 views
1 answer
6
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
1 answer
7
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
1 answer
8
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
4 answers
9
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$.Which of the following is TRUE? $\left\{\left[1,-1,0\right]^T,\left[1,0,-1\right]^T\right\}$ is a ... is a linearly independent set, but it does not span $X$ and therefore is not a basis of $X$. $X$ is not a subspace of $R^3$. None of the above
commented Sep 6 in Linear Algebra 5.1k views
1 answer
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
answer edited Aug 31 in TIFR 14 views
1 answer
11
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
1 answer
12
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
answered Aug 31 in TIFR 16 views
1 answer
13
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
1 answer
14
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
1 answer
15
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
1 answer
16
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
1 answer
17
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
1 answer
18
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
2 answers
19
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 156 views
2 answers
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.
answer edited Aug 27 in Numerical Ability 429 views
6 answers
21
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
2 answers
22
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
3 answers
23
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
1 answer
24
If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is $\log \frac{1+x}{1-x}$ $\frac{x}{1-x} + \log(1+x)$ $\frac{1}{1-x} + \log(1-x)$ $\frac{x}{1-x} + \log(1-x)$
commented Aug 22 in Others 169 views
4 answers
25
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
3 answers
26
How many solutions does the following system of linear equations have? $-x + 5y = -1$ $x - y = 2$ $x + 3y = 3$ infinitely many two distinct solutions unique none
commented Jun 21 in Linear Algebra 2.5k views
4 answers
27
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)
edited Apr 18 in Numerical Ability 941 views
3 answers
28
For $x \in \{0,1\}$, let $\lnot x$ denote the negation of $x$, that is $\lnot \, x = \begin{cases}1 & \mbox{iff } x = 0\\ 0 & \mbox{iff } x = 1\end{cases}$. If $x \in \{0,1\}^n$, then $\lnot \, x$ denotes the component wise negation of $x$ ... $g(x) = f(x) \land f(\lnot x)$ $g(x) = f(x) \lor f(\lnot x)$ $g(x) = \lnot f(\lnot x)$ None of the above.
commented Apr 18 in Digital Logic 1.8k views
2 answers
29
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 175 views
10 answers
30
The following are the starting and ending times of activities $A, B, C, D, E, F, G$ and $H$ respectively in chronological order: $ a_s \: b_s \: c_s \: a_e \: d_s \: c_e \: e_s \: f_s \: b_e \: d_e \: g_s \: e_e \: f_e \: h_s \: g_e \: h_e $ ... scheduled in a room only if the room is reserved for the activity for its entire duration. What is the minimum number of rooms required? $3$ $4$ $5$ $6$
commented Apr 6 in Algorithms 4.7k views
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